Men with Hats

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?

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.