# Men with Hats

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Suppose N people (and their hats) attend a party (in the 1950s). For fun, the guests mix their hats in a pile at the center of the room, and each person picks a hat uniformly at random. What is the probability that nobody ends up with their own hat?**mickeymousemodels**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

EstimateProbability <- function(n.people, n.simulations=10000) { NobodyGotTheirHat <- function(n.people) { people <- 1:n.people hats <- sample(people, size=n.people, replace=FALSE) return(all(people != hats)) } mean(replicate(n.simulations, NobodyGotTheirHat(n.people))) } CalculateProbability <- function(n.people) { InclusionExclusionTerm <- function(i) { return(((-1) ^ (i + 1)) * choose(n.people, i) * factorial(n.people - i) / factorial(n.people)) } 1 - sum(sapply(1:n.people, InclusionExclusionTerm)) } x.max <- 40 xs <- 1:x.max dev.new(height=6, width=10) plot(xs, sapply(xs, EstimateProbability), pch=4, ylim=c(0, 0.60), main="Men with Hats: N hats uniformly assigned to N people", xlab="N", ylab="probability that nobody ends up with their own hat") lines(xs, sapply(xs, CalculateProbability), col="firebrick", lwd=2) mtext("What is the probability that nobody ends up with their own hat?") legend("topleft", "True probability", bty="n", lwd=2, col="firebrick") legend("topright", "Estimate from 10k simulations", bty="n", pch=4) savePlot("men_with_hats.png") # The probability converges to e^-1 as N -> Inf exp(1) ^ -1

As in my earlier puzzle post, the solution is an application of the inclusion-exclusion principle. What’s fascinating about this particular puzzle is that the probability settles down not at zero or one, but instead converges to e^-1 as the number of people (and hats) grows large. I don’t know about you, but I never would have guessed.

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