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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?

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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