The lost boarding pass, and other practical problems

TL;DR

This paper discusses classic probabilistic puzzles involving randomness, focusing on the lost boarding pass problem, providing elementary solutions, generalizations, and connections to advanced distributions like Poisson–Dirichlet.

Contribution

It offers simplified proofs and generalizations of the lost boarding pass problem, highlighting probabilistic intuition and independence properties.

Findings

The probability the last passenger sits in their assigned seat is 1/2.

Occupancy statuses of different seats are independent.

Connection to Poisson–Dirichlet distribution is discussed.

Abstract

The reader is reminded of several puzzles involving randomness. These may be ill-posed, and if well-posed there is sometimes a solution that uses probabilistic intuition in a special way. Various examples are presented including the well known problem of the lost boarding pass: what is the probability that the last passenger boarding a fully booked plane sits in the assigned seat if the first passenger has occupied a randomly chosen seat? This problem, and its striking answer of $\frac12$, has attracted a good deal of attention since around 2000. We review elementary solutions to this, and to the more general problem of finding the probability the $m$th passenger sits in the assigned seat when in the presence of some number $k$ of passengers with lost boarding passes. A simple proof is presented of the independence of the occupancy status of different seats, and a connection to the Poisson--Dirichlet distribution is mentioned.