A Little Sampling Puzzle

June 18, 2011
By

(This article was first published on mickeymousemodels, and kindly contributed to R-bloggers)

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 <- function(i) {
((-1) ^ (i + 1)) * choose(10, i) * ((10 - i) / 10) ^ 20
}

1 - sum(sapply(1:9, g))

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

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