☄ alt2005 reveals the math behind roulette and answers questions

Coin wrote: =casebetween(0;36)

The generator in EXCEL is definitely without adaptive

For earlier versions of Excel =ROUNDBOTTOM(RAND()*37,0)

Coin wrote: Regarding the 2/3 law. (alt2005)

As I have said here many times, the so-called 2/3 law is not a law at all, but a consequence of school combinatorics. Over 10 years ago I made a program that calculates all possible variants of numbers falling out on a segment of 37 spins, and calculates their probabilities.
Even then it dawned on me that 2/3 is a completely useless thing, and even harmful. Now if not 24-25 different numbers came up, but significantly less, then you could bet on repeats. In this case, obviously you would lose less and win more. Or, on the contrary, if more numbers came up, then you would need to bet on those that have not yet come up and also be in chocolate. But since exactly 24-26 of them come up, then whether you bet on those that have come up or on those that have not come up, you will still be in the minus over a large interval.

About 2 years ago I started to seriously study distances. Numbers are repeated, and the distance is the number of spins between repetitions. I took statistics of about 20K spins from the Wiesbaden archive (there are a ton of them) and for each number I calculated the distance to its next repetition. Then I simply sorted these distances in ascending order. I got this curve (I have posted this graph 100500 times). The Y-axis shows the distance values, the X-axis simply shows their quantities on a segment of 20K spins. What catches the eye? That there are more small distances than large ones. Distance 1 is approximately the largest of all, distance 2 is a little smaller (plus or minus), distance 3 is even smaller, etc.
Took another sample - same thing. Generated 40K random numbers in Excel - same thing again. There are practically no differences. And this is easy to explain mathematically.
The probability of repeating the number immediately on the next spin (i.e. distance = 1) is 1/37
The probability of repeating a number only through a spin (i.e. distance = 2) is 36/37* 1/37. Let me explain for those who are not on good terms with the theory of probability, everything is simple here. Let's say the number 10 came up on spin 1. The probability of repeating the number 10 only on the 3rd spin (through 2 spins) means that it did not come up on the 2nd spin (P=36/37), and came up on the 3rd (P = 1/37).
Then everything is as simple as pie. The probability of a repeat only on the 4th spin (d = 3) P4 = 36/37 * 36/37 * 1/37. In general, the probability of a repeat only on the (N+1)th spin = (36/37)^N * 1/37. This thing is called a geometric distribution, because the probability decreases in geometric progression with a coefficient of 36/37.
At about point 25, the probability of the number coming up at least once is equal to the probability of it not coming up at all. Let me repeat: in 25 spins, the probability of a number coming up is the same as not coming up. That is, ½. If mathematically, then approximately 36/37 ^ 25 = 1 - 36/37 ^ 25. This means that in 25 throws, in half the cases, the player will guess the number at least once. And this is clearly visible on the curve. This also means that in 25 times, about half of the numbers will come up (18-19). And not necessarily in a row. On any 25 spins.
It also doesn't matter from which reference point to expect a certain number to appear. Not necessarily from its appearance. From any reference point.

The 2/3 rule is basically the same thing, only for 37 spins. Look at the curve. The distances from 1 to 37 take up about 2/3 of the entire graph. Plus or minus the spread.
This means that distances from 1 to 37 occur in 2/3 of cases.
This means that in 2/3 of cases the number will come up at least once in the interval 37.
This means that approximately (more precisely, no more than) 2/3 of the numbers fall out on this interval (in 94% of cases)
Everything clearly corresponds to the theory. But this curve means a drain. It corresponds to a negative MO. And since the 2/3 law also lies on it, it is also a drain.

Therefore, all the tales about how you can use the 2/3 law to win are, as Olynes said, bullshit for the feeble-minded. Those who need to, will sing. And those who didn't get into it, can continue to believe in this crap... to the delight of the scammers. The ostrich hides its head in the sand, but its ass still sticks out. And they will kick it mercilessly. It will hurt, especially if the ass and head have swapped places.
Although the ass sticking out of the sand can also be useful. You can stick strategies there based on 2/3, hot numbers, cold ones, "equalization laws", "waves", etc. If they all don't fit at once - no big deal. You can shove them there one by one. Put it in - take it out, put it in - take it out. Erotica!

I hope you are not tired of this scribbling. So I will continue. All reasoning about mathematical roulette, flat betting. No progressions.

So, the distances have a geometric distribution (GD). This is the only distribution that has no memory. What does this mean? Let's look at the graph. If we stand at the beginning, then 50% of the distances are from 1 to 25. Let's try to skip 25 spins and start with the 26th. What do we see? The same thing. The spins up to the 26th can be forgotten. On the remaining part of the curve, 50% of the distances are from 26 to 50, the rest of the distances are longer. If the 26th spin is considered the 1st (first), then everything is one to one - in half the cases these are distances from 1 to 25.

Let's try to start with the 51st spin. Again the same. Distances 51-75 are about half. Nothing has changed.
What does this mean? Well, you can wait as long as you want for the number not to come up. No matter where you start betting on it, it will still come up in half the cases after the next 25 throws. Only half the cases. This means a loss. As sad as it is.

The problem is that any number "tends" to come up as soon as possible. There is nothing good about this. On the contrary, if the probability of a number repeating itself were greater the further, the more – players would be in chocolate. It would be possible to simply wait until it does not come up for a long time, and start betting on it (which is what Vano tried to do). But then there would be no roulette as a business.

I asked myself: under what conditions would the probabilities of guessing a number no earlier than some spin be the same at any step? The answer is not complicated. Simple arithmetic of about 5th grade of elementary school.

The probability of guessing the number is 1/37, everyone knows that.
In order for the probability of guessing the number only at the 2nd step to be the same 1/37, it is necessary that
36/37 * x = 1/37, hence x = 1/37 : 36/37 = 1/36.

In order for the probability of guessing the number only at the 3rd step to be the same 1/37, it is necessary that
36/37 * 35/36 * x = 1/37, hence x = 1/37 : (36/37 * 35/36) = 1/35. Here 36/37 is the probability of not falling on the 1st step, 35/36 is the probability of not falling on the 2nd step (35/36 = 1 - 1/36)

In order for the probability of guessing the number only at the 4th step to be the same 1/37, it is necessary that
36/37 * 35/36 * 34/35 * x = 1/37, hence x = 1/37 : (36/37 * 35/36 * 34/35) = 1/34. Here 36/37 is the probability of not getting it on the 1st step, 35/36 is the probability of not getting it on the 2nd step (35/36 = 1 - 1/36), 34/35 is the probability of not getting it on the 3rd step.
In short: the probability of getting it right (guessing) only at the N-th step is 1 / (38-N)

But it is not enough to draw formulas, you also need to understand their meaning. And it is very simple: on the 1st step there are 37 numbers, on the 2nd 36, on the 3rd 35, etc. That is, the numbers that have come up should not come up again. Like in a deck of cards - the cards that have come up do not return to the deck.

But roulette is not a deck of cards. The fact that a number has just come up does not mean that it will fall asleep and will not come up on the next spin. The player starts each spin from zero. The curve above illustrates this. So what? So you need to change its appearance. Either press it at the beginning, or, conversely, bend it at the end. Simply put - as long as you maintain the ratio of 25 distances in half of the cases (or, equivalently, 37 distances in 2/3 of the cases), you are doomed to lose. No strategies and methods will help if they do not change the appearance of this curve. If someone is building illusions - forget it.

And there are no strategies until someone comes up with one. But he won't blab about it. So, maybe there already are. For example, I just barked at the standard strategies. I expressed an opinion on how not to play. But I didn't say how to play.
Theorever also didn't exist at one time.

Distribution of distances is a mathematical phenomenon, and statistically confirmed a thousand times. Unlike the so-called adaptive, which no one has ever seen. And even the most amiable Mr. Shpilevoy himself will hardly be able to say from specific statistics whether there is an adaptive or not.

I need to refresh the concept of "distance distribution" in order to discuss
which I will do with pleasure