sampling the mean
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A challenge found on the board of the coffee room at CEREMADE, Université Paris Dauphine:
When sampling with replacement three numbers in {0,1,…,N}, what is the probability that their average is (at least) one of the three?
With a (code-golfed!) brute force solution of
mean(!apply((a<-sample(0:n,3e6,rep=T),3)),2,mean)-apply(a,2,median))
producing a graph pretty close to 3N/2(N+1)² (which coincides with a back-of-the-envelope computation):
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