# A Little Sampling Puzzle

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Suppose you have 10 objects from which you take a sample of size 20 (with replacement, or you’re in trouble). What’s the probability that each object was chosen at least once? Getting an answer via simulation is pleasantly easy:

f <- function(n=10, k=20) { x <- 1:n x.sample <- sample(x, size=k, replace=TRUE) return(length(unique(x.sample)) == n) } num.simulations <- 100000 table(replicate(num.simulations, f())) / num.simulations

You should see a number close to 0.215, which is confirmed by the analytic solution:

g

The second term is the probability that at least one object was not sampled. Enjoy!

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