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Your question makes assumptions with which I disagree. I do not think that strength means choosing winning moves more frequently in theoretically won positions. The positions encountered in chess are …

The proportion of girls in one family is a biased estimator of the proportion of girls in a population consisting of many families because you are underweighting the families with a large number of ch …

This result holds less obviously for Brownian motion with constant drift, not just $0$ drift. It is critical that the starting point is centered on the interval and it fails otherwise. Stern, F. An In …

There are quite a few extensions of the Central Limit Theorem to dependent random variables whose dependence is controlled. This includes the case of a sequence of sums of identically distributed rand …

This specific game seems very familiar, and I'm sure I have seen it discussed before along with a discussion of the optimal strategy, although I can't remember where. It might have been the Project Eu …

The minimum value is simply $2\alpha-1 = 0.365379$ where $\alpha = \Phi(1)-\Phi(-1) = P(|X|<1)$ where $X \sim N(0,1)$. This can be achieved by translating the percentile of $X$ (considering the percen …

The Clairvoyant Game Here is a well-known toy problem (the Clairvoyant Game) that doesn't converge: Suppose your hand is face-up. You have no hidden information. You don't know whether your opponent' …

Here is an answer to the updated question: Suppose that there are two betting rounds. Darth Vader has three types of hands. Type 1 wins with probability 1. Type 2 is a draw that hits (becomes a winni …

Getting all-in while behind It is not just when you are ahead that you might want to get all-in against someone who has an information advantage. Suppose the pot is $1$ and the effective stack depth …

This is a combination of the answer Gerry Myerson gave on MSE, the paper linked there, and the comments here. The largest possible minimum $m$ is $(n-1)^2+1 = n^2-2n+2$. This was proved by Wielandt, …

See A balls-and-colours problem and Another colored balls puzzle although those don't talk about the two-dimensional distribution. These suggest looking at the count of pairs of pebbles in different b …

This is related to the coupon-collector problem. These random variables have been studied by many people, although I don't recall a particular name for them. See, for example, Anna Pósfai's thesis (ab …

Shuffles like the overhand shuffle or riffle shuffle are not just random walks, they are symmetric random walks because you apply a random permutation drawn from the same distribution no matter what t …

The lower bound argument I gave for $\sqrt{n}$ points in a square works here, too. I have tried to simplify it. The idea is to use the union bound: The probability that a random path with $m=\lfloor \ …

There is little room for improvement over brute force because the average number of collisions is a very slowly growing function. The Erdős multiplication table problem asks how many numbers are pro …