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If $a_1,a_2,\dots$ are IID random bits (correction as per Anthony Quas: these "bits" are $+1$ and $-1$ with equal probability), then with probability 1, the set of natural numbers $n$ such that $a_1+a …

I've thought of an approach to variance reduction that surely can't be new, but I haven't been able to find it published anywhere; I'd appreciate some leads. Consider some sort of random variable $X$ …

Consider critical edge-percolation in the induced subgraph of the square grid with vertex set {$(i,j) \in Z \times Z:\ i+j \geq 0$}, and let $p_n$ be the probability that the cluster containing $(0,0) …

Divide the unit circle into three arcs, and let $z$ be a point in the open unit disk. Is there a simple formula for the probability that Brownian motion started at $z$ will hit one particular arc rat …

Part of what I do is study typical behavior of large combinatorial structures by looking at pseudorandom instances. But many commercially available pseudorandom number generators have known defects, …

Let $X_1,X_2,\dots$ be iid draws from the uniform distribution on $\{1,2,...,m\}$, and let the random variable $N$ be the minimum $j$ such that $X_j = X_i$ for some $i<j$. I'm aware that the expected …

Starting from the length-1 list whose only entry is 1, iterate the process of replacing the last (and largest) entry in the list of length $n$ (call that entry $m$) by the two numbers $mU_n$ and $m(1- …

Given disjoint nonempty subsets $X_1, X_2$ of the state space of a finite irreducible Markov chain, there are unique measures $\mu_1$ on $X_1$ and $\mu_2$ on $X_2$ such that (a) starting from a $\mu_1 …

Does anyone know of any work on the following model or variants thereof?: Finitely many chips are distributed on the integers at time 0. To find the distribution at time $t+1$, take all the chips at …

If one picks $a+b$ points uniformly at random (and independently) in $[0,1]$, coloring $a$ of them Red and $b$ of them Blue, then reading the points from left to right one gets a uniform random sequen …

How many steps on average does a simple random walk in the plane take before it visits a vertex it's visited before? If an exact formula does not exist (as seems likely), then I'm interested in good …

More than one combinator(ial?)ist has asked me to recommend a good book to learn probability from, and I never know what to say; the probability theory that I use in my research up was mostly learned …

In the case where $\sum_{n \in \mathbb{Z}} \mu(n) = \sum_{n \in \mathbb{Z}} \nu(n) = 1$, this setup is reminiscent of the theory of martingales, so I'm tagging this question pr.probability as well as co.combinatorics …

In the case where $\sum_{n \in \mathbb{Z}} \mu(n) = \sum_{n \in \mathbb{Z}} \nu(n) = 1$, this setup is reminiscent of the theory of martingales, so I'm tagging this question pr.probability as well as co.combinatorics …

Consider two independent copies of IID random walk on ${\bf Z}$ starting from $0$, and let $N_1(t)$ (resp. $N_2(t)$) denote the number of times, up to time $t$, that the first (resp. second) walker ha …