for those who understand mathematics

Correspondence between two people about the probability of 50/50. Will someone decipher it for ordinary people?

Man No. 1
The General game in this case can be considered as a dynamic (step-by-step) game, where the step corresponds to the game being played.
If in the General game as a criterion of optimality the mathematical expectation of the winning value is determined, then it is reduced to a sequence of one-step games in accordance with the Bellman principle of optimality. If in these one-step games players adhere to optimal strategies, if, in addition, the conditions do not change and information about previous games can not be used to correct strategies, the overall game is reduced to a one-step game. In this case, applying each time the optimal strategy of a one-step game, the player implements the optimal strategy in the overall dynamic game.

Under the given conditions, the game is not described, because the players do not have strategies (if not to talk about the harmonization of rates). Therefore, in the described "game" there is no criterion of optimality, but there is a set of indicators (parameters) of a random process, some of which you would like to have more or less. But nothing can be done: there are no strategies – there is no possibility.
If we talk about the {+1, -1} - process, in this case, its parameters can be calculated according to the Bernoulli scheme with independent tests. If we calculate the total win / loss from the beginning of the "work", such a process will be a random walk with discrete time and discrete States.
On the question of for what random variables (local, integral,...) to calculate the variance, or something else, it is possible to respond according to your wishes. Without specifying the principle of using the calculated parameters (the principle of action on them) to talk about the feasibility of some calculations is meaningless.

Step-by-step {+1, -1} - process (win/lose) is a Bernoulli process. The process describing the total win from the initial moment is a random walk generated by this Bernoulli process. Therefore, we can say that the probability calculations of all events of this walk are based on the process with independent tests (Bernoulli calculations).
Since the value of a random sequence-walk is the sum of equally distributed random variables, then with a large number of steps in accordance with the Central limit theorem, its distribution approaches the Gaussian one (which you use).
Is it justified to continue to use the Gaussian approximation or the Gaussian distribution to roll with {+1,-1}-equally probable? - Decide for what process you want - for the new (old forgotten) or the process continues with the same starting point. Here everything is obvious-if the process continues, the use of Gaussian approximation is no less justified than before.If we used the Gaussian approximation for the initial site, then we should continue in the same spirit.

As an example, for a sequence of n tests (you call them games), calculate the probabilities of different number of wins (losses) and the number of wins not more than a given value in accordance with the binomial distribution, for example, for n=3, 4, 10, 20, 50 and 100. For these values of n, you can calculate the mean values, variances, Mat. waiting for modules...etc.
3. To realize that for n = 100 a) you do not have enough computing resources and b) the results of calculations are becoming more predictable, i.e. for large values of n the values you are looking for can be accurately determined based on the normal (Gaussian) distribution (read about the convergence of Bernoulli and other distributions, for example, in Wikipedia).


The standard normal distribution is a centered normal (Gaussian) distribution with normalized variance. Convolution of distributions corresponds to the distribution of the sum of independent random variables. The variances of random variables add up, therefore, given that the variance of the {-1.1} - distribution is not zero (equal to 1), it is easy to understand that the variance of the convolution will be more than one (equal to 2), i.e. convolution will no longer be a standard normal distribution.

It is also appropriate to think about the accuracy of the approximation of the distribution of the sum of random variables a) pure Gaussian distribution or b) a mixture of two or more Gaussians, Note that it does not matter at what steps you have replaced {-1,1}- random values on Gaussian, it is important how many of the total number of n steps used Gaussian, and how many {-1,1}- random variables.

Get a random walk ? - Yeah, random walk. Bernoulli's formulas are used to calculate the probability of k events (for example, wins) in n trials (games). This probability will be the same as the probability of k losses in n games.
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How to calculate the maximum deviation of a point from zero at a certain (n) step?
The formulas of Bernoulli. They allow us to calculate the probability of any transition from state i (the available sum) to state j (the estimated sum) in k steps. In this case, the probability of transition from i to j depends only on the difference |i-j| and the number of steps and does not depend on what was before.

Man No. 2

Gaussian approximation in this case is used for calculation of the probabilities of the sum of a large number bernoullis terms to get in the specified set. You said it yourself, but it is not quite true: not the distribution of the sum is close to Gaussian, but the distribution of the sum after normalization:

(2*S(n) - n)/sqrt{n}.

The validity of this approximation can be tested on the berry - Esseen inequality: the difference between the distribution functions of the centered and normalized sum and the standard normal random variable does not exceed 0.5 / sqrt{n}:

| P((2*S(n) - n)/sqrt{n} < x) - f(x) | <= 0.5/sqrt{n}.

Accordingly, when the 10,000 trials, the accuracy from replacing the real distribution function for a normal will be no more than 0.005.

Of course, it is a mixture with weights 1/2 of two normal distributions with unit variance and minoritarijami 1 and -1.
But the question does not become clearer from this. Why, in the context of a simple random walk, add the standard normal with the Rademacher distribution?

QUESTION:HOW TO CALCULATE THE MAXIMUM POSSIBLE DEVIATION OF THE MARKER FROM ZERO AT A CERTAIN STEP?

The maximum deviation of the marker from zero at the step n is equal to n. This happens with a probability of 1/2^{n-1}.

If n is relatively large, then for any fixed x (independent of n)
P (|S(n)| > x) = 2P(S (n) > x) ~ 2F (- x/sqrt{n}), where f (x) is the distribution function of the standard normal law.

If you are interested in the distribution of the first moment of time when the walk reached the point x, i.e. Y (x)=min {k: S (k) >= x}, then for n and x of the same parity
P(Y(x) = n) = (x/n)*P(S(n)=x) = (x/n)*C_n^{(n-x)/2} * (1/2^n).

For large n for any fixed x (whose parity is the same as for n)
P(Y(x) = n) ~ (x/n*sqrt{n})* sqrt{2/pi};
P(Y(x) >= n) ~ (x/sqrt{n})* sqrt{2/pi}.

Calculate the maximum deviation of a point from zero at a certain (n) step is possible by Bernoulli formulas?
Bernoulli's formula calculates, as you further said, the probability. The probability at this step to have some specific deviation from zero, of course, can be calculated by the Bernoulli formula. It above and recorded.

I'm suspecting that this is just a bunch of baloney

googling some of the expressions, they turn out to be real, so I'm not sure. But usually these conversations between mathematicians are chock full of equations, not words

one guy gives himself away, perhaps, with "minoritarijami" ... something close to a made up word? see link and decide for yourself

https://www.google.com/search?q=minoritarijami&lr=lang_en&sa=X&ved=0ahUKEwiVydfI4fTcAhUC4YMKHRDoBSgQuAEIJg&biw=1366&bih=631

Quote: odiousgambit

I'm suspecting that this is just a bunch of baloney

Or homework.


We don’t do homework.

Quote: DJTeddyBear

Or homework.


We don’t do homework.

Says every adult fast food worker I've met.

I have now read all of the original post. Got a mild headache doing it.

Is this a continuation of a discussion in one of his previous threads? Is he responding to a mustangsally post in his previous thread?

Obviously, English is not his first language, but even so, he seems like a very poor communicator. OP knows some math, perhaps self taught, but he is flailing about in this post. Does he actually ask a question anywhere that he doesn't answer in the next breath?

Can someone give me the dummies version of the question being asked?

Quote: Wizard

Can someone give me the dummies version of the question being asked?

I think he is analyzing a random walk on a line (each step is either +1 or -1) as a surrogate for analyzing a series of coin flips.

Then he says:
"Since the value of a random sequence-walk is the sum of equally distributed random variables, then with a large number of steps in accordance with the Central limit theorem, its distribution approaches the Gaussian one (which you use)."

Then he asks a question which I'll restate in my own words: Consider the probability distribution of the number of Heads outcomes after a series of n coin flips and also after n+1 coin flips. How can the probability distribution of the outcomes for both n flips and n+1 flips both be Gaussian when the n+1 flip was a binary (non-Gaussian) event?

Underlying his question is, apparently, an uneasiness with normal distributions in probability theory and a comfort with analyzing gambling as a series of trials with Bernoulli equations - a la mustangsally.

"minoritarijami" = mathematical expectation

I thought the op's question was: reduce these intentionally obtuse and perhaps obfuscated mathematical descriptions of a game into common English.

Quote: Wizard

Can someone give me the dummies version of the question being asked?

You have no idea how happy I am that YOU’RE the one asking. 😁

Quote: gordonm888

I think he is analyzing a random walk on a line (each step is either +1 or -1) as a surrogate for analyzing a series of coin flips.

Then he says:
"Since the value of a random sequence-walk is the sum of equally distributed random variables, then with a large number of steps in accordance with the Central limit theorem, its distribution approaches the Gaussian one (which you use)."

Then he asks a question which I'll restate in my own words: Consider the probability distribution of the number of Heads outcomes after a series of n coin flips and also after n+1 coin flips. How can the probability distribution of the outcomes for both n flips and n+1 flips both be Gaussian when the n+1 flip was a binary (non-Gaussian) event?

Underlying his question is, apparently, an uneasiness with normal distributions in probability theory and a comfort with analyzing gambling as a series of trials with Bernoulli equations - a la mustangsally.

This question sounds familiar. I think it was asked more directly in another thread regarding strategy.

Edit: Perhaps here

Revised:
The easiest way to accept {+1,-1} for the outcome of each game , then the process of winning/losing can be represented by {+1,-1}-sequence.

Step-by-step {+1, -1} - process( win/lose) is a Bernoulli process (with a large number of steps it approaches Gaussian, which after centering and normalization becomes centered and normalized, standard normal).The process describing the total gain from the initial moment is a random walk, with discrete time and discrete States (generated by the Bernoulli process). Therefore, the calculations of the probabilities of all events of the walks are based on a process with independent trials (Bernoulli calculations).

Use the Gaussian approximation or convolve the Gaussian distribution with {+1,-1}- equal ? There will be a mixture of two shifted by (+1,-1) Gaussians, convolution will no longer be the standard normal distribution. Decide for what process you want - for the new (old It doesn't matter) or the process continues with the same starting point. Here everything is obvious-if the process continues, the use of Gaussian approximation is justified.


It is also appropriate to think about the accuracy of the approximation of the distribution of the sum of random variables a) pure Gaussian distribution or b) a mixture of two or more Gaussians, it does not matter at what steps replaced {-1,1}- random variables on Gaussian, it is important how many of the total number of n steps used Gaussian, and how many {-1,1}- random variables.

Attention to the question:
how to calculate a new (old It doesn't matter )process?

"A mixture of two or more Gaussians, it does not matter at what steps replaced {-1,1}- random variables on Gaussian, it is important how many of the total number of n steps used Gaussian, and how many {-1,1}- random variables. "- how to understand this?