Absent-minded passengers

TL;DR

This paper analyzes the distribution of incorrectly seated passengers in a boarding process where the first passenger chooses randomly and subsequent passengers sit in their assigned seats if available, otherwise at random, extending known results about the last passenger's seat probability.

Contribution

It introduces a detailed study of the distribution of mis-seated passengers and explores scenarios with multiple absent-minded passengers, expanding understanding of this boarding process.

Findings

Last passenger's seat probability is 1/2.

Distribution of incorrectly seated passengers is characterized.

Multiple absent-minded passengers case is analyzed.

Abstract

Passengers board a fully booked airplane in order. The first passenger picks one of the seats at random. Each subsequent passenger takes his or her assigned seat if available, otherwise takes one of the remaining seats at random. It is well known that the last passenger obtains her own seat with probability $1/2$. We study the distribution of the number of incorrectly seated passengers, and we also discuss the case of several absent-minded passengers.