# Questions tagged [pr.probability]

Questions in probability theory

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### Can the “mutual independence” condition in the Lovász local lemma be weakened?

The Lovász local lemma, as stated in Corollary 5.1.2 here, is given as follows. Lemma. Let $A_1, \ldots, A_k$ be events such that each $A_i$ has probability at most $p$ and such that each $A_i$ is ...
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### The noise distribution on $F_2^n$: probability of landing in a subspace versus a coset

Let $V = F_2^n$ be the $n$-dimensional vector space over the field of two elements. The $\epsilon$-noise distribution on $V$, denoted $\mu_\epsilon$, is a probability distribution on $V$ for which ...
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### Percolation probabilities

I have a finite undirected graph with a probability $p_e$ given for each edge $e$. This gives a random graph by removing each edge e with probability $1-p_e$ independently of the others. I'm ...
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### Proving properties of Random Graphs

I am asking the question on a slightly abstract level and it may depend on the specifics but it would be great to have related references or ideas. Consider the random graph model $G_{n,p}$ where its ...
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### Subspace-evasive set performance in the random case

A subspace evasive set is defined as a large subset of a vector space which has small intersection with any $k$ dimensional affine space. That is, it "evades" all affine subspaces of small enough ...
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### Maximizing the number of heads in $N$ tosses by choosing which coin to toss

Assume you have two coins $A,B$ with biases $P_A,P_B$ respectively. We would like to make $N$ coin tosses and get the maximal number of heads possible. Unfortunately, we know $P_B$, but $P_A$ is ...
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### A flowchart for concentration bounds

When I teach tail bounds, I use the usual progression: If your r.v is positive, you can apply Markov's inequality If you have independence and also bounded variance, you can apply Chebyshev's ...
Setting Consider repeating the following process on the numbers $N=\{1, 2, 3, \ldots, n\}$: Pick an integer $k \in N$, uniformly at random. Pick a subset of $k$ elements from $N$, uniformly at ...