☄ alt2005 reveals the math behind roulette and answers questions

zen.yandex.ru/media/id/5a3bc6e5256d5ca91...4a43aad43600affce4d0
I wonder what the opinions will be.

I liked this post by Alta. You invited everyone to discuss the article beautifully.
He can be gallant (when he wants to)

But nevertheless, the fact remains a fact; already in comment #378, Alt demonstrates an inattentive reading of the problem statement, which is far from cool for a mathematician.
I don't even know...
Either he "galloped through Europe" and ran through this article without really delving into what the young researcher was trying to convey to the public...
...or he specifically wrote it this way to give me the pleasure of catching him in inaccuracies.

I will put this moment in Alt in the center of attention:

...When he considers 1 million players playing, say, once, this is essentially what happens - everyone bets 100, and the [total balance] / [total turnover] will be about 5%.

There should be no words "let's say" here in principle. The author sets the condition.
That's the point, they didn't place the bets once, but 60 times, and not a hundred constantly, but the remaining balance.
It's not as simple as Alt shot back in his comment.

The guy who wrote the article was surprised by the right (downward) trend, and I was surprised by the left linear (upward), and even strictly 5%,
The ascending one can happen at a short distance and at the very beginning,
but will it be exactly 5%, that’s the question.

I will express my thoughts consistently.

Look, the author divides the study into two parts.
The first is ensemble play, the second is individual play.
But in both the first and second cases, the current balance is set.

I'll show the first screenshot


One minute to throw. Result of five: two tails, heads, tails, heads.
When you win, your current balance increases by 50%, which is equivalent to a multiplication by 1.5
when you lose, the current balance decreases by 40%, which is equivalent to multiplying by 0.6 (or dividing by 1/0.6 = 1.666667)

1) 100x0.6 = 60
2) 60x0.6 = 36
3) 36x 1.5 = 54
4) 54x0.6 = 32.4
5) 32.4 x 1.5 = 48.6
If you look at the graph, that's exactly how it is.
There are no permanent hundred.
The simulator that the author uses is programmed to bet on balance balances.

Then he makes 60 throws (or in our terminology 60 spins)
9 episodes (like 9 people)...
20 episodes (like 20 people)...
1000 episodes (like 1000 people)...
I'll make a short stop at a thousand.

Shows an upward trend


But did you notice how he got it?
He did it in a communist way. He stops after each throw, takes money from the rich, takes the last crumbs from the poor and (from everyone else), and then distributes it evenly among everyone. And it turns out that everyone is doing well. On average.
That's it, there is the first point on the graph. And so are all the others.
Notice how he was genuinely surprised when he saw the beginning of the balance collapse: "It's just a pity that at the end the chart somehow fell down."
This is a natural phenomenon: under such payment conditions, the balance tends to zero over the long term.
100x1.5 x 0.6
Eagles and tails in the long run 50/50
100x0.9x0.9x0.9............
Remaining bets are always multiplied by 1.5 when winning,
and are divided by 1/0.6 = 1.6666666666666667 in case of loss.
In order for the game to stop being unprofitable, it is necessary that when the remainder is multiplied by 1.5 when winning,
at least divisible by 1.49925 in case of loss (i.e. 50% and 33.3%)
The loss in case of loss should be less than 1/3 of the bet amount
And in this condition the loss is more than 1/3
150 - 40% = 150 - 60 = 90
150/3 = 50, 50 < 60
But with this point (the tendency of balance to zero over a long distance) Alt has already demonstrated everything perfectly on his graph.
....
Next the author moves on to a million players.

...Each ensemble had its own trajectory, which for ease of distinction was painted in a unique color, as shown in the picture of 20 trajectories.

Yes, this guy is an artist. 100 shades of red, 100 shades of black... 100 shades of gray... (and so on down the palette)

Next, everything is as in the example with a thousand, it displays the average for each spin and plots this point on the graph

Writes:

A clear linear trend with income growth of 5%. The game, as intuition suggested, is profitable

But he doesn’t attach files with calculations.
I would like to look at these calculations.
Because without them, it’s a drawn line.
It seems that he artificially adjusted the linear upward trend to 5%.
he needs to demonstrate it so that there are no discrepancies with the expected value.
But he calculated the expected value when it’s always 100, and that’s a different game.
But he doesn't understand this. It's obvious that he doesn't understand.

And he also writes: "The expected value of a random variable (in our example, the next heads or tails) is calculated using the formula for mathematical expectation."

The value of a random variable is not "heads and tails", but money. And the expectation is calculated in money (not in heads and tails). It can also be deduced as a percentage.

The checkmate finds the expectation for one game, but plays another.

Where did this strict 5% increase come from?
There may be an increase, it's a short distance: 60 spins in total. But exactly 5%?
And in a game where the balance tends to zero

My opinion is this.
Or he didn’t calculate any averages, but simply drew this line.
Or he did calculate it, and it is the result of a uniquely pumped up dispersion,
and it swung, just as if on order: the average ones lined up like soldiers, at a certain angle, just as the author needed.

Even if we admit that this happened by chance,
A special case is not proof.

Always put two black and one white in one urn, and one white in the other urn!

By the way, I can throw in another problem, I brought it to CGM a long time ago.

There are 4 balls: 2 black and 2 white. And there are 2 urns. The player places the balls in the urns as he wishes. Then the urns are mixed (the player does not see this), and he draws one ball from any urn (without looking into it). How to place the balls in the urns to have the highest probability of drawing a white ball? (Of course, if one of the urns is empty, and the player chooses it, he will be forced to draw from the other)
There is probably a solution on the Internet, but it’s more interesting to solve it yourself.

In general, Alattissa's comments are literate and correct.

Although I will make some small additions.

But nevertheless, the fact remains a fact, already in comment #378 Alt demonstrates an inattentive reading of the problem statement, which is far from cool for a mathematician.

1) Alt, actually a software developer, not a mathematician. Although he graduated from the mathematics department of MAI, where he was hammered with this science for almost 6 years. I re-took the theory test 5 times, but I learned a few things, for example, how to solve problems.

The guy who wrote the article was surprised by the right (downward) trend, and I was surprised by the left linear (upward), and also strictly 5%,
The ascending one can happen at a short distance and at the very beginning,
but will it be exactly 5%, that's the question.

2) If the rates are constant (flat), then it will work. But betting on the remainder and calculating ROI is of course incorrect.
Which is shown in the quote.

Alatissa wrote:

When you win, your current balance increases by 50%, which is equivalent to multiplying by 1.5
when you lose, the current balance decreases by 40%, which is equivalent to multiplying by 0.6 (or dividing by 1/0.6 = 1.666667)

1) 100x0.6 = 60
2) 60x0.6 = 36
3) 36x 1.5 = 54
4) 54x0.6 = 32.4
5) 32.4x 1.5 = 48.6

Even if we assume that there are exactly as many heads as tails (N of each), what do we get? Each such pair of heads-tails gives the result 1.5 * 0.6 = 0.9. No matter how you rearrange it. And with N such double trials, we get 0.9 ^ N, which obviously tends to 0 as N grows.

In general, it is pointless to calculate ROI on progressions. Let's take a simple martin. We bet all the way on red, starting with 1 USD. If we lose, we double the bets. Then the sum of the bets (turnover) = 1 + 2 + 4 + 8 + 16 + ... = 2^n - 1, where n is the length of the series up to and including the win. And this win in the case of red will be 2^(n-1). That is, the balance in case of a win always increases only by 1 = 2^(n-1) (win) - (2^(n-1) - 1) (placed before), or rather by the initial bet. At the same time, the turnover grows like an avalanche
And how can ROI be calculated here? If we consider each such series separately, then at the point of winning
ROI = 1/ (2^n - 1) ->0, and at other points until that moment it will simply be -100%.

The author sets the condition.
That's the point, they didn't place the bets once, but 60 times, and not a hundred constantly, but the remaining balance.
It's not that simple, as Alt shot back in his comment.

When I started reading the article, I immediately felt something was wrong, and immediately decided to check in Excel. So I didn't really go into it, yes. What the author writes there is not so important. The main thing is that if a million players bet 100 once, it is equivalent to one player betting a million times 100, without relying on the previous balance. Then 5% will quickly pop up. That's what the file says.

for an ensemble with a million players. They need to reorganize and put not the remainders, but one constant (100 or another amount) to each. Then there will be an ascendant.
But the conditions don’t say whether this can be done.

The author does not take these nuances into account.

It's not that I thought of it, rather it was my first intuitive guess after I posted a video from a movie on the Mira forum, where a teacher (actor Kevin Spacey) offers a task about choosing doors, and one of the students explains why, he changed his choice when the presenter opened one door with a scooter... In the case of balls, in one urn the probability of choosing a white ball will be 33.3%, in the other 100%, respectively, the probability of choosing a white ball from two urns will be 66.7%... For me, probability theory remains a theory that is not applied in business processes at all. However, oddly enough, probability theory sets a certain trend for the target audience involved in business processes. And since there is no business without people, your topic is more than interesting! It helps to reveal the cyclical nature of thinking inherent in any human mind.
Adaptive is emotionless and its choice is always based on variable replacement. This fact helped me build a matrix with a betting protocol and get the first result based on the adaptive choice of the top 2 priority. Thanks Mira again!