☄ alt2005 reveals the math behind roulette and answers questions

In the case of a deck of cards (with replacement) - geometric distribution p.
In the case of a deck without card returns, the hypergeometric distribution is p.
Don't confuse me.

For example, let's say there is a basket with balls.
one ball with sticker #27, the remaining 36 are white.

The ball was taken out, looked at, returned, until it was caught with a sticker - game over.
(The probability of getting a ball with a sticker each time is constant 0.027)
All over again.
The number of unsuccessful hits is recalculated until the first successful one.
X is the number of these unsuccessful hits,
The probabilities of these x's are distributed according to a geometriclaw.

Second game.
one ball with a sticker, the other 36 are white.
The ball was taken out, looked at, returned(!), until it fell out with a sticker, but the game continues.
You are allowed to get the ball limited number of times.
For example, 100 times.
During these hundred attempts, the ball with the sticker can fall out:
1) never
2) once
3) twice
4) three times
etc.
The number of successful hits over a limited period is recalculated.
A completely different random variable is taken for x.
The probabilities of these x's are distributed according to the binominal law.

Third game.
one ball with a sticker, the other 36 are white.
They took out the ball, looked at it, but didn’t return it (!) and so on until they come across one with a sticker.
Here, another random variable is taken as x.
and the probability distribution of these x's is hypergeometric.

To solve each problem, carefully look at what is taken as x (X), and only then the distribution is determined.

Let's stop at the binomial version of the game, Alt2005 said that he knows how to become a millionaire at roulette.

click writes:

Let's stop at the binomial version of the game, Alt2005 said that he knows how to become a millionaire at roulette.

If I understood him correctly, he wants to bring together geometric and hypergeometric.

Vladimir wrote:
4. Please tell or explain to the slow-witted where YOU or I can use the information provided here above?
Better in practice!

[moderator]
How to use the information - everyone decides for himself. Did you want the grail to be presented to you on a tray? There are no such waiters in the world. Nowhere on the Internet will you be given a winning strategy. Perhaps "win-win" systems for $5 per kilogram.
To move on to practice, you need theory. Will you get into a plane that was assembled on the knees of craftsmen in a garage who knew nothing about aviation?

Alt2005, in mathematics and statistics everything has long been proven! I don't think that you are the ones who will turn the knowledge in these sciences upside down!

Those who considered the Earth a pancake on 3 whales, also thought that everything had already been proven. All that remains is to measure the length of the trunks of the elephants on which those whales lie - and that's it, we have learned the pinnacle of the universe ))

I am asked questions - I answer. And the answer is for everyone who wants to read it.
[moderator]

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Shpilevoy wrote: many qualitative transitions occur only after quantitative accumulations ))))
For example, I really liked the idea of Klick to compare the probability of winning the current bet with the probability of the accumulated chain of Losses...
for example, I clearly understand that a single number can easily remain in the cold for more than 100 spins, and in some rare cases, 1000 is not the limit (and without the help of an adaptive)
with the help of the theory of faith I understand the boundaries of "normality" of what I expect
I assess the chances of success of bets

Boris, I once liked this idea too. But, alas....it has nothing in common with what I see in simulations (both on RNG and on real archives of Wiesbaden and Bad Homburg). No past influences future drops, if we are talking about mathematical, not physical roulette. This is the essence of geometric distribution of repetitions (GR), which has no memory...including on real roulette
There was a participant on CGM named Vano. He came up with a whole "theory of limit deviations" and based on it they made a program, I think "magicSpins". The idea was that there is some limit to the number not coming up, after which the probability of its appearance should increase. How exactly it should increase - he himself did not know. In short, you have to wait until something does not come up for a very long time. This is equivalent to losses.
Well, Vano had no idea about GR. Many knowledgeable people tried to prove to him that he was, to put it mildly, wrong)) But he didn't want to listen. So he lost everything and got into debt. He tried to sell that program, but also without much success. Angel bought it (who is now a moderator on CGM), and then he cursed both the program itself and its seller))

There are no extreme deviations. And I have not seen any boundaries of normality. There are 3 sigmas, but this is vague, I would say conventional. It can be considered that the number can sleep no more than 300 spins (I have never seen more than 600 non-drops in a row). But this fact does not give anything, because you do not know where to start betting on this number. If you start early - you may not have time to wait and lose, if you start late - you will miss the drop, and you will have to start all over again.

I conditionally calculate the max. interval of non-occurrence of an event as N = 2 * pi / P
where pi = 3.1415...
P is the probability of an event.

For the number this interval will be 2 * 3.14 * 37 ~ 232 spins (in my calculations I assumed 334 spins, it is more convenient for me)
For sixline 2 * 3.14 * 37/6 ~ 39 spins

If something jumps out of the interval - I consider it just outliers. They certainly do not affect the overall picture.