Every guest's hat gets handed back at random — what are the odds nobody gets their own?
N gentlemen check their hats at a party. At the end of the night, the attendant, having lost track of who owns what, hands every hat back completely at random, one per guest. As the number of guests grows, does the chance that not a single person gets their own hat back rise toward certainty, fall toward zero, or settle somewhere in between?
Reveal the answer
It settles at roughly 37%, and stays almost exactly there no matter how large the party gets. The probability converges to 1/e (about 0.368) as the guest count increases, a result French mathematician Pierre Rémond de Montmort worked out in 1708. Such a random arrangement with no one in their original position is now called a 'derangement.'
— Pierre Rémond de Montmort, Essay d'analyse sur les jeux de hazard — 1708; solved 1713