VLS Roulette Forum

* Preface
* Uncertainty ... or?
* Probability
* Independent spins
* Three results events - Intro
* All three-results events - Part 1
* All three-results events - Part 2
* Rare events
* Hitting the second or third trial
* Missing the first trial
* Increasing the probability
* Calculate a chain of trials
* More on chains
* Mixed chains
* Deviation from the average
* The standard deviation - Part 1
* The standard deviation - Part 2
* Recorded results
* Fair empirical studies
* Math vs. reality
* Checking my claims
* Favorable situations - Part 1
* Favorable situations - Part 2
* ... where are they?
* It will even out in the long run
* The wake-up
* Misdirected intuition
* The end
* PS - Hit or win
* PPS - Misdirected intuition; a classic
* Appendix - The files
* Appendix - Table #1
* Appendix - Table #2
* Appendix - Table #3

Before you start reading, I'd like to point a few things out...

* On a personal level:

- English is not my first language. I have spell-checked the text but those programs don't know that I mis-spelled "whole" as "whale", for example. Also they can't tell if I have chosen or inflected words correctly. Point me at any peculiarity and I'll correct it. Thanks.

- I'm not a teacher. I have never been. In this article I have tried to be as clear as I possibly can be, starting with the very, very basics and advancing from there in a slow, slow manner so everyone can follow - the advanced math in these cases are not really advanced at all. And as I REASON you through most of it instead of handling complicated formulas, I really hope I have explained everything in a clear and concise way.


* On a contents level:

The text doesn't express my personal OPINION or something.
Everything I write is the accepted way to calculate general probabilities and roulette probabilities in particular.
Although many have tried to prove this way wrong during the centuries roulette has existed, no one has so far been able to do it. Not one single person!
Not logically. Not mathematically. And not empirically given a sample-size of significance.
Not anyone.
Not ever.

(That I know of.)

The below is not what I THINK is correct; it is not an OPINION.
It is the consensus and - so far - it's accepted as correct among mathematicians and other serious researches in gambling and probability theories. But it's still theories - not laws.
It is however theories that are proven by empirical studies of the real world.

If anyone throughout the centuries had really proven the math, the logic and/or the results of empirical studies to be wrong, the math-world (and in reality our whole world) wouldn't be what we today experience. And if someone in the future proves it, they are in for a Nobel Math Prize - at the very, very least.

So if you think I'm wrong I am willing to discuss it - if you FIRST prove your claims. And I mean prove - not the usual "I have seen..." or "I know..." thing.
A proof is not a few visits to the casino where you've seen some phenomenon. A proof is something anyone is able to replicate anytime on any random sample of at least the same size as yours, and have approximately the same results as you have. Every time. Any time.
If you think I'm wrong; please remember: You are not only suggesting that I'm wrong; you're suggesting that fundamental probability math and the mathematical society as a whole is wrong.
So show your proof before you start a debate, please.
(I show you mine...)


* On a reader's level:

  You will need a few things before you start...
  + A calculator for the four rules of arithmetic and also for Square Roots.
  + Basic math knowledge to the level that you understand and are able to handle and solve fractions like: (4/37)+(12/37)x(18/37)+(7/37)/(12/37), at least on your calculator.
  + Basic knowledge of the square-root is also recommended when we do a bit more advanced calculations. (It's necessary that you at least can handle the Square Root key on your calculator...)
  + A little more than half a brain as A LOT (all of it, in fact) is based on very basic logics... logics that will seldom be spelled out but is there all the time, in the background. All the time...
  + An attention-span that lasts for a bit more than just a few paragraphs. However this post is one of the longest ones...


* And on the wheel-level:

Everything in this text is for single-zero wheels.
There are 37 numbers. For double-zero wheels you have to remember that there are 38 numbers and thus change all "37" to "38" when you calculate and take ZeroZero into account at all times. Otherwise the results will be skewed beyond recognition and the total will not be 100%. Not only does it give you trouble math-wise: Because there's ZeroZero there are lower probabilities for everything - it is "1 / 38" instead of "1 / 37"...
Avoid such wheels if you can.


I hope you will enjoy the reading.

In the title I stress "easier" because it is not easy to understand - but not THAT hard...

And learn at least something new.
If you do; I'm satisfied.

I am aware that it's a looong text with lots of (simple) math involved - and you should be aware of that fact too. But I have been trying to write in a light way and in very small doses so I hope I can keep your interest to the end.

The best for you, I think, is to read this in the small portions I post it and digest and recapitulate each part before you go on.

What you actually learn is worth a lot more than the quantity of words you just read...
And what you learn here can be worth money - saved money.

/KFS

... Hmmm ...

...

No...

Probably I shouldn't begin with probability at all, but with CERTAINTY.

Roulette certainty.
Yes.

So I ask you this question:
Can you tell me if there is any event in the game of roulette that is a certainty and that also can be shown to be, without a doubt, a certainty?
Preferably mathematically.

.
.
.

There are TWO (or, rather three):

The FIRST is if you don't make any bet at all...
- Ridiculous!
- No, not at all: It is a certainty that you cannot hit any number at all. Absolutely certain. Isn't it?
- Well, yes. But how, then, is that certainty shown mathematically?

Look at it this way: There are 37 numbers and you bet 0 numbers. You bet, or "cover", 0 numbers of 37.
0 / 37
Enter that into your calculator and what's the result?
0 Zero
Meaning: No, Zero, 0% chance to hit ANY number. That is certainly a certainty.
(Show me the casino that will give you chips when you bet nothing...)

To hit a number when you bet nothing is a 0% chance and a certainty that you don't.
Is there anything that can make you win...? No. It is absolutely a certainty.
And the calculator showed you the probability for you to hit a selected number: 0.

A probability of 0 is the same as certainty that something will NOT happen.


The SECOND event is when you cover all 37 numbers.
Then it is, of course, a certainty that the winning number will be one that you cover.
And you can look at it this way: You cover 37 of the 37 numbers of the wheel.
37 / 37
Enter that into your calculator and what's the result?
1
This "1" says that you have 100% chance to hit the winning number. And you will of course, if you're betting every number there is, 37 of 37 possible.
(We usually multiply the result of a division by 100 and talk about per cent, so the "1" in this case becomes "100%" just as 0.25 becomes 25%)

To hit a number when you cover every one is a 100% chance and ALSO a certainty.
Is there anything that can make you miss...? No. It is absolutely a certainty.

A probability of 1 is the same as certainty that something WILL happen.


Now to the THIRD event and that's a little special because it's the event that has ALREADY happened:

Past results.

Statistics, actually. But often involved when discussing roulette math. (More on that later)
You look at the marquee or your collection of spins and you see a "25" there.
That "25" was the result of that spin at that time at that table. Is there any doubt about that result? Could it be something else?
No, of course not (logics!).
So that is one result, "25", and there can only be one result - it cannot be "4" in some other situation, can it?
The PAST result "25" is 1 result out of 1 possible.
1 / 1
Enter that into your calculator and what's the result?
1
This "1" says that it is a 100% chance that the "25" WAS the hitting number at that spin. And of course it was.

Past results most certainly are certainties.
Is there anything that can make it change...? No. It is absolutely a certainty.

A probability of 1 is the same as certainty that something HAS happened.

Or, as we saw before, WILL happen, whichever happens first...


So now we know this about CERTAINTY:
It can be 0 or it can be 1 in figures, on the calculator. Or 0% or 100% in words.
And nothing in-between.

Usually, when we talk about past spins we talk about STATISTICS.
Statistics are certainties because it reflects events that have already happened and thus cannot be changed.

It is a certainty that it HAS happened.

But...
Will it happen again?? Do we know??

SO ... let's talk about ... <drum-roll here> ... UN-certainty!

Uncertainty. It may happen or it may not happen. Maybe it will happen... It will probably happen... Or not.

Probability.

Now, we do play the roulette, so why don't we place a chip somewhere?
So you place it on a single number - the "13".
What's your chance that the result will be exactly that number? That you hit?

There are 37 numbers and you cover one. No more and no less. 1 number of 37.
1 / 37
If you enter that into your calculator you get the result 0.027027027027
Multiply by 100 and you get 2.7027027 which is the percentage.  2.7%

Likewise, if you place your chip on - for example - the double street 1-6, you cover six numbers. No more and no less.
The chance that you will cover the the result will be 6 / 37 because you cover six numbers. Out of the 37 possible.
And 6 / 37 is 0.162162162 or in other words 16.2%

Last let's look at the RED (or Black, Low, High, Even, Odd) numbers.
If you count them you'll find there are 18 in total and therefore, if you place your chip on Red, you cover 18 of the 37 possible numbers.
18 / 37 is 0.486486486 or 48.6%


The chance that the result will be your selected number(s) is obviously increasing by the increasing number of numbers you cover.
And of course it is - the more numbers you cover the closer you come to certainty that is 37 covered numbers (or 37 / 37 or 100%).
36 numbers are as close as you can get, without betting all, and it is 36 / 37 or 97.3%.

- Hey! There is, then, a risk of missing the number!
- So...We are UN-certain now, are we? We are not 100% sure we will hit with our 36 numbers.

And that UN-certainty is 1 / 37 or 2.7%.
The chance for 36 numbers to hit is 36 / 37 (97.3%) and to miss is 1 / 37 (2.7%) because there is only one number we haven't covered.

We cover 36 numbers with a 36/37 chance to hit and a 1/37 chance to miss.
We cover 36 numbers with a 97.3% chance to hit and a 2.7% chance to miss.
Add them together and you get 37/37 or 100% - certainty.
And that's correct: You know for certain that you will either hit or miss your numbers (that's logics!). Or? Can there be anything else? No. Of course not.

Once again we have seen that 37/37 or 1/1 or 100% means certainty.


And this is what we do when we deal with chances and probabilities:
We do "Reality Checks" to see that all chances to hit and all chances to miss added together equals 100%, is a certainty.

If it's not 100% there is something wrong in our calculation, because we cannot hit AND miss in neither less nor more than the bets we actually do. Of course.
If we hit in 60% and miss in 30% - what will happen in the other 10%?
Or if we hit 60% and miss 50% - where did the extra 10% come from?

Reality Check = 100%.

Always. That's for certain.


Now you know how to calculate the chance (probability) of a hit on the next spin.
And I have already given you a clue on how to calculate the chance of a miss...

As the chance to a hit, covering 36 numbers, is 36 / 37 the chance of a miss must be 1 / 37 as the sum can only be 37 / 37 = 100%

So the chance of a 6-number bet to miss has to be (37 - 6) / 37...
The parenthesis show you the trick: Subtract the known figure (the numbers you cover, in this case) from the total (37) and you get the chance of the other (the uncovered numbers, in this case): 31 / 37 or 83.8%

And the same for Red: (37 - 18) / 37 = 19 / 37 = 51.4%

To prove that it is mathematically correct we do our "Reality Check" and add the hit-% to the miss-% and we should end at 100%.
Do we, in these examples?

And so you know how to calculate everything regarding ONE bet, but only one...
This is valid no matter what, really.
As long as the wheel and ball are fair and the dealer doesn't aim for certain parts of the wheel (with more success than random) it is valid.
Because the wheel cannot remember any past results and cannot influence any future ones and - most important - all 37 numbers are there each and every time so the chance is equal for all numbers in all spins. Always.

We talk about "Independent trials" - each trial has the same probability every time and is, in reality, not dependent on any past results.

Some people claim that because their selected bet hasn't turned up for so-or-so long time, it is "due". But think about it (logics now): How can the ball or wheel have any influence whatsoever on the result?
Do they discuss the matter:
"Red hasn't come for 20 spins so now we have to make it hit!"
"No, the number 24 hasn't hit since we opened - it is really due to come now."
?

If this was the case, empirical studies over significant sample-sizes would have shown that, many, many moons ago.
A lot of people, myself included, have tried endlessly to find proof by studying recorded roulette-spins.
None have so far, to my knowledge, found any evidence. Unfortunately.
(But a lot more on empirical studies later)


So the theory of "Independent trials" is here to stay - at least for a while longer. (Quite a while, I'd think but it IS a theory...)
And this is important, as the rest of this article is based on that phenomenon - that the wheel and ball can NOT influence future results by themselves.

If you think they can, you shall know that I can see your point if you say that mathematics is BS in roulette (you can't think both are correct):
If the wheel has decided to select a certain number to hit, no math, as we know it, can be used. We have to have a way to quantify or "parameterize" the mood of the wheel. And we haven't...
I mean... What if the wheel all of a sudden doesn't care? Or is absent-minded and forgets which number was due? Will the ball be in charge???

As long as there are no formulas including such, we have to cope with the math as it is.
And as I said: No empirical studies, over significant sample-sizes, has ever shown the probability theories or the math wrong. Ever.


But please; read on even if you think future results are influenced in any way by the past - maybe you will learn something you have use for, in the future...?

Now, I will use a lot of fraction-math here.
The reasons are that fractions are EXACT and clearly show relations. In only a few cases a decimal-number is not needed to be rounded when divided by 37.

To show the hit-probability for a 6-number bet we can use two ways:
p = 6 / 37 (this is exact)
or
p = 0.162162162 (this is rounded) or 16.2% (this is even more rounded)

The first example entered into a calculator gives the second as the result.
The first example also tells us the EXACT RELATION: It is 6 (somethings) of 37 (somethings).

The second example lacks that exact information. It could as well be 16,216,216,211,199,999 of 100,000,000,000,000,000 because it's rounded.
And rounding is bad especially if we start getting really small figures - the rounding errors will cause us trouble in the long term.

I will use the fraction notation a lot, not so much the decimals.


Now let's focus on THREE RESULTS...

How many combinations are possible?

First you have to take every number 0 - 36 and combine them each with every number 0 - 36 and so you have (37 x 37) 1,369 combinations. This is for two results.
Now combine each and every one of those results with all numbers 0 - 36 and you end up with (1369 x 37 or 37 x 37 x 37) 50,653 three-numbers sequences.
(Did you really do it?)

For three results we have 50,653 combinations and there are no more possible to find. And there can't be less, either, if we want them all. 50,653... This is 100% of all there are.

But we cannot use 50,653 series to show math and probability in a clear way. That's impossible. So first of all we will call those combinations the "Low-Level" combinations. They really are on the lowest level - single numbers.
LL combinations. 50,653 of them...

I could, of course, use only Red and Black for example - excluding the Zero - but that wouldn't be fair. The calculated probabilities wouldn't be true, because it's a simplification.
And it wouldn't be roulette - more like coin-tossing.

And this is about roulette probability.

So we have to use a middle-way - please take a deep breath and hold on to your hat. Here we go:

We will use "High-Level" (HL) combinations: the "even" bets Red and Black, that each have a 18 / 37 probability to hit PLUS the Zero that has a probability of 1/37 to hit.
We will use combinations like "R-B-B" or "R-0-0" or "0-B-R" or... you understand, I guess.

Red + Black + Zero = (18/37) + (18/37) + (1/37) = 37/37 = 100% = Nothing missing.
This will bring down the amount of combinations considerably; from 50,653 "LL" to 27 "HL".

That is because every HL combination in reality contains more than only one of the LL-combinations.
Let's find out how many.

And don't even think about skipping this section - it is VITAL that you KNOW this later!


We can check this one for example: "R-R-R".
In this group you will find...

1  1  1  1  1  1 ... 16 16 16 18 18 18 ... 36 36 36 36 36 36 <--- All the 18 red numbers combined with...
1  1  1  1  1  1 ... 36 36 36  1  1  1 ... 36 36 36 36 36 36 <--- ... all the 18 red numbers combined with...
1  3  5  7  9 12 ... 32 34 36  1  3  5 ... 25 27 30 32 34 36 <--- ... all the 18 red numbers

Each "R" is in reality 18 numbers so what you have to do, to find the number of possible combinations, is to multiply 18 by itself three times (18 x 18 x 18) = 5,832.
This is the same calculation you did to find how many possible combinations there were using ALL numbers - you multiplied 37 by itself three times.
That was three groups of 37 numbers, now you have three groups of 18 numbers. 18 x 18 x 18. OK?

This particular HL combination contains 5,832 LL combinations of the possible 50,653.

Another group is "B-R-B" in which you will find...

2  2  2  2  2  2 ... 17 17 17 20 20 20 ... 35 35 35 35 35 35 <--- All the 18 black numbers combined with...
1  1  1  1  1  1 ... 36 36 36  1  1  1 ... 36 36 36 36 36 36 <--- ... all the 18 red numbers combined with...
2  4  6  8 10 11 ... 31 33 35  2  4  6 ... 26 28 29 31 33 35 <--- ... all the 18 black numbers

Is there a difference in numbers if the three are "B-R-B", compared to "R-R-R"?

No, each group have the same 18/37 chance each to hit so this HL combination also contains 5,832 LL of the 50,653.
In fact: ALL HL combinations that contains only R and/or B is the same. Here they are:

R-R-R (18 x 18 x 18 = 5,832)
R-R-B
R-B-R
R-B-B
B-R-R
B-R-B
B-B-R
B-B-B

There are 8 of them so they together contain 8 x 5832 = 46,656 LL combinations.
Reality Check: 50,653 - 46,656 = 3,997

We are missing 3,997 combinations!
Where are they?

This is roulette. There is a ZERO also!

"R-R-0" - there is a HL with a Zero. In this group you will find:

1  1  1  1  1  1 ... 16 16 16 18 18 18 ... 36 36 36 36 36 36 <--- All the 18 red numbers combined with...         
1  3  5  7  9 12 ... 32 34 36  1  3  5 ... 25 27 30 32 34 36 <--- ... all the 18 red numbers combined with...
0  0  0  0  0  0 ...  0  0  0  0  0  0 ...  0  0  0  0  0  0 <--- ... the Zero

How many LL combinations does it contain?
Red (and Black) is 18 numbers and Zero is... well, only 1 number of the 37.
So the calculation is: 18 x 18 x 1 = 324. Each one of those HL that contains one (and only one) Zero contains 324 LL of the 50,653 possible combinations.

Another group is "B-0-R" and there you will find:

2  2  2  2  2  2 ... 17 17 17 20 20 20 ... 35 35 35 35 35 35 <--- All the 18 black numbers combined with...
0  0  0  0  0  0 ...  0  0  0  0  0  0 ...  0  0  0  0  0  0 <--- ... the Zero that is combined with...
1  3  5  7  9 12 ... 32 34 36  1  3  5 ... 25 27 30 32 34 36 <--- ... all the 18 red numbers

And all those "one-zero" groups are:

R-R-0 (18 x 18 x 1 = 324)
R-B-0
B-R-0
B-B-0
R-0-R
R-0-B
B-0-R
B-0-B
0-R-R
0-R-B
0-B-R
0-B-B

There are 12 of these so they altogether contain 12 x 324 = 3,888 LL of the original 50,653.

Reality Check: 46,656 + 3,888 = 50,544 - still not 50,653
Still missing some...

There are also HL combinations with two Zeros like "0-B-0" in which you will find:

0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 <--- The Zero that is combined with...
2  4  6  8 10 11 13 15 17 20 22 24 26 28 29 31 33 35 <--- ... all the black numbers combined with...
0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 <--- ... the Zero

That's all: 18 combinations in this group. It has only 1 result with 18 numbers but two with only one number (the Zeros).
Therefore the calculation is: 18 x 1 x 1 = 18 for all such groups.
And they are:

R-0-0 (18 x 1 x 1 = 18)
B-0-0
0-R-0
0-B-0
0-0-R
0-0-B

6 of them, each having 18 combinations, give 6 x 18 = 108 LL.

Reality Check: 46,565 + 3,888 + 108 = 50,652.
100% = 50,653.

ONE is missing!


This one: "0-0-0"
As it is only one number in each result the calculation is quite simple: 1 x 1 x 1 = 1


And there you are: 27 HL combinations containing each and every one of the 50,653 possible LL.
And this is the complete table, showing how many each contains and their probability to hit:

B-B-B  -  5,832 p = 5832/50653 = .115136 (11.5%)
B-B-R  -  5,832
B-B-0  -    324 p = 324/50653 = .006396 (0.64%)
B-R-B  -  5,832
B-R-R  -  5,832
B-R-0  -    324
B-0-B  -    324
B-0-R  -    324
B-0-0  -     18 p = 18/50653 = .000355 (0.036%)
R-B-B  -  5,832
R-B-R  -  5,832
R-B-0  -    324
R-R-B  -  5,832
R-R-R  -  5,832
R-R-0  -    324
R-0-B  -    324
R-0-R  -    324
R-0-0  -     18
0-B-B  -    324
0-B-R  -    324
0-B-0  -     18
0-R-B  -    324
0-R-R  -    324
0-R-0  -     18
0-0-B  -     18
0-0-R  -     18
0-0-0  -      1 p = 1/50653 = .00002 (0.002%)
---------------
    Sum: 50,653 = (37 x 37 x 37) = Correct

I will refer to this table a lot later and therefore it is VITAL that you understand what each group contains.

Please recapitulate if you are uncertain of anything... (there is a high probability that you are)

As each combination in the above table is three results, the total of results are 3 x 50,653 = 151,959!

- Hey! Why bother to have that 0-0-0? It happens only once in 152K spins? That's a rare event if I've ever heard of one! I will never bet 152 thousand times in my life!
- First of all it HAS to be there - without it the table would be incomplete and not be covering 100% of all possible combinations.

Second: That particular combination has the exact same probability to hit as the combination 25-3-31. You wouldn't exclude that one, would you?
Or any other three-numbers combination? They are all 1/50653 chances. 1-2-3 or 7-34-22 or 36-36-36... All of them are equal - the 0-0-0 is no exception in that context. It's just painted green on the wheel.

The ONLY reason 0-0-0 is distinguished in the TABLE, is because 25-3-31 is contained inside the R-R-B combination and the other examples inside their respective groups.
0-0-0 doesn't fit anywhere (it's bullied because it's green!) but it has exactly the same probability as any other combination.
Why should we exclude it?
Why should we bully any...one?


As a side-note: The 0-0-0 combination doesn't come once in 152K spins, it comes in 1/50653 combinations of three spins, and that's quite a difference. Suppose this sequence of spins:

R-B-0-0-0-R-B-B-R

There is a 0-0-0 combination, obviously. But then you split the sequence into 3-results events:

R-B-0
0-0-R
B-B-R

None of these is a 0-0-0 combination... So the probability to find a 0-0-0 combination in a three-results event is a lot lower (1/3, actually) than simply find it anywhere.
In three-results combinations: 1 in 50,653 COMBINATIONS (times 3 = 151,959 results in total)
Anywhere: 1 in 50,653 SINGLE RESULTS when we treat the results as one looong continual chain.

(Isn't math beautiful?)


Also; 0-0-0 has a probability (mathematical average) to happen like 2.16 times a year at a 300 spins/day table.
Maybe, just maybe, you happen to be there at the time that happens - why not, really?
And you know what they say:


If s**t can happen, s**t will happen.
- But when it hits the fan, probability theory is like having an umbrella.

OK now. Order in class!

Let's go back to the table.

We saw before that the probability to hit at THE NEXT trial, if we cover 18 numbers, is always 18/37. Now; how about the trial after that? The second trial? Is it the same probability?
We can use the table for that but first we have to decide where to bet. Let's say we are going to bet Red.

We also HAVE to check that the 18/37 for the FIRST Red-bet trial is correct:
To do that, you look for all combinations that contains a "R" in the first position. Note the number of LL combinations and add them together:

R-B-B  -  5,832
R-B-R  -  5,832
R-B-0  -    324
R-R-B  -  5,832
R-R-R  -  5,832
R-R-0  -    324
R-0-B  -    324
R-0-R  -    324
R-0-0  -     18
---------------
    Sum: 24,642 / 50,653 possible. 24642 / 50653 = 18 / 37. We were correct.

- OK that's fine but what about the second?
- Do the same thing: Collect all combinations where "R" is in the second position.

B-R-B  -  5,832
B-R-R  -  5,832
B-R-0  -    324
R-R-B  -  5,832
R-R-R  -  5,832
R-R-0  -    324
0-R-B  -    324
0-R-R  -    324
0-R-0  -     18
---------------
    Sum: 24,642 / 50,653 possible. 24642 / 50653 = 18 / 37. Correct?

And for the third position:

B-B-R  -  5,832
B-R-R  -  5,832
B-0-R  -    324
R-B-R  -  5,832
R-R-R  -  5,832
R-0-R  -    324
0-B-R  -    324
0-R-R  -    324
0-0-R  -     18
---------------
    Sum: 24,642 / 50,653 possible. 24642 / 50653 = 18 / 37. Correct?

Yes, it's correct. What we've seen here is that it doesn't matter if you bet at the first, second or third attempt of three: The probability to hit is always 18 / 37.

But that's on one IMPORTANT condition: That we have THREE TRIALS LEFT.

Remember this!

- So if we DON'T have all three left... Like, what if I miss the first bet - it's Black or Zero? Then for sure Red will have at least a little higher probability to hit.
- That's a good question. We will check that.

What combinations can possibly end our three-trials combination AFTER the first trial?
These ones:

B-B
B-R
B-0
R-B
R-R
R-0
0-B
0-R
0-0

That's all there are!

- But what about the first, lost, trial? It isn't there!
- That's OK, I can put it there. It was a non-Red, right?
- Yes. 24.
- OK, here's the table including the number of LL combinations within each HL:

24-B-B  -  324 (1 x 18 x 18 = 324)
24-B-R  -  324
24-B-0  -   18 (1 x 18 x 1 = 18)
24-R-B  -  324
24-R-R  -  324
24-R-0  -   18
24-0-B  -   18
24-0-R  -   18
24-0-0  -    1 (1 x 1 x 1 = 1)
--------------
   Sum:  1,369 / 1369 possible LL combinations = 100%. Correct.

- How can that be correct? It's not 50,653 as it was above!
- First of all: The 24 is a past spin and is KNOWN. If you remember the beginning of this article; a known result is a CERTAINTY. Because it cannot be anything else.
So it cannot possibly be ANY Black or the Zero as it WAS, you said it yourself, number 24...

It was number 24 and it is one number of one possible. 1 / 1. The probability is 100% or p = 1.
And so, the calculation for the number of LL in 24-B-B is: 1 x 18 x 18 = 324.
The Zero is one number out of 37 so for the combination 24-B-0 the calculation is: 1 x 18 x 1 = 18.

Now, this is important that you understand: A known result is a certainty and therefore it cannot have any other probability but p = 1.
The probability to hit cannot be 1/2 or something else. Right? It really did hit.
I mean; can it be possible that all of a sudden it DIDN'T hit and what will happen then? It was just last spin and it WAS number 24... And now it's ...
No, it can not be anything else but the 24 we saw hit.

A certainty is a certainty and it's 100% sure - no more and no less - so p = 1.

So, Because we have only TWO FUTURE events, the total of POSSIBLE combinations is calculated as: 1 x 37 x 37 = 1369.

And your question was...?

Aaah - will the chance to hit Red increase (at least a little bit) at the second attempt if the first was missed?
Just look at the table for [two trials (remember there are only two left in our three-trials series - one, number 24, is gone and is not in the future) and count the number of LL in each HL combination where "R" is in the second position.

24-R-B  -  324
24-R-R  -  324
24-R-0  -   18
--------------
      Sum: 666 / 1369 = 18 / 37

It's the same old 18 / 37 chance, I'm afraid. And it doesn't matter if you bet Black instead - check the table.
(Of course it is another thing if you bet Zero because it's one single number instead of 18. Can you do the calculation for a 0-hit instead of this Red-hit?)


Now, I hope you understand why we have to put "24" in the beginning of our three-trials combinations.
The "24" isn't part of "R-R-B", for example. It IS part of "B-R-B" (and some others) and if we use them (still calculating the chance for Red to hit in the second trial) we would use

B-R-B  -  5,832
B-R-R  -  5,832
B-R-0  -    324

But that would be very wrong as "B-R-B" also includes 2, 4, 6, 8, 10, 11... - not only 24 - as a start in all its 5,832 LL combinations. And we know that 24 hit - not 11 or 4.
So we have to eliminate all combinations with another number but 24 in the first position, to have the correct number of possible combinations in the TWO REMAINING trials: 1 x 37 x 37 = 1,369.


For the third trial, it's exactly the same: If the second result was also a non-Red, 33 for example, the only combinations that can finish our three-trials series are:

24-33-B  -  18
24-33-R  -  18
24-33-0  -   1
--------------
       Sum: 37 / 37 possible numbers to finish the series.

So for both Red and Black it is a 18 / 37 chance to hit that last, third, trial after the two first have been seen.
I mean; the end-result cannot possibly be 12-33-1 or 24-12-2 or 11-15-3, can it?
No, of course the end-result must start with 24-33 and nothing else, regardless the last result.


SO, the answer to your question is:
No, the probability does NOT increase if you miss one or more bets.


Have you noticed something? Until now, I've only talked about the NEXT trial.

First, before anything happened, I showed that there is a 18/37 probability for Red to hit at the first trial - the "next" trial in that context.
Then I showed you that the probability to hit the second trial (the now "next") is also 18/37 and that is REGARDLESS if you saw the "24" or not.
And the same goes for the third trial when it is the "next": The probability to hit is 18/37. And it doesn't matter if you missed the two first trials or not.

The fact is, that you can expand this for as long as you like and to any roulette bets you like; the probability to hit at the next trial is always the same:

"The number of covered single numbers divided by 37".
It's never anything else.

And the risk to miss is always "(37 minus the number of covered single numbers) divided by 37".
If your calculations are correct, the two results added together equals 37/37 or 1.0 or 100%.

Always.

Before, you have learned to how to calculate the probability for ONE trial - now we will calculate the probabilities for a CHAIN of trials.

- Yes! What's the chance to hit three Reds in three trials?

You just have to look in the three-trials table and see how many LL there are in the "R-R-R" HL. No other HL group gives you three Red numbers in a row, right?
5,832 LL combinations of the possible 50,653. So the chance to hit ALL THREE is 5832 / 50653 = .115136 or 11.5%
We already did the calculation when we calculated the number of LL: 18 x 18 x 18 = 5832. Remember?

It's the same for all combinations of only Rs and Bs, for example "R-B-R" or "B-B-R" or any other, as long as they don't include any Zeros.


- So what's the probability to have one hit in the three trials if we exit right after the hit?

We can do this in two ways:
* The long and tiresome way, or
* The short and quick one

The former includes lots of adding while the latter includes some, but not a lot, of logic and multiplication. And one subtraction.
Your choice...

Joke; I'll do them both.

The logic way: If you don't care when you hit, during the three trials, and you will be satisfied by the first hit...
In that case, there is only one occasion we need to calculate: When you miss all three attempts.
Think about it: If you don't miss ALL, obviously you MUST have hit once... (logic!)
Right?
The probability to miss one trial is 19/37 so for three trials the calculation is: (19/37) x (19/37) x (19/37) = 6859 / 50653
You have 6,859 chances to miss all three events, so you have (50,653 - 6,859 =) 43,794 / 50,653 chances to hit. (Like 86.5%)

The long way: Check the three-results table and add the LL values from all HL combinations that include one or more "R":
(The "X" shows the eXit-point)

B-B-R X  -  5,832 (3 bets to eXit)
B-R X B  -  5,832 (2 bets to eXit)
B-R X R  -  5,832 (2 bets)
B-R X 0  -    324 (2 bets)
B-0-R X  -    324 (3)
R X B-B  -  5,832 (1)
R X B-R  -  5,832 (1)
R X B-0  -    324 (1)
R X R-B  -  5,832 (1)
R X R-R  -  5,832 (1)
R X R-0  -    324 (1)
R X 0-B  -    324 (1)
R X 0-R  -    324 (1)
R X 0-0  -     18 (1)
0-B-R X  -    324 (3)
0-R X B  -    324 (2)
0-R X R  -    324 (2)
0-R X 0  -     18 (2)
0-0-R X  -     18 (3)
-----------------
      Sum: 43,794  / 50,653 (Exactly the same as the above result)

Some simple logic can help a lot, sometimes ;)

Also: If you want to know the probability for you to hit IF you miss the first time (but you are not SURE to miss as the first result has NOT come yet)... how's that calculated?

Remember in this calculation, the first trial of the three has no result YET - we are BEFORE the first spin. (Otherwise it's of course the same conditions and result as above, when 24 hit the first trial)

Let's do it the long way first: We have the following HL combinations WITHOUT a "R" in the first position but WITH a "R" at the second or third (or both).

B-B-R X  -  5,832 (3 bets to eXit)
B-R X B  -  5,832 (2 bets)
B-R X R  -  5,832 (2)
B-R X 0  -    324 (2)
B-0-R X  -    324 (3)
0-B-R X  -    324 (3)
0-R X B  -    324 (2)
0-R X R  -    324 (2)
0-R X 0  -     18 (2)
0-0-R X  -     18 (3)
-----------------
      Sum: 19,152 / 50,653 (Some 37.8%)

That's the probability for you to miss at the first trial AND THEN hit (at least) one of the two remaining.
IF you STILL have ALL THREE trials WAITING for you!

- Can this be solved by some logic, then?

- Well, look at it this way: You ALWAYS have a 18/37 probability to hit the next - in this case the first - trial. That is equal to 24,642 of the 50,653 possible combinations. (This was calculated above)
We also calculated the probability to hit ANYTIME during the three trials and that was 43,794 / 50,653.
So if we exclude all those first-trial hits, because we calculate to miss that one, we get 43,794 - 24,642 = 19,152...
19,152 / 50,653.

The same as above...

Now, with a little bit of logic thinking and a calculator, you can use what you have learned above to make all those calculations for all the roulette-bets there are.
Instead of using the 18/37 fraction, you change the "18" to the "number of covered single numbers" in your bet.

Each trial can have their own bets:
Trial 1: Bet a Dozen (12/37)
Trial 2: Bet a 6-Street (6/37)
Trial 3: Bet Low (18/37)
Trial 4: Bet a Split (2/37)

What's the probability to hit ALL four?
Solution: (12/37) x (6/37) x (18/37) x (2/37) = 2,592 / 1,874,161 (Some 0.14%)

What's the probability to hit AT LEAST once?
First trial: 12/37 +
Second trial: (25/37) x (6/37) +
Third trial: (25/37) x (31/37) x (18/37) +
Fourth trial: (25/37) x (31/37) x (19/37) x (2/37) =

That is: (12/37) + ((25x6) / 37x37)) + ((25x31x18) / (37x37x37)) + ((25x31x19x2) / (37x37x37x37))
= 1,358,786 / 1,874,161 to be exact.

:)

Or 72.5% rounded.


- Can you PLEASE explain?

Of course...

The first trial: The bet is a Dozen so you have a probability of 12/37 to hit. And a 25/37 probability to miss. Nothing special.

The second trial: First of all you lost the first trial; that's a 25/37 probability (see above). Then you have a 6/37 to hit the second trial.

*** The probability to hit at this trial is: The probability to MISS the previous trial times the probability to HIT this trial.
So the probability to hit at this trial is (25/37) x (6/37) = 150 / 1369.

*** The probability to miss at this trial is: The probability to MISS the previous trial times the probability to MISS this trial.
So the probability to miss at this trial is (25/37) x (31/37)

Do you understand the "25 / 37" - why is it there?
It's because you had lo lose the first trial to come to this, the second, trial. That was a 25/37 probability and therefore it has to be included.
It's the 25/37 results in the first trial that didn't hit.

A loss here therefore happens in (25/37) x (31/37) as that many results didn't hit here.
The 25/37 left from the first trial and now 31/37 left from here.
As they are combined, you have to multiply them.

REMEMBER:
The probability to hit at this trial is: The probability to MISS the previous trial times the probability to HIT this trial.
The probability to miss at this trial is: The probability to MISS the previous trial times the probability to MISS this trial.


The third trial: First of all you lost the second trial and the probability to do that was (25/37) x (31/37). See above.
Now you bet Low, that has a 18/37 probability to hit and therefore the calculation is:
To hit:&nbsp; (25/37) x (31/37) x (18/37)
To miss: (25/37) x (31/37) x (19/37)


The fourth trial: Now you missed at the third trial as well and that was, as we saw above, a probability of (25/37) x (31/37) x (19/37).
And now you bet a 2-numbers split that has a 2/37 chance to hit. Calculations:
To hit: (25/37) x (31/37) x (19/37) x (2/37)

That is how the probability of 1,358,786 / 1,874,161 was calculated.
And if we have 1,358,786 chances to hit, there has to be 515,375 chances to miss (as 1,874,161 - 1,358,786 = 515,375)


The quicker (logic) way:
If you quit at any time you hit, you only have to calculate by the probability of a miss on ALL the four trials - if you don't miss all four, you obviously must have hit... right? So, for this four-misses event the calculation is:

(25/37) x (31/37) x (19/37) x (35/37) = 515,375 / 1,874,161.

And therefore the chance to hit is (1,874,161 - 515,375 =) 1,358,786 / 1,874,161

The same exactly.

Probably correct...

----------------------------------

As a complement to this document I have written a small probability calculator in JavaScript.
It is a HTML-page so it is easy to open it in your web-browser.

Download it from the members download area here:
nolinks://vlsroulette.com/downloads/?sa=view;id=167 (nolinks://vlsroulette.com/downloads/?sa=view;id=167)

You can calculate the probabilities to...

* Hit "this" trial
* Miss "this" trial
* Hit at least one trial
* Miss all trials


in as long a chain of trials you like with different bets for each.
Bet from one number to all - both 0- and 00-wheel.

Save the output by a simple copy-and-paste.

- YEAH YEAH YEAH! 18/37 is a theoretical thing! If I bet Red 37 times, will I hit 18 times? No! So what's good with all this???

The BAD thing is, what you correctly point out, that it's pure theoretical. It's the MATHEMATICAL AVERAGE for a large sample of results.

The GOOD with all the above is that you've learn the basics of how to calculate your true odds for every single bet in every possible situation.
And from here, we can dig a bit deeper into probability theory; you NEED to know that basic stuff before we can go on.


And yes; you suppose a sample of 37 results and you don't think there is exactly 18 Reds, 18 Blacks and one Zero there, like math says it should be.
And of course there aren't (probably). One or the other will probably be dominant. And if you check a lot of 37-spins samples you will find that sometimes Red is dominant and sometimes Black (probably not the Zero, though).

18 / 37 is just a mathematical average. It's the kind of figure you will have from examining a lot of 37-results samples.
But it's an average and can as such be used to measure the deviation from it.

The DEVIATION from the average.

If there are 22 Reds in that sample, the deviation for Red is +4 hits.
If there are only 12 Blacks the deviation for Black is -6 hits.
(And if both are true, the deviation for the Zero is +2 hits).

Red has +4 because 18 was expected but there were 22 hits.
Black has -6 because 18 was expected but there were only 12 hits.
Zero has +2 because only one was expected but there were 3 hits.

Now, mathematicians have invented something they call STANDARD DEVIATION (SD for short) that can be used for measuring the deviation in order to see if it is RANDOM or not.
Through count-less (literally) empirical studies and other means they have found it to be reliable and it is in fact the norm.
(Maybe a subjective one...)


One SD is a certain part of the whole. To show what I mean as an ILLUSTRATION (not a true example):
Suppose coin-tossing 100 times while recording the results. Now suppose doing that 1000 times. You have a sample of 1000 100-trials events.
Now, when you check those results (that all should be 50 heads and 50 tails according to math) you find that the vast majority has 47 to 53 heads. The deviation is +-3 hits for the majority.
You could very well call this "My Deviation" (MD for short) and now you can categorize all 1000 samples by their MD - your own "ruler" for success... or failure.

One MD is equal to three hits...

Some have 51 Heads and that's 0 MD as it didn't break the "barrier" of 53 hits.
Some have 54 Heads and that's +1 MD as it's more than 53 hits but not more than 56.
56 hits is the next level, the +2 MD barrier, so having 56 heads is still +1 MD while 57 is +2 MD.
And so on. And of course also the negative way for negative MD.

Now, that was "My Deviation"...

What mathematicians have found and agreed upon, is a formula to calculate their "Standard Deviation" (SD for short).
And that wouldn't be worth a rat's a*s if it wasn't for the fact that their SD have some very interesting characteristics...

As I said; count-less (No! More!) of empirical studies have been done and they show this, and it can be replicated at any time by anyone (given there is a significant size of sample available) - and not just on roulette; anything measurable!

And with those characteristics as a basis... This is the consensus:

Breaking the 1 SD barrier is said to be equal to a 16% probability that the result is random
Breaking the 2 SD barrier is said to be equal to a 5% probability that the result is random
Breaking the 3 SD barrier is said to be equal to a 1% probability that the result is random
Breaking the 4 SD barrier and beyond is said to be a very small probability that the result is random

I bold "random" as that is what the SD is measuring: If the results are random (within normal fluctuation) or if they are "static" (have the ability to consistently stay positive or negative).

Now, here's where those characteristics I talked about come into play:
When an operation (a betting- or selection-method or any other thing that gives a measurable result on a random set of results) is at a positive SD and manages to STAY there, in many smaller samples, it may very well start to CLIMB towards +3 SD AND BEYOND given larger and larger samples.

For example:
You have a lot of 100-results samples for an "even" chance and you find yourself consistently being at 50 hits. Every 100-spins sample. If you check, you will find it is a positive SD of +0.27 in a 100-spins sample.
Now, if you take 10 such samples you get 500 hits in a 1000-results sample. That SD is at +0.85 - it is growing!
Ten 1000-spins samples having 500 hits each is 5,000 hits in 10,000 results. That is a nice +2.7 SD. Closing in at +3 SD...
And ten SUCH samples, each having 5,000 hits, is 50,000 hits in 100,000 results and THAT, my friend, is +8.55 SD! It grows a lot!

"Breaking the 4 SD barrier and beyond is said to be a very small probability that the result is random"
Think about that.

Now, to break the +3 SD barrier in no way means that your "operations" are MONETARY profitable... You saw it above as the "even" bet had a 0 profit in all samples but was at +8.55 SD.
(This is probability theory - not economics...)

The break-even, money-wise, for a single "even" bet at +3 SD goes at approximately 12,350 bets and 6,175 hits. (The "en prison" or "le Partage" rules are not considered)
For a single dozen or column bet, the figures are around 24,420 bets giving 8,140 hits.
A 6-number street bet breaks even at +3 SD just around 60,600 bets giving 10,100 hits.
Etc.
So just because you break the +3 SD barrier doesn't mean that your method is monetary a sound method. You have to break it quite quickly if you want to be profitable...

(So THAT is what we all are looking for: A method for roulette that consistently gives us so many hits that it quickly break the +3 SD barrier and goes beyond.)

But that was money-talk; we're talking probability theory and they are very separate subjects...


Now, we were talking about the SPREAD of results, the deviation from the average.
A random deviation is agreed to never break the +-3 SD barrier and so the mathematical "prediction" for the results in a sample, is the expected mathematical average of the sample +-3 SD. Not an exact number any more, but a spread.

And here's how the SD is calculated:

SD = SQR(n * p * (1 - p))


Yeah, I know; it's a formula. But it's not really complicated - let me explain...

We have some letters there:
n = number of bets or predictions - we do NOT measure results for something not giving us results, like skipped spins.
p = the probability to hit - 18/37 or 1/37 or quite anything, really.
And then there is the "SQR", meaning the SQuare Root - check your calculator for the Square Root key - it's necessary if you want to do these calculations...

First look at the end: "(1 - p)". 1 minus the probability to hit. Do you remember? This is the way to calculate the probability for a miss.

In other words: the Standard Deviation is equal to "the Square Root of (number of spins times Probability to hit times Probability to miss)"

Supposing a Red bet for 1000 times, we solve the formula this way:
For Red to hit is p = 18/37 so the probability to miss is (1 - 18/37) = 19/37.
So now we can insert values for "n", "p" and "(1-p)"...

1000 * (18 / 37) * (19/37) = 249.82

And last you calculate the SQuareRoot of this 249.82 = 15.8

One SD = 15.8 hits.
The mathematical average is (18 / 37) * 1000 placed bets = 486.4 and so we add and subtract 15.8 to/from 486.4 to find the barriers:

+0 SD: 486.4 hits
+1 SD: 486.4 + 15.8 = 502.2 hits which means that if you have 503 hits you've broken the +1 SD barrier and has left the 0 SD level.
+2 SD: 502.2 + 15.8 = 518.0 hits which means that if you have 519 hits you've broken the +2 SD barrier and has left the +1 SD level.
+3 SD: 518.0 + 15.8 = 533.8 hits which means that if you have 534 hits you've broken the +3 SD barrier and has left the +2 SD level.
534 or more hits in a 1000-results sample... That's great.

The other way around for the negative SD:

-0 SD: 486.4 hits
-1 SD: 486.4 - 15.8 = 470.6 hits which means that if you have 470 hits you've broken the -1 SD barrier and has left the 0 SD level.
-2 SD: 470.6 - 15.8 = 454.8 hits which means that if you have 454 hits you've broken the -2 SD barrier and has left the -1 SD level.
-3 SD: 454.8 - 15.8 = 439.0 hits which means that if you have 439 hits you've broken the -3 SD barrier and has left the -2 SD level.
(Don't use that method...)

So for a 1000 results sample I (math) would estimate the number of hits being like 440 - 533, if the bet is Red or any other "even" bet.
And all of a sudden we have a practical way to measure results - not a theoretical fraction like 18/37 but an acceptable spread for the results.
Because if your results, using your betting-method or selection on one 1000 results sample, are within the boundaries of +-3 SD there is, really, no surprise. It is regarded random - could happen just anytime to anyone. But if it's consistently in the positive you should get more and larger samples - that's my recommendation.

- OK OK OK! ENOUGH IS ENOUGH! You only talk math and probability and deviation and formulas and fractions that means - frankly - ABSOLUTELY NOTHING at the casino. The spins tell the truth!

Oooh... You want to do some REAL "reality checks" - EMPIRICAL STUDIES!
Glad you asked.

I have downloaded Spielbank Weisbaden results (table #3), April - Dec 2003, complete 2004, 2005, 2006 and 2007 plus Jan - July of 2008.
A total of 655,941 results over a period of 1,903 open days. [See the file-index in the Appendix]
(That was the most I could download from one table without trouble...)


- But no - they are past results, they are not valid! The only valid results are when you go to the casino and put money on the table!
- You mean that past spins cannot be used for empirical studies in this context?
- Yes! No! Old spins are old spins...
- So suppose I go to a casino, I bet and at the same time I note the results, counting Red hits. I find them to be 135 and that's a valid study in your opinion. But if I bring those results home and count the Red hits and find them to be 135 - then it's not valid any more??? Or do you mean that I will maybe find only 134?? Or 136???

I can see your point if I'm trying to develop a method to beat roulette. Then it would be all too easy to adopt that method to those results. That is called "backwards engineering" and is in reality a useless way to develop systems because the next sample of results may differ a lot from the first one.
But here I'm not backwards engineering anything - I simply count the occurrences of Red hits. Or whatever. They are the same if I'm recording at the casino or if I study the records at home, aren't they?


- But you don't know if there are errors! They may influence the end-result. You can only trust spins that you have recorded yourself.
- If there are errors, do you think they are very common? I don't - but of course it's impossible to know for us. We weren't there.

But! If they are few, they certainly "drown" in the 655,941 results sample and thus cannot influence the end result at all because what we look for occurs maybe thousands of times.
And should there be more than just a few you have to remember: Errors work both ways. Meaning that probably are not all errors noted as black numbers but as both red and black. They will probably even out at least to a degree and thus not influence the end result too much. Because what we look for occurs maybe thousands of times.

But you are correct: Spins that are verified are better. Can you get hold of a substantial amount of such spins; please let me know and I'll use them.

- "Substantial amount" - yeah... Why do you need millions and millions of spins. Not ONE living person will EVER make a gazillion bets in their whole life!
- You are correct there. Absolutely. But we're not interested in how many bets you will place in your life-time.

We are interested in FAIR studies.

Each and every possible combination shall have the possibility to be found in the sample.
That's only fair, isn't it?

And the more possibilities (more results) the more secure you can be that the result is not some fluke.
If what you count occurs only once in one 500 results sample - what does that tell you? That it is a 1/500 (0.2%) probability to hit always?
Maybe it was just a fluke and in the next 500 results it occurs ten times. Now it's suddenly a 1/50 (a whole 2%) probability... Or? How could you possibly tell?
Which sample contains the fluke? Maybe both. The figures are really too small to be sure.

Now suppose you check a 5,000,000 results sample and the same combination occurs 54,869 times, you have a much better figure to work with: 54869/5000000 (1.1% rounded).
Now, if something hits like 55,000 times in a sample and you have done nothing wrong when you count them, you can be rather sure that the result is representative for your combination.
You can have confidence in the results as you ran a FAIR study.

My PERSONAL opinion is that we would need absolutely NO LESS than (37 x 37 x 37) x (37 x 3) for a study of THREE-results, single-number bet, events. That's 5,622,483 events = 16,867,449 results!

Why? Because this amount gives every 3-spins event the possibility to occur 37 x 3 (111) times in average. That isn't much in terms of statistical security I admit that, but I wrote "no less than" and I mean it. 111 times in average is actually not a reliable amount - something like 11,111 possibilities would be a lot better.

This shows a problem when we want to empirically study something - who's got a results-collection of a large enough size?
My sample is, as I said, 655,941 results and gave 218,010 three-results events... Each LL combination has the possibility to occur 4.3 times... Let's hope it's enough as we check HL combinations.

The triple Zero (or any other three-numbers combination) should, mathematically, occur only 4.3 times in this size and kind of a sample - do you now realize how small it is?

Now you know why the larger sample is preferred before the smaller. And the larger the sample, the more realistic is the end result.

I mean; to empirically investigate the frequency of the combination "R-B-R" you wouldn't just use three spins, would you?
If they happen to be "B-R-B" your combination has a probability of 0 to hit at any time.
You obviously have to use a larger sample than that. Would you settle with any less than eight as there are eight possible combinations of R and B in three spins?

No, because if you randomly collect 8 three-results events, you can of course not be sure that there are all the eight different R/B combinations present - and that wouldn't be FAIR.
So you want a larger sample.
And so on. And so on...

And the sample should be random because we want a FAIR study, don't we? If so, the only good thing to do is to have as large a sample as possible. The bigger is ALWAYS the better.
(In this context...)


Whatever you think about Wiesbaden results: They give a rather significant sample size.
(And I have no other of this size - far from it)

Sooo...

Do you remember the HL-table above, for three results?
We calculated the mathematical amount of LL in each HL, for example that the HL group "B-B-B" contained 5,832 LL combinations out of the 50,653 possible combinations.

In a perfect world, if we had 50,653 three-results events, there would be exactly 5,832 "B-B-B"s in that sample. But the world isn't perfect.
But how close to perfect is it?


In order to run the first "Math vs Reality" check I took the spins, starting April 1st 2003, of each day, starting at the first result and recorded them as "B" for Black, "R" for Red or "0" for Zero, in sequences of three.
If at the end of the day there were one or two spins left, they were discarded. The reason being that we don't know what happens while the casino is closed.

In the end I had 218,010 such 3-results events in the 1903 day-samples. [See the file-index in the Appendix]

As my combined (total) sample contains 218,010 three-results events each group would occur approximately 4.3 times their total of LL-combinations. Should. Mathematically. But do they?
I have checked.
(An empirical study!)

In the below table you can, for each group, see how many combinations they contain ("Combs"), the mathematical average number of occurrences there SHOULD be in the sample ("Math Ave") and the Standard Deviation ("StdDev").
You can also see the not-breaking -3 to +3 SD span, how many occurrences that was REALLY found in the sample ("Real") and the "broken" SD barrier ("SD") - if any.
Finally I show the difference between the Math average and the real number of occurrances, both in numbers ("Diff") and in percent ("Diff %"), compared to the Math average.

Group   Combs   Math Ave   StdDev   -3 SD   +3 SD     Real  SD      Diff   Diff %
---------------------------------------------------------------------------------
B-B-B =  5832   25100.86   149.03   24654 - 25547    25298  +1   +197.13    +0.78
B-B-R =  5832   25100.86   149.03   24654 - 25547    25116  +0    +15.13    +0.06
B-B-0 =   324    1394.49    37.22    1283 -  1506     1421  +0    +26.50    +1.90
B-R-B =  5832   25100.86   149.03   24654 - 25547    24921  -1   -179.86    -0.71
B-R-R =  5832   25100.86   149.03   24654 - 25547    25123  +0    +22.13    +0.08
B-R-0 =   324    1394.49    37.22    1283 -  1506     1407  +0    +12.50    +0.89
B-0-B =   324    1394.49    37.22    1283 -  1506     1389  -0     -5.49    -0.39
B-0-R =   324    1394.49    37.22    1283 -  1506     1467  +1    +72.50    +5.19
B-0-0 =    18      77.47     8.80      52 -   103       65  -1    -12.47   -16.09
R-B-B =  5832   25100.86   149.03   24654 - 25547    25159  +0    +58.13    +0.23
R-B-R =  5832   25100.86   149.03   24654 - 25547    24966  -0   -134.86    -0.53
R-B-0 =   324    1394.49    37.22    1283 -  1506     1387  -0     -7.49    -0.53
R-R-B =  5832   25100.86   149.03   24654 - 25547    25021  -0    -79.86    -0.31
R-R-R =  5832   25100.86   149.03   24654 - 25547    25041  -0    -59.86    -0.23
R-R-0 =   324    1394.49    37.22    1283 -  1506     1369  -0    -25.49    -1.82
R-0-B =   324    1394.49    37.22    1283 -  1506     1381  -0    -13.49    -0.96
R-0-R =   324    1394.49    37.22    1283 -  1506     1446  +1    +51.50    +3.69
R-0-0 =    18      77.47     8.80      52 -   103       72  -0     -5.47    -7.06
0-B-B =   324    1394.49    37.22    1283 -  1506     1419  +0    +24.50    +1.75
0-B-R =   324    1394.49    37.22    1283 -  1506     1367  -0    -27.49    -1.97
0-B-0 =    18      77.47     8.80      52 -   103       88  +1    +10.52   +13.58
0-R-B =   324    1394.49    37.22    1283 -  1506     1424  +0    +29.50    +2.11
0-R-R =   324    1394.49    37.22    1283 -  1506     1428  +0    +33.50    +2.40
0-R-0 =    18      77.47     8.80      52 -   103       71  -0     -6.47    -8.35
0-0-B =    18      77.47     8.80      52 -   103       84  +0     +6.52    +8.42
0-0-R =    18      77.47     8.80      52 -   103       77  -0     -0.47    -0.60
0-0-0 =     1       4.30     2.07       0 -    10        3  -0     -1.30   -30.29
---------------------------------------------------------------------------------
        50653  218010.00 (when all decimals are incl. Here: 218009.88)

I would say that the mathematical distribution is pretty close to the real. Wouldn't you agree?


I also gave examples of combinations that have the same probability as "0-0-0" and therefore should appear approximately the same number of times:

Group      Combs   Math Ave   StdDev   -3 SD   +3 SD     Real  SD      Diff   Diff %
------------------------------------------------------------------------------------
0- 0- 0 =     1       4.30     2.07       0 -    10        3  -0     -1.30   -30.29
1- 2- 3 =     1       4.30     2.07       0 -    10        9  +2     +4.70  +109.30
7-34-22 =     1       4.30     2.07       0 -    10        0  -2     -4.30  -100.00
25- 3-31 =     1       4.30     2.07       0 -    10        9  +2     +4.70  +109.30
36-36-36 =     1       4.30     2.07       0 -    10        5  +0     +0.70   +16.30

(Such small figures - "Real" - in only one collection are NOT to be taken too serious as they are not statistically sure, but they give an indication...)

That was how many times they appeared within the 3-results sample.
How many times did they appear if we used the results as continual sequences per day (break at the end of the day but we calculate the sample as 655,941 results):

Group      Combs   Math Ave   StdDev   -3 SD   +3 SD     Real  SD      Diff   Diff %
------------------------------------------------------------------------------------
0- 0- 0 =     1      12.95     3.60       3 -    23        9  -0     -3.95   -30.50
1- 2- 3 =     1      12.95     3.60       3 -    23       13  +0     +0.05    +0.39
7-34-22 =     1      12.95     3.60       3 -    23       14  +0     +1.05    +8.11
25- 3-31 =     1      12.95     3.60       3 -    23        8  -2     -4.95   -38.22
36-36-36 =     1      12.95     3.60       3 -    23       11  -0     -1.95   -15.06

(Such small figures - "Real" - in only one collection are NOT to be taken too serious as they are not statistically sure, but they give an indication...)


BTW, for the reader's knowledge: I was not personally the one who did the actual job making these tables - it was a computer-freak friend of mine who did.

For the next "Math vs Reality" check, let's consider what I claimed in the beginning:

If you make a 18/37 bet, you will have the probability to hit in 18/37 - no matter if you bet on the first, second or third trial.
You will always hit in 18/37 - as a mathematical average.


The math therefore also says that 18/37 three-results events in a sample will have a "R" in the first position, 18/37 will have a "R" at the second position and 18/37 will have it in the third.
As the sample-size (the three-results samples) is 218,010...

... the 18/37 Mathematical average = 106,058.92 (rounded) ...
... the SD = 233.37 and so ...
... the not-breaking -3 to +3 SD span is 105,359 - 106,759.


That's the math. What about reality?

Black in position 1: Found 106,207  +0 SD  +148.08 = +0.14%
                  2: Found 106,221  +0 SD  +162.08 = +0.15%
                  3: Found 106,096  +0 SD   +37.08 = +0.03%

Red in position 1: Found 105,842  -0 SD  -216.92 = -0.20%
                2: Found 105,805  -1 SD  -253.92 = -0.24%
                3: Found 106,031  -0 SD   -27,92 = -0.03%

Quite similar, I'd say... Not a quarter of one percent wrong either way...


More "Math vs Reality": What about only making a new bet (second or third) after a miss?

I claimed this to be true:

A: Making a first 18/37 bet of three, there is a 18/37 probability to hit at the first trial.
     So 18/37 of the 3-results events in the sample will do that. (As we actually just saw above)
     And 19/37 of the 3-results events in the sample will miss at the first trial.

B: Making a second 18/37 bet of three, there is a 18/37 probability to hit at the second trial, for the 19/37 that missed the first trial.
     So (18/37) x (19/37) of the 3-results events will do that.
     And (19/37) x (19/37) of the 3-results events will miss also at the second trial.

C: Making a third 18/37 bet of three, there is a 18/37 probability to hit at the third trial, for the (19/37) x (19/37) that missed the first AND the second trials.
     So (18/37) x (19/37) x (19/37) of the 3-results events will do that.
     And (19/37) x (19/37) x (19/37) of the 3-results events will miss also the third trial.

These are the Mathematical averages (in fractions). I will add and subtract 3 SD in order to make a mathematical "prediction" of the number of occurrences. And we will be able to see if math and probability differ very much from reality.
What do you think?


I used the following sample-sizes:

1: The first 100 three-results events of the file
2: The first 500
3: The first 1,000
4: The first 5,000
5: The first 10,000
6: The first 50,000
7: The first 100,000
8: All 218,010 three-results events of the file


The below table shows the results in the same format as above, for each trial (A - C) and the sample-sizes 1000, 10000 and all. And for both Black and Red...
[For a COMPLETE table incl all sample-sizes: See the Appendix - Table #1]


Bet   SampSize   Trial     Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %   Did NOT hit
------------------------------------------------------------------------------------------------------------
BLACK     1000       1       486.48    15.80      440 -    533      509  +1    +22.51    +4.62          491
           491       2       249.81    11.32      216 -    283      243  -0     -6.81    -2.72          248
           248       3       128.28     8.11      104 -    152      116  -1    -12.28    -9.57          132
               No hits       135.41     8.11      111 -    159      132  -0     -3.41    -2.51

BLACK    10000       1      4864.86    49.98     4715 -   5014     4871  +0     +6.13    +0.12         5129
          5129       2      2498.17    35.81     2391 -   2605     2542  +1    +43.82    +1.75         2587
          2587       3      1282.84    25.66     1206 -   1359     1264  -0    -18.84    -1.46         1323
               No hits      1354.11    25.66     1277 -   1431     1323  -0    -31.11    -2.29

BLACK   218010       1    106058.91   233.37   105359 - 106759   106207  +0   +148.08    +0.13       111803
        111803       2     54462.68   167.23    53961 -  54964    54386  -0    -76.68    -0.14        57417
         57417       3     27967.32   119.83    27608 -  28326    27910  -0    -57.32    -0.20        29507
               No hits     29521.06   119.83    29161 -  29880    29507  -0    -14.06    -0.04


Bet   SampSize   Trial     Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %   Did NOT hit
------------------------------------------------------------------------------------------------------------
RED       1000       1       486.48    15.80      440 -    533      461  -1    -25.48    -5.23          539
           539       2       249.81    11.32      216 -    283      270  +1    +20.18    +8.07          269
           269       3       128.28     8.11      104 -    152      138  +1     +9.71    +7.57          131
               No hits       135.41     8.11      111 -    159      131  -0     -4.41    -3.25

RED      10000       1      4864.86    49.98     4715 -   5014     4855  -0     -9.86    -0.20         5145
          5145       2      2498.17    35.81     2391 -   2605     2514  +0    +15.82    +0.63         2631
          2631       3      1282.84    25.66     1206 -   1359     1249  -1    -33.84    -2.63         1382
               No hits      1354.11    25.66     1277 -   1431     1382  +0    +27.88    +2.05

RED     218010       1    106058.91   233.37   105359 - 106759   105842  -0   -216.91    -0.20       112168
        112168       2     54462.68   167.23    53961 -  54964    54374  -0    -88.68    -0.16        57794
         57794       3     27967.32   119.83    27608 -  28326    28027  +0    +59.67    +0.21        29767
               No hits     29521.06   119.83    29161 -  29880    29767  +0   +245.93    +0.83

This was only ONE study performed on only ONE sample and other samples may give different results, but only within the boundaries of +-3 SD.
How I know? From personal experience (years...) plus results from all serious studies I have ever seen or read about. To date.

I ask:
* Do you think math is very different from reality?
* Do we really need to do such tiresome, time-consuming empirical studies or can we use a calculator?

- BUT NO! You can't just blindly look for Red - or whatever! In a casino you bet when the table is FAVORABLE to you.
- And when is that, then?

- Listen. I'll use your own math to show you a really GOOD method, based on a COMBINATION OF SEVERAL FAVORABLE situations: A method that will, in average -  yes, but you use averages too - bring you AT LEAST 98 units per 106 events of 7 trials each. (See; I can use your vocabulary)
- And how's that done?

- I'll use your OWN math: The probability for an "even" bet like Red to miss seven tries is calculated as:
(19/37) x (19/37) x (19/37) x (19/37) x (19/37) x (19/37) x (19/37) = well, something like 1/106.
So the probability to miss seven trials is one single event in every 106. In average.
Is this correct?

- Yes, it's absolutely correct. 100%...

- YES! I knew it! Now, here's my SPECIAL TWIST to find FAVORABLE SITUATIONS: I WAIT until I have SEEN FOUR NON-RED spins in a row and THEN I start betting Red. The first bet is 1 unit.
If that bet is lost because Red didn't hit I bet 2 units on Red. I have now bet a total of 3 units.
If that bet is also lost I make a third bet that is 4 units. On Red. I have now bet 7 units.

- So if all three bets are lost I have lost 7 units.
But this will happen ONLY ONCE in 106 - in average, that's your OWN math - and if any one of my three bets hits I have won 1 unit and that will happen in the other 105 cases.

So I win one unit 105 times and lose 7 units once = 98 units in profit for every 106 times I try it: Wait for four non-Reds and then bet Red for up to three times.

Plus and Minus that "3 SD" you mathboyzzz like to use as a safety-net. Now I'll do that too. Hah! And so I have calculated the Standard Deviation like:

3 SD = 3 x SQR(trials x (1/106) x (105/106))

...because the chance to HIT a seven-spins MISS = 1/106... Got that?
(Y'see - I learn!)
SO! YOUR OWN math shows that I will win. And win A LOT!

I have ANOTHER thing working for me here as well: When Red hasn't shown for four sprins, it GENERALLY comes within three more.
I have seen this in the casino sooo many times it must be considered a truth.
HAH!

And a last thing: Empirical studies! They show that events even out over time and therefore Red SHOULD hit more than average after not been seen for a while. As you have collected sequences that have no Reds AT LEAST in the first four results, there is a BIG difference between Red and Black so it HAS to even out in the last three results.

There you are!
What do you say? Do you DARE check it? I've got YOUR math on MY side now!


- OK... So you want me to collect 7-spins sequences where there is no Red within the first four.Then check the next three for Red.
I will collect them in the same manner as the 3-results sequences: I'll start at the first result each day and if there are any left-overs so I cannot collect a complete 7-results sequence at the end of the day I will discard the last spins. Each and every day.

- Exactly! And do it for Black as well!

I looked into the file and...
...for the four non-Reds, I found a total of 21,355 events in the file.
...for the four non-Blacks, I found a total of 21,157 events in the file.

This is due to how I collected them: I looked for the four non-Red, recorded them and the next three results. Then I started at the NEXT result, looking for another sequence of four non-Reds.
It is possible to find more events if I, after recording one event, went BACK to the result IMMEDIATELY AFTER the first one of the last recorded event.

That way gives us a considerably larger sample-size and that would be an advantage but has the disadvantage that many of the seven-spins events are part of each-other and thus may influence the end-result in a non-preferred way.
We have to settle with the "discrete" events.


- You know; I'll make my own prediction for this study... Actually, I will predict that the results from this study will be very similar to the THREE-results study above... Remember that one? That's my "prediction" for this study... Exactly the same as above...


SampSize         Math Ave  StdDev   -3SD   +3SD
-----------------------------------------------
  100   No hits     13.54    2.56      5 -   21 ... You say:   0 -   3
  500   No hits     67.70    5.73     50 -   84 ... You say:   0 -  11
1000   No hits    135.41    8.11    111 -  159 ... You say:   0 -  18
5000   No hits    677.05   18.14    622 -  731 ... You say:  27 -  67
10000   No hits   1354.11   25.66   1277 - 1431 ... You say:  66 - 123
21157   No hits   2864.90   37.33   2752 - 2976 ... You say: 158 - 241
21355   No hits   2891.71   37.50   2779 - 3004 ... You say: 160 - 243

Who do you think is correct - our figures differ A LOT...?

.
.
.

And NOW <drum-roll here> the REEESULTS:

[For COMPLETE tables incl all three sample-sizes: See the Appendix - Table #2]

No RED in the first four:
Bet   SampSize          Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %
-------------------------------------------------------------------------------------------
RED        100 No hits     13.54     2.56        5 -     21       15  +0     +1.45   +10.77
RED        500 No hits     67.70     5.73       50 -     84       64  -0     -3.70    -5.47
RED       1000 No hits    135.41     8.11      111 -    159      131  -0     -4.41    -3.25
RED       5000 No hits    677.05    18.14      622 -    731      649  -0    -28.05    -4.14
RED      10000 No hits   1354.11    25.66     1277 -   1431     1323  -0    -31.11    -2.29
RED      21355 No hits   2891.71    37.50     2779 -   3004     2882  -0     -9.71    -0.33

Bet   SampSize          Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %
-------------------------------------------------------------------------------------------
BLACK      100 No hits     13.54     2.56        5 -     21       10  -0     -3.54   -26.15
BLACK      500 No hits     67.70     5.73       50 -     84       66  -0     -1.70    -2.51
BLACK     1000 No hits    135.41     8.11      111 -    159      128  -0     -7.41    -5.47
BLACK     5000 No hits    677.05    18.14      622 -    731      639  -0    -38.05    -5.62
BLACK    10000 No hits   1354.11    25.66     1277 -   1431     1330  -0    -24.11    -1.78
BLACK    21355 No hits   2891.71    37.50     2779 -   3004     2849  -0    -42.71    -1.47

No BLACK in the first four:
Bet   SampSize          Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %
-------------------------------------------------------------------------------------------
BLACK      100 No hits     13.54     2.56        5 -     21       15  +0     +1.45   +10.77
BLACK      500 No hits     67.70     5.73       50 -     84       73  +0     +5.29    +7.81
BLACK     1000 No hits    135.41     8.11      111 -    159      132  -0     -3.41    -2.51
BLACK     5000 No hits    677.05    18.14      622 -    731      676  -0     -1.05    -0.15
BLACK    10000 No hits   1354.11    25.66     1277 -   1431     1331  -0    -23.11    -1.70
BLACK    21157 No hits   2864.90    37.33     2752 -   2976     2840  -0    -24.90    -0.86

Bet   SampSize          Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %
-------------------------------------------------------------------------------------------
RED        100 No hits     13.54     2.56        5 -     21       10  -0     -3.54   -26.15
RED        500 No hits     67.70     5.73       50 -     84       60  -0     -7.70   -11.38
RED       1000 No hits    135.41     8.11      111 -    159      118  -0    -17.41   -12.85
RED       5000 No hits    677.05    18.14      622 -    731      679  +0     +1.94    +0.28
RED      10000 No hits   1354.11    25.66     1277 -   1431     1365  +0    +10.88    +0.80
RED      21157 No hits   2864.90    37.33     2752 -   2976     2913  +0    +48.09    +1.67


Again, I ask:
* Do you think math is very different from reality?
* Do we really need to do such tiresome, time-consuming empirical studies or can we use a calculator?

- But then your math DOESN'T WORK!!! Why did I miss so many? You said that my calculations were correct!
- No, I said that your calculation that "Red wouldn't hit for seven spins in 1/106" (approximately) was correct. And I say again that it is.
BUT!
Then, after that, you said that you were going to WAIT for four spins without a Red... You didn't ask me about that.


What you didn't realize was that you all of a sudden had four KNOWN results. Do you remember what I said about known results, in the beginning?
A known result is a certainty and as such we always calculate it as 1/1.

Look here: ALL the sequences that hadn't got a Red in the first four trials, could be said to look like this if we put all of them in one big HL (High-Level - remember?) combination:

Result 1: B0 <--- Black or Zero hit here
Result 2: B0 <--- The same
Result 3: B0
Result 4: B0
Result 5: ? <--- Result not known yet
Result 6: ?
Result 7: ?

Now this HL group consists of a lot of LL 4-number results, for example "17-24-0-8" or "33-11-2-29"... And this is where you are when you start betting.
Now: By which three-result combinations can that four-spins, non-Red, sequence you have just seen, end to make it a complete seven-spins sequence?
What are the possible results AFTER "17-24-0-8" or "33-11-2-29"?

- Well, "B-B-B", "B-B-R", "B-B-0"... Wait a minute! I remember this...?
- Yes, we've seen this before: All of a sudden we're back at this table:

B-B-B  -  5,832 p = 5832/50653 = .115136 (11.5%)
B-B-R  -  5,832
B-B-0  -    324 p = 324/50653 = .006396 (0.64%)
B-R-B  -  5,832
B-R-R  -  5,832
B-R-0  -    324
B-0-B  -    324
B-0-R  -    324
B-0-0  -     18 p = 18/50653 = .000355 (0.036%)
R-B-B  -  5,832
R-B-R  -  5,832
R-B-0  -    324
R-R-B  -  5,832
R-R-R  -  5,832
R-R-0  -    324
R-0-B  -    324
R-0-R  -    324
R-0-0  -     18
0-B-B  -    324
0-B-R  -    324
0-B-0  -     18
0-R-B  -    324
0-R-R  -    324
0-R-0  -     18
0-0-B  -     18
0-0-R  -     18
0-0-0  -      1 p = 1/50653 = .00002 (0.002%)
---------------
     RC: 50,653 = (37 x 37 x 37) = Correct

Are OTHER combinations or other numbers of LL in this HL possible, to end our seven-spins sequence after the first four results are known...?
No, of course not - there are no other or no less - and that's why I could simply use the same figures I used to "predict" the number of lost 3-trials events. Because there were only three trials left AFTER you had seen the first four.

None of these combinations (not HL nor LL) can be excluded or changed in any way because you have happened to see a four non-Red or non-Black sequence, right?
And if none of them can be excluded or changed, no other combination can have a higher probability than usual to hit, can it? Will the sum still be 100% if that is the case?

Or the four spins you have just seen; can they in some way influence the results of the coming three?
("Hey Ball" says the wheel, "Red didn't hit for four spins, now jump to a red number - it is due, you know")
No, of course not.

The sad truth is: When you HAVE seen the first four results of seven, only three results are FUTURE and so the seven results can only end in (37x37x37) ways.

Please study this empirically on a sufficient number of your own results (remember what I said about a fair study?) if you don't believe me...

- But it happens so often in the casino. I have seen it a gazillion times! We ALL have.
- No, in reality not. You're a victim of something called "selective memory" - you notice things that you are looking for or that is special to you but you DON'T remember all the gray dull stuff that happens all the other time.
Do this: Bring pen and paper to the casino and record every time the Red comes within three spins and every time it doesn't, after not been seen for four spins. It will not take long before you realize things: When you record, the phenomenon doesn't occur that often at all.

Do you think the results are influenced by the fact that you or someone else are recording or do you think the results were, in fact, the same all the time?


- But it should even out! At least a little and that SHOULD have been noticeable in the study. The Blacks are so many more in the first four...
- Well, actually probability theory doesn't say that anything WILL even out. It says that you can calculate a PROBABILITY for it. A high probability? Let's use an example for this:

Suppose that you, after seeing 10 spins have counted 6 Reds, 3 Blacks and 1 Zero. Now you have a 3-hits difference between Red and Black. For this to even out you will need 3 more Blacks than Reds. So to even out you will need AT LEAST 3 spins - more if they are not all Blacks. Now, do you know the probability to have MORE Black hits than Red, in three spins? (As that's when you will at least start to even out - and at least by one hit)

Yes you do! You can see that in the three-trials table above. Look for all combinations that have more "B"s than "R"s.

B-B-B  -  5,832
B-B-R  -  5,832
B-B-0  -    324
B-R-B  -  5,832
B-0-B  -    324
B-0-0  -     18
R-B-B  -  5,832
0-B-B  -    324
0-B-0  -     18
0-0-B  -     18
---------------
         24,354 / 50,653 - less than 50%...

Because the probability to hit is only 18/37 - not 50% - there is a greater chance that you will STAY or even be MORE BEHIND, than that you will at least START to even out...
But if you look at it in PER CENT it will probably be correct: A difference of 10 in 200 is 5% but a difference of 50 in 2000 spins is half; only 2.5%. In this case you have to ask yourself: Do I hit in per cent or do I hit in numbers? Is the per cent DEcreasing? Is the number of hits really INcreasing?

Which question is more interesting to you???

I used this example as we already had done all the maths. It is, however, possible to extend it to any bet and any difference and the results will always be the same:
If the probability to hit is LESS than 50%, the chance to even out IN NUMBERS is also less than 50%...

The opposite is of course also true:
If the probability to hit is HIGHER than 50%, the probability to even out IN NUMBERS is also above 50%.

Ask yourself: Can I have a higher probability to hit, just because I try to even out?

I did a special study to show that the "even-out" theory is, in fact, a myth if you have less than 50% probability to hit:

On a day-basis^* I checked numbers that hadn't hit for at least 74 spins. When they hit for the first time after that (the "wake-up hit") I counted how many times and when it hit again, within the next 37 spins.
The theory being that a number that hasn't hit for a long time should even out.

It is also to show that saying things like:
"We have all seen sleepers hit like crazy after waking up, haven't we? It happens all the time"...
...in reality is the result of a selective memory.

There were a total of 52,408 "wake-ups" in the file, of which 33,401 had at least one hit within the next 37 spins.

An excerpt of the results:
[For a COMPLETE table incl all spins: See the Appendix - Table #3 (Also shows distribution of the first and all hits)]

How long before the FIRST hit comes AFTER the wake-up hit? (37 spins max wait)

Spin   Math ave   StdDev    -3SD to +3SD     Real  SD      Diff  Diff %
------------------------------------------------------------------------
   1    1416.43    37.12    1306 -  1527     1443  +0    +26.56   +1.87
   2    1378.15    36.61    1269 -  1488     1394  +0    +15.84   +1.15
   3    1340.90    36.12    1233 -  1449     1343  +0     +2.09   +0.15
.
.
.
  35     557.98    23.30     489 -   627      543  -0    -14.98   -2.68
  36     542.90    22.98     474 -   611      524  -0    -18.90   -3.48
  37     528.23    22.67     461 -   596      548  +0    +19.76   +3.74
------------------------------------------------------------------------
                                      Sum:  33401
0 hits 19016.31    22.67   18948 - 19084    19007  -0     -9.31    -0.04
------------------------------------------------------------------------
                                      Sum:  52408

Frankly: Nothing hits like crazy here. Math average estimated 9.31 losses too little - estimating in the 19,000 misses range!
Four hundredths of one percent wrong.

(To really see that nothing out of the ordinary happens, you should study the complete table in the Appendix)


So (for the last time) I ask you:
* Do you think math is very different from reality?
* Do we really need to do such tiresome, time-consuming empirical studies or can we use a calculator?

(Well... that second question is really asked by my friend who is making these tables...)


^*) What I mean by "day-basis" is that I ended everything at the end of the day and discarded everything that was not complete. The reason to discard was that I didn't want "broken" 37-spins results. A number that hits for the first time after 180 spins and gets only 20 possibilities to occur because the day ends - that isn't fair.
And I want to do this in a fair way.

To sum it all up:
You did one of the most common mistakes in the business... You mixed STATISTICS and PROBABILITY - you mixed certainty and UNcertainty...
And you simply cannot handle those two the same way, like you did.

The method to wait some spins in order to detect something (like 4 non-Red spins, a "wake-up hit" or actually whatever) in order for something else or the same to happen because it is "due to happen", is generally called "Gambler's fallacy", partly because that's what it is - a true fallacy, as you could see for yourself - and partly because mostly (ignorant and/or greedy) gamblers fall for it.

It is often seen in gambling forums (I heard it's called "Forum mathematics" by some...) posted by members who don't know how the logics, math and reality work and so they do it out of ignorance.
Have mercy on them...

The figures you get, calculating like you did, are very enticing and the math looks really sound, I admit that.
But nevertheless it has nothing in common with reality.

This kind of enticing logics and math has even got an official name ("Gambler's fallacy" aside) and that is "Misdirected intuition", named by C G Hempel (1905 - 1997).
(A translation from my language - maybe not the exact English name. Please enlighten me...)


Misdirected intuition...
Says it all.


And here's a friendly warning:
There is one group of people who FREQUENTLY use and defend "Misdirected intuition / Gambler's fallacy" arguments and the adherent skewed logics and math: Roulette Systems Peddlers!
As I said; the "probabilities" are enticing and the logic and math looks sound so of course they use it! But do you think that the peddlers don't know it's a fallacy?
Not in a long shot. They know.

Because they HAVE tested their systems/methods in a FAIR way, haven't they?
Using a sample of a size that makes it fair? Sure... (Ask them how big the sample was...)

Either they have, and in that case they KNOW they are LYING.
OR they sell a system/method they haven't studied properly. And how great is that?
Have mercy on their customers...


Misdirected intuition...
Something is due to happen. Says who?

Logics? No.
Logics says "There are 37 numbers each spin with equal chances to hit."

Math? No.
Math says "p = covered bets / 37"

Fair empirical studies? No.
Fair empirical studies say: "We agree with Logics and Math... +-3 SD, that is"

Roulette system peddlers? Yes.
I wonder why...

Now, my friend, we are at the end of this... We've returned to our 3-results table and it means that that's all - from now on it's only variations of the theme.
You can use the same logics and math on any kind of bet and any sample-size of significance - they are the same in all situations.


I really hope I have been able to enlighten you a little and shown that logics, math and reality in fact are close to the same. (Well, plus/minus 3 SD...)
And I hope you've learn how you can use logics and math to calculate the probabilities of different events and be comfortable with the results you get - whatever they are. And that you DO it, of course.
Also; I hope that you know how to interpret the results you get correctly.

And thus it's also possible for you to, in a very simple way, decide if a system/method is at least worth a study.
And you can do FAIR empirical studies and know how to interpret the results in a correct way, because you understand the value of a fair study that gives fair results. I hope.

And last, but not least, I hope I have given you an arsenal of weapons against those systems peddlers who's income from "roulette" goes via your wallet.
At least you can see through their "99.9% winning rate" promises in no-time.
(You can do even better: Simply betting on the 1st dozen and High for six times gives you a 99.995% chance to hit!
Sell it to them!)


As a final hint for you, that will help you A LOT through the every-day basic calculations that we've done above:
Use Grabb's tools! You find them here:
nolinks://vlsroulette.com/grabb/ (nolinks://vlsroulette.com/grabb/)

The "Hit or Sleep" and the "Standard Deviation" tools are super for those simpler calculations you do every day.

For the more complicated, though, you'll still need your calculator.


- But... but... WAIT! You can't stop now, because now I've done all sorts of calculations on all sorts of bets and I have studied recorded spins empirically and I simply CANNOT find anything that is winning... That's no fun. Do I have to give up roulette???

- No, of course not. If you like to play, why not play? The difference from before is that you now know that it's a LEISURE-game. It's a MATH GAME with the odds in favor of the casino, and I told you in the beginning that the math works - and will work - "as long as the wheel and ball are fair and the dealer doesn't aim for certain parts of the wheel (with more success than random)...".
That may be something to remember.

Maybe you should, while you are leisure-playing, study the wheel, ball and dealer in close detail to see if maybe, just maybe, there is something making the game not-so random. Is the dealer shooting the ball in a consistent way? Is it predictable? Does the ball hit only a few deflectors? Is the wheel level? Is the wheel and ball in perfect condition? Is that spot grease?

Ask yourself these and more questions - be curious. And why not? You are just leisure-playing anyway, so you can think more of the future games than on the present - and maybe you'll go from playing a math-based game to a physics-based game.

Methods based on logics and math cannot - WITHOUT LUCK - give you a real advantage as REALITY shows.
Maybe methods based on physics can?

But that's a completely different story.


Good Luck, my friend, and I hope YOU will find that method that for ever breaks the +3 SD barrier.

See You On-Board!
Kon-Fu-Sed

VLS forum member


If s**t can happen, s**t will happen.
- But when it hits the fan, probability theory is like having an umbrella.

As you may have noticed, I didn't use the words "winning" and "losing" - only "hitting" and "missing" - with the exception of a few paragraphs.
This is on purpose to have no confusions about what I say.
In my terms a "hit" is when a number/bet I select is also the result of the spin. Otherwise it's a "miss".
A "win" on the other hand is when a number/bet gives a positive net after the pay-out. Otherwise it's a "loss". Or "even".

The difference can be shown clearly as if I bet 1 unit on all the 37 numbers, the result will always be a "hit".
But as I bet 37 units and I get only 36 back I cannot "win" that way - only have a "loss" of 1 unit.
So I can "hit" and "lose" the same spin.
No confusion.
That's why.

Regarding "Misdirected intuition"...
The literature is full of examples of it and a real classic is this one (not roulette-related, though):

Three men enter a hotel and ask for a three-bed room.
- It's $25, says the clerk. Shall we have a look?
They go to the room and the three men are satisfied so they give the clerk one $10 bill each.
The clerk gives five $1 back.
The three men keep $1 each and give the clerk the remaining $2 as a tip.

So the three men gave the clerk $10 each and got $1 back; they payed $9 each.
3 x $9 = $27.
The clerk got $2 as a tip.
$27 + $2 = $29.

Where did the 30th $ go?

The files that were used are included in a 820K zipped archive. Download it from the member's area:
nolinks://vlsroulette.com/downloads/?sa=view;id=165 (nolinks://vlsroulette.com/downloads/?sa=view;id=165)

The archive contains the following files:

*** AllSpinsDate.txt

This file contains all the spins from Wiesbaden I used.
It is formatted in this fashion:

01.04.2003
23
1
25
.
.
.
10
15
35
02.04.2003
12
3
31
.
.
.
19
25
21
18.04.2003
19.04.2003
19
25
28
.
.
.

First there is the date and then comes all the numbers from that date on one line each, until a new date.
If there are no spins one date it is immediately followed by a new date: As can be seen above, there were no spins 18.04.2003.

The file contains 655,941 spins distributed over 1,903 days that had at least one spin.


*** BR33.txt

This file contains all the three-results events.
It is formatted in this fashion:

RRR
BRB
R0B
BBR
RBB
.
.
.

One three-results event on each line.
There are 218,010 events.


*** NB47.txt

This file contains all the seven-results events that have no Blacks in the first four results.
It is formatted in this fashion:

RRRRBRR
RRRRBRB
R0RRBRB
RRRRBRR
RR0RBBR
.
.
.

One seven-results event on each line.
There are 21,157 events in the file.


*** NR47.txt

This file contains all the seven-results events that have no Reds in the first four results.
It is formatted in this fashion:

0BBBRRB
BBBBRBR
BBB0RRB
BBBBR0R
BBBBBBB
.
.
.

One seven-results event on each line.
There are 21,355 events in the file.

Appendix - TABLE #1


Table 1A - Sample: Three-results events
Sample sizes: First 100, 500, 1000, 5000, 10000, 50000, 100000 and 218010 of the file

Each result is "B", "R" or "0"
Three trials to hit
Exit after a hit


Bet   SampSize   Trial     Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %  Did NOT hit
-----------------------------------------------------------------------------------------------------------
BLACK      100       1        48.64     4.99       34 -     63       54  +1     +5.35   +10.99           46
            46       2        24.98     3.58       15 -     35       22  -0     -2.98   -11.93           24
            24       3        12.82     2.56        6 -     20       16  +1     +3.17   +24.72            8
               No hits        13.54     2.56        5 -     21        8  -0     -5.54   -40.92

BLACK      500       1       243.24    11.17      210 -    276      263  +1    +19.75    +8.12          237
           237       2       124.90     8.00      101 -    148      115  -1     -9.90    -7.93          122
           122       3        64.14     5.73       47 -     81       58  -1     -6.14    -9.57           64
               No hits        67.70     5.73       50 -     84       64  -0     -3.70    -5.47

BLACK     1000       1       486.48    15.80      440 -    533      509  +1    +22.51    +4.62          491
           491       2       249.81    11.32      216 -    283      243  -0     -6.81    -2.72          248
           248       3       128.28     8.11      104 -    152      116  -1    -12.28    -9.57          132
               No hits       135.41     8.11      111 -    159      132  -0     -3.41    -2.51

BLACK     5000       1      2432.43    35.34     2327 -   2538     2427  -0     -5.43    -0.22         2573
          2573       2      1249.08    25.32     1174 -   1325     1278  +1    +28.91    +2.31         1295
          1295       3       641.42    18.14      587 -    695      609  -1    -32.42    -5.05          686
               No hits       677.05    18.14      622 -    731      686  +0     +8.94    +1.32

BLACK    10000       1      4864.86    49.98     4715 -   5014     4871  +0     +6.13    +0.12         5129
          5129       2      2498.17    35.81     2391 -   2605     2542  +1    +43.82    +1.75         2587
          2587       3      1282.84    25.66     1206 -   1359     1264  -0    -18.84    -1.46         1323
               No hits      1354.11    25.66     1277 -   1431     1323  -0    -31.11    -2.29

BLACK    50000       1     24324.32   111.76    23990 -  24659    24263  -0    -61.32    -0.25        25737
         25737       2     12490.86    80.08    12251 -  12731    12617  +1   +126.13    +1.00        13120
         13120       3      6414.23    57.39     6243 -   6586     6358  -0    -56.23    -0.87         6762
               No hits      6770.57    57.39     6598 -   6942     6762  -0     -8.57    -0.12

BLACK   100000       1     48648.64   158.05    48175 -  49122    48642  -0     -6.64    -0.01        51358
         51358       2     24981.73   113.26    24642 -  25321    25090  +0   +108.26    +0.43        26268
         26268       3     12828.46    81.16    12585 -  13071    12700  -1   -128.46    -1.00        13568
               No hits     13541.15    81.16    13297 -  13784    13568  +0    +26.84    +0.19

BLACK   218010       1    106058.91   233.37   105359 - 106759   106207  +0   +148.08    +0.13       111803
        111803       2     54462.68   167.23    53961 -  54964    54386  -0    -76.68    -0.14        57417
         57417       3     27967.32   119.83    27608 -  28326    27910  -0    -57.32    -0.20        29507
               No hits     29521.06   119.83    29161 -  29880    29507  -0    -14.06    -0.04


Table 1B - Sample: Three-results events
Sample sizes: First 100, 500, 1000, 5000, 10000, 50000, 100000 and 218010 of the file

Each result is "B", "R" or "0"
Three trials to hit
Exit after a hit

Bet   SampSize   Trial     Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %  Did NOT hit
-----------------------------------------------------------------------------------------------------------
RED        100       1        48.64     4.99       34 -     63       43  -1     -5.64   -11.61           57
            57       2        24.98     3.58       15 -     35       29  +1     +4.01   +16.08           28
            28       3        12.82     2.56        6 -     20       14  +0     +1.17    +9.13           14
               No hits        13.54     2.56        5 -     21       14  +0     +0.45    +3.38

RED        500       1       243.24    11.17      210 -    276      221  -2    -22.24    -9.14          279
           279       2       124.90     8.00      101 -    148      131  +0     +6.09    +4.87          148
           148       3        64.14     5.73       47 -     81       81  +2    +16.85   +26.28           67
               No hits        67.70     5.73       50 -     84       67  -0     -0.70    -1.04

RED       1000       1       486.48    15.80      440 -    533      461  -1    -25.48    -5.23          539
           539       2       249.81    11.32      216 -    283      270  +1    +20.18    +8.07          269
           269       3       128.28     8.11      104 -    152      138  +1     +9.71    +7.57          131
               No hits       135.41     8.11      111 -    159      131  -0     -4.41    -3.25

RED       5000       1      2432.43    35.34     2327 -   2538     2435  +0     +2.56    +0.10         2565
          2565       2      1249.08    25.32     1174 -   1325     1277  +1    +27.91    +2.23         1288
          1288       3       641.42    18.14      587 -    695      625  -0    -16.42    -2.56          663
               No hits       677.05    18.14      622 -    731      663  -0    -14.05    -2.07

RED      10000       1      4864.86    49.98     4715 -   5014     4855  -0     -9.86    -0.20         5145
          5145       2      2498.17    35.81     2391 -   2605     2514  +0    +15.82    +0.63         2631
          2631       3      1282.84    25.66     1206 -   1359     1249  -1    -33.84    -2.63         1382
               No hits      1354.11    25.66     1277 -   1431     1382  +0    +27.88    +2.05

RED      50000       1     24324.32   111.76    23990 -  24659    24392  +0    +67.67    +0.27        25608
         25608       2     12490.86    80.08    12251 -  12731    12383  -1   -107.86    -0.86        13225
         13225       3      6414.23    57.39     6243 -   6586     6445  +0    +30.76    +0.47         6780
               No hits      6770.57    57.39     6598 -   6942     6780  +0     +9.42    +0.13

RED     100000       1     48648.64   158.05    48175 -  49122    48665  +0    +16.35    +0.03        51335
         51335       2     24981.73   113.26    24642 -  25321    24790  -1   -191.73    -0.76        26545
         26545       3     12828.46    81.16    12585 -  13071    12788  -0    -40.46    -0.31        13757
               No hits     13541.15    81.16    13297 -  13784    13757  +0   +215.84    +1.59

RED     218010       1    106058.91   233.37   105359 - 106759   105842  -0   -216.91    -0.20       112168
        112168       2     54462.68   167.23    53961 -  54964    54374  -0    -88.68    -0.16        57794
         57794       3     27967.32   119.83    27608 -  28326    28027  +0    +59.67    +0.21        29767
               No hits     29521.06   119.83    29161 -  29880    29767  +0   +245.93    +0.83

Table 2A - Sample: Seven-results events without Red in the first four
Sample sizes: First 100, 500, 1000, 5000, 10000 and 21355 of the file

Each result is "B", "R" or "0"
Three trials to hit
Exit after a hit

Bet   SampSize   Trial     Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %  Did NOT hit
-----------------------------------------------------------------------------------------------------------
RED        100       1        48.64     4.99       34 -     63       48  -0     -0.64    -1.33           52
            52       2        24.98     3.58       15 -     35       21  -1     -3.98   -15.93           31
            31       3        12.82     2.56        6 -     20       16  +1     +3.17   +24.72           15
               No hits        13.54     2.56        5 -     21       15  +0     +1.45   +10.77

RED        500       1       243.24    11.17      210 -    276      226  -1    -17.24    -7.08          274
           274       2       124.90     8.00      101 -    148      132  +0     +7.09    +5.67          142
           142       3        64.14     5.73       47 -     81       78  +2    +13.85   +21.60           64
               No hits        67.70     5.73       50 -     84       64  -0     -3.70    -5.47

RED       1000       1       486.48    15.80      440 -    533      467  -1    -19.48    -4.00          533
           533       2       249.81    11.32      216 -    283      252  +0     +2.18    +0.87          281
           281       3       128.28     8.11      104 -    152      150  +2    +21.71   +16.92          131
               No hits       135.41     8.11      111 -    159      131  -0     -4.41    -3.25

RED       5000       1      2432.43    35.34     2327 -   2538     2361  -2    -71.43    -2.93         2639
          2639       2      1249.08    25.32     1174 -   1325     1302  +2    +52.91    +4.23         1337
          1337       3       641.42    18.14      587 -    695      688  +2    +46.57    +7.26          649
               No hits       677.05    18.14      622 -    731      649  -0    -28.05    -4.14

RED      10000       1      4864.86    49.98     4715 -   5014     4759  -2   -105.86    -2.17         5241
          5241       2      2498.17    35.81     2391 -   2605     2545  +1    +46.82    +1.87         2696
          2696       3      1282.84    25.66     1206 -   1359     1373  +3    +90.15    +7.02         1323
               No hits      1354.11    25.66     1277 -   1431     1323  -0    -31.11    -2.29

RED      21355       1     10388.91    73.04    10170 -  10608    10326  -0    -62.91    -0.60        11029
         11029       2      5334.85    52.34     5178 -   5491     5326  -0     -8.85    -0.16         5703
          5703       3      2739.51    37.50     2627 -   2852     2821  +2    +81.48    +2.97         2882
               No hits      2891.71    37.50     2779 -   3004     2882  -0     -9.71    -0.33


Table 2B - Sample: Seven-results events without Red in the first four
Sample sizes: First 100, 500, 1000, 5000, 10000 and 21355 of the file

Each result is "B", "R" or "0"
Three trials to hit
Exit after a hit

Bet   SampSize   Trial     Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %  Did NOT hit
-----------------------------------------------------------------------------------------------------------
BLACK      100       1        48.64     4.99       34 -     63       49  +0     +0.35    +0.72           51
            51       2        24.98     3.58       15 -     35       30  +1     +5.01   +20.08           21
            21       3        12.82     2.56        6 -     20       11  -0     -1.82   -14.25           10
               No hits        13.54     2.56        5 -     21       10  -0     -3.54   -26.15

BLACK      500       1       243.24    11.17      210 -    276      262  +1    +18.75    +7.71          238
           238       2       124.90     8.00      101 -    148      125  +0     +0.09    +0.07          113
           113       3        64.14     5.73       47 -     81       47  -3    -17.14   -26.72           66
               No hits        67.70     5.73       50 -     84       66  -0     -1.70    -2.51

BLACK     1000       1       486.48    15.80      440 -    533      505  +1    +18.51    +3.80          495
           495       2       249.81    11.32      216 -    283      255  +0     +5.18    +2.07          240
           240       3       128.28     8.11      104 -    152      112  -2    -16.28   -12.69          128
               No hits       135.41     8.11      111 -    159      128  -0     -7.41    -5.47

BLACK     5000       1      2432.43    35.34     2327 -   2538     2500  +1    +67.56    +2.77         2500
          2500       2      1249.08    25.32     1174 -   1325     1247  -0     -2.08    -0.16         1253
          1253       3       641.42    18.14      587 -    695      614  -1    -27.42    -4.27          639
               No hits       677.05    18.14      622 -    731      639  -0    -38.05    -5.62

BLACK    10000       1      4864.86    49.98     4715 -   5014     4958  +1    +93.13    +1.91         5042
          5042       2      2498.17    35.81     2391 -   2605     2485  -0    -13.17    -0.52         2557
          2557       3      1282.84    25.66     1206 -   1359     1227  -2    -55.84    -4.35         1330
               No hits      1354.11    25.66     1277 -   1431     1330  -0    -24.11    -1.78

BLACK    21355       1     10388.91    73.04    10170 -  10608    10458  +0    +69.08    +0.66        10897
         10897       2      5334.85    52.34     5178 -   5491     5327  -0     -7.85    -0.14         5570
          5570       3      2739.51    37.50     2627 -   2852     2721  -0    -18.51    -0.67         2849
               No hits      2891.71    37.50     2779 -   3004     2849  -0    -42.71    -1.47


Table 2C - Sample: Seven-results events without Black in the first four
Sample sizes: First 100, 500, 1000, 5000, 10000 and 21157 of the file

Each result is "B", "R" or "0"
Three trials to hit
Exit after a hit

Bet   SampSize   Trial     Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %  Did NOT hit
-----------------------------------------------------------------------------------------------------------
BLACK      100       1        48.64     4.99       34 -     63       45  -0     -3.64    -7.50           55
            55       2        24.98     3.58       15 -     35       32  +1     +7.01   +28.09           23
            23       3        12.82     2.56        6 -     20        8  -1     -4.82   -37.63           15
               No hits        13.54     2.56        5 -     21       15  +0     +1.45   +10.77

BLACK      500       1       243.24    11.17      210 -    276      240  -0     -3.24    -1.33          260
           260       2       124.90     8.00      101 -    148      135  +1    +10.09    +8.07          125
           125       3        64.14     5.73       47 -     81       52  -2    -12.14   -18.93           73
               No hits        67.70     5.73       50 -     84       73  +0     +5.29    +7.81

BLACK     1000       1       486.48    15.80      440 -    533      475  -0    -11.48    -2.36          525
           525       2       249.81    11.32      216 -    283      263  +1    +13.18    +5.27          262
           262       3       128.28     8.11      104 -    152      130  +0     +1.71    +1.33          132
               No hits       135.41     8.11      111 -    159      132  -0     -3.41    -2.51

BLACK     5000       1      2432.43    35.34     2327 -   2538     2386  -1    -46.43    -1.90         2614
          2614       2      1249.08    25.32     1174 -   1325     1272  +0    +22.91    +1.83         1342
          1342       3       641.42    18.14      587 -    695      666  +1    +24.57    +3.83          676
               No hits       677.05    18.14      622 -    731      676  -0     -1.05    -0.15

BLACK    10000       1      4864.86    49.98     4715 -   5014     4821  -0    -43.86    -0.90         5179
          5179       2      2498.17    35.81     2391 -   2605     2563  +1    +64.82    +2.59         2616
          2616       3      1282.84    25.66     1206 -   1359     1285  +0     +2.15    +0.16         1331
               No hits      1354.11    25.66     1277 -   1431     1331  -0    -23.11    -1.70

BLACK    21157       1     10292.59    72.70    10075 -  10510    10260  -0    -32.59    -0.31        10897
         10897       2      5285.38    52.09     5130 -   5441     5353  +1    +67.61    +1.27         5544
          5544       3      2714.11    37.33     2603 -   2826     2704  -0    -10.11    -0.37         2840
               No hits      2864.90    37.33     2752 -   2976     2840  -0    -24.90    -0.86


Table 2D - Sample: Seven-results events without Black in the first four
Sample sizes: First 100, 500, 1000, 5000, 10000 and 21157 of the file

Each result is "B", "R" or "0"
Three trials to hit
Exit after a hit

Bet   SampSize   Trial     Math Ave   StdDev    -3 SD    +3 SD     Real  SD      Diff   Diff %  Did NOT hit
-----------------------------------------------------------------------------------------------------------
RED        100       1        48.64     4.99       34 -     63       54  +1     +5.35   +10.99           46
            46       2        24.98     3.58       15 -     35       24  -0     -0.98    -3.92           22
            22       3        12.82     2.56        6 -     20       12  -0     -0.82    -6.45           10
               No hits        13.54     2.56        5 -     21       10  -0     -3.54   -26.15

RED        500       1       243.24    11.17      210 -    276      249  +0     +5.75    +2.36          251
           251       2       124.90     8.00      101 -    148      135  +1    +10.09    +8.07          116
           116       3        64.14     5.73       47 -     81       56  -1     -8.14   -12.69           60
               No hits        67.70     5.73       50 -     84       60  -0     -7.70   -11.38

RED       1000       1       486.48    15.80      440 -    533      495  +0     +8.51    +1.74          505
           505       2       249.81    11.32      216 -    283      264  +1    +14.18    +5.67          241
           241       3       128.28     8.11      104 -    152      123  -0     -5.28    -4.11          118
               No hits       135.41     8.11      111 -    159      118  -0    -17.41   -12.85

RED       5000       1      2432.43    35.34     2327 -   2538     2471  +1    +38.56    +1.58         2529
          2529       2      1249.08    25.32     1174 -   1325     1248  -0     -1.08    -0.08         1281
          1281       3       641.42    18.14      587 -    695      602  -2    -39.42    -6.14          679
               No hits       677.05    18.14      622 -    731      679  +0     +1.94    +0.28

RED      10000       1      4864.86    49.98     4715 -   5014     4906  +0    +41.13    +0.84         5094
          5094       2      2498.17    35.81     2391 -   2605     2492  -0     -6.17    -0.24         2602
          2602       3      1282.84    25.66     1206 -   1359     1237  -1    -45.84    -3.57         1365
               No hits      1354.11    25.66     1277 -   1431     1365  +0    +10.88    +0.80

RED      21157       1     10292.59    72.70    10075 -  10510    10297  +0     +4.40    +0.04        10860
         10860       2      5285.38    52.09     5130 -   5441     5253  -0    -32.38    -0.61         5607
          5607       3      2714.11    37.33     2603 -   2826     2694  -0    -20.11    -0.74         2913
               No hits      2864.90    37.33     2752 -   2976     2913  +0    +48.09    +1.67

Waiting for a wake-up hit AT LEAST 74 spins

The hit after at least 74 spins without a hit, is the wake-up hit.
How many hits are there in the next 37 spins after the wake-up hit?

The sample:
All recorded wake-ups = 52408
            with hits = 33401
            (tot hits = 52478)
         without hits = 19007


Table 3A - How long before the FIRST hit comes AFTER the wake-up hit? (37 spins max wait)

Example:
The Math average says that the FIRST HIT after the wake-up will come at the fourth spin 1304.66 times in this sample.
One Standard deviation = 35.62 and the interval -3SD to +3SD is 1198 to 1411 times.
The real number of occurrencies found in the file was 1318 times and that was not outside the 0 SD range.
The difference between the real number and the Math average was +13.33 or +1.02% for the real number.

Spin   Math ave   StdDev    -3SD to +3SD     Real  SD      Diff  Diff %
-----------------------------------------------------------------------
   1    1416.43    37.12    1306 -  1527     1443  +0    +26.56   +1.87
   2    1378.15    36.61    1269 -  1488     1394  +0    +15.84   +1.15
   3    1340.90    36.12    1233 -  1449     1343  +0     +2.09   +0.15
   4    1304.66    35.62    1198 -  1411     1318  +0    +13.33   +1.02
   5    1269.40    35.14    1164 -  1374     1316  +1    +46.59   +3.67
   6    1235.09    34.66    1132 -  1339     1185  -1    -50.09   -4.05
   7    1201.71    34.19    1100 -  1304     1111  -2    -90.71   -7.54
   8    1169.23    33.72    1069 -  1270     1160  -0     -9.23   -0.78
   9    1137.63    33.26    1038 -  1237     1140  +0     +2.36   +0.20
  10    1106.88    32.81    1009 -  1205     1189  +2    +82.11   +7.41
  11    1076.97    32.37     980 -  1174     1079  +0     +2.02   +0.18
  12    1047.86    31.93     953 -  1143     1072  +0    +24.13   +2.30
  13    1019.54    31.49     926 -  1114     1074  +1    +54.45   +5.34
  14     991.98    31.06     899 -  1085     1019  +0    +27.01   +2.72
  15     965.17    30.64     874 -  1057      961  -0     -4.17   -0.43
  16     939.09    30.22     849 -  1029      918  -0    -21.09   -2.24
  17     913.70    29.81     825 -  1003      870  -1    -43.70   -4.78
  18     889.01    29.41     801 -   977      942  +1    +52.98   +5.95
  19     864.98    29.01     778 -   952      892  +0    +27.01   +3.12
  20     841.60    28.61     756 -   927      789  -1    -52.60   -6.25
  21     818.86    28.22     735 -   903      880  +2    +61.13   +7.46
  22     796.73    27.84     714 -   880      748  -1    -48.73   -6.11
  23     775.19    27.46     693 -   857      759  -0    -16.19   -2.08
  24     754.24    27.08     673 -   835      694  -2    -60.24   -7.98
  25     733.86    26.72     654 -   814      754  +0    +20.13   +2.74
  26     714.02    26.35     635 -   793      683  -1    -31.02   -4.34
  27     694.73    25.99     617 -   772      696  +0     +1.26   +0.18
  28     675.95    25.64     600 -   752      648  -1    -27.95   -4.13
  29     657.68    25.29     582 -   733      655  -0     -2.68   -0.40
  30     639.90    24.95     566 -   714      619  -0    -20.90   -3.26
  31     622.61    24.61     549 -   696      662  +1    +39.38   +6.32
  32     605.78    24.27     533 -   678      610  +0     +4.21   +0.69
  33     589.41    23.94     518 -   661      555  -1    -34.41   -5.83
  34     573.48    23.62     503 -   644      608  +1    +34.51   +6.01
  35     557.98    23.30     489 -   627      543  -0    -14.98   -2.68
  36     542.90    22.98     474 -   611      524  -0    -18.90   -3.48
  37     528.23    22.67     461 -   596      548  +0    +19.76   +3.74
-----------------------------------------------------------------------
                                      Sum:  33401
0 hits 19016.31    22.67   18948 - 19084    19007  -0     -9.31   -0.04
-----------------------------------------------------------------------
                                      Sum:  52408


Table 3B - At what spin does the FIRST hit of TOTAL X hits come, AFTER the wake-up hit? (37 spins max wait)

  X      1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32   33   34   35   36   37       Sums
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  1    529  510  516  512  566  489  485  541  544  592  546  537  530  546  524  538  493  583  534  505  572  499  514  485  536  524  540  508  507  516  562  526  488  556  514  513  548      19528
  2    541  555  516  509  484  454  397  407  411  408  371  377  400  339  316  281  297  267  273  228  247  199  203  171  190  141  135  126  132   91   89   78   63   49   29   11            9785
  3    266  239  225  225  194  181  176  160  142  160  121  131  121  104   97   81   58   76   70   53   53   43   39   34   24   16   21   13   14   11   11    5    4    3                      3171
  4     84   68   70   61   54   51   48   42   35   21   32   22   17   27   20   18   19   16   12    3    7    5    3    3    3    2         1    2    1         1                                 748
  5     18   15   13    9   16    8    4    8    8    8    9    5    6    2    4         3         3         1    2         1    1                                                                    144
  6      5    6    1    1    1    2    1    2                             1                                                                                                                            20
  7           1    2    1    1                                                                                                                                                                          5
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Sums  1443 1394 1343 1318 1316 1185 1111 1160 1140 1189 1079 1072 1074 1019  961  918  870  942  892  789  880  748  759  694  754  683  696  648  655  619  662  610  555  608  543  524  548      33401


Table 3C - At what spin did HIT # X come, AFTER the wake-up hit? (37 spins max wait)

  X      1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16   17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32   33   34   35   36   37       Sums
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  1   1443 1394 1343 1318 1316 1185 1111 1160 1140 1189 1079 1072 1074 1019  961  918  870  942  892  789  880  748  759  694  754  683  696  648  655  619  662  610  555  608  543  524  548      33401
  2          34   93  117  133  164  196  240  237  263  282  307  355  351  368  375  406  456  413  462  478  451  495  473  497  464  550  539  501  495  548  525  487  551  489  557  521      13873
  3                     1    5   13   13   21   19   44   35   41   45   50   80   66   73   88   85  126  123  140  141  142  173  159  183  183  186  194  232  234  225  225  261  243  239       4088
  4                               1         1         5    1    3    3    1    7    7    9   11   22   12   22   21   21   40   31   39   32   43   45   62   41   50   81   76   65   83   82        917
  5                                                        2                        1    1    2    2    1    3    4    3    7    3    3    7    7    5    3   13   10   14   15   22   17   24        169
  6                                                                                                                    4    1                   2         1    3              3    5    3    3         25
  7                                                                                                                                                            1                   1         3          5
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Sums  1443 1428 1436 1436 1454 1363 1320 1422 1396 1501 1399 1423 1477 1421 1416 1367 1359 1499 1414 1390 1506 1364 1423 1357 1458 1348 1468 1422 1392 1374 1500 1429 1362 1478 1386 1427 1420      52478