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Random pigeonholing 100 pigeons p1,…,p100 fly into 500 labelled holes h1,…,h500. Each pigeon picks a hole...

Random pigeonholing

100 pigeons p1,…,p100 fly into 500 labelled holes h1,…,h500. Each pigeon picks a hole uniformly at random and independently from the choices of the other pigeons.

  1. What is the probability that at least one hole contains at least 2 pigeons.
  2. What is the probability that at least one hole contains at least 3 pigeons.
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