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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 …

Here is a counterexample. Since this question is more restrictive than the sequel, it is a counterexample to that, too. Let $\{1,2,3,4\}$ have equal probability. Let $1 \in A_{1,n},A_{2,n}$. Let $2 …

Here is a counterexample. Let the $0-1$ random variables $X_n$ follow a Pólya urn model: Choose $U$ uniformly on $[0,1]$ and then let the $X_i$ be independent Bernoulli trials with success rate $U$. $ …

Although this has been studied before, I find it interesting to apply basic techniques. Here are two. First, let $p_i$ be the probability that the first $i$ batches don't cover everything, and let $I …

These random walks are recurrent when $d\le 2$ and transient when $d \ge 3$. That behavior happens for a wide variety of random walks. The expected number of returns to the origin is $$\sum_{n=1}^\in …

Here is a direct argument. Suppose independent $X_1,X_2 \sim X$, and $X_1+X_2$ is uniform on $[0,1]$. $X$ is supported on $[0,1/2]$. For any $0 \lt \alpha \lt 1/4$, $\alpha = P\left(X_1+X_2 \in [ …