Tim Roberts | Developer

One Hundred Passengers Puzzle

A puzzle involving 100 passengers boarding a flight and taking random seats. Can you guess who sits where?

Can you guess who sits where?

Introduction

This puzzle had me scratching my head for over a week. I tried solving it mathematically but couldn’t get it right. After checking forums, I still didn’t understand it. Finally, I created a Python simulation and now brought it to JavaScript for everyone to see.

The Puzzle

One hundred people are in line, boarding an airplane with one hundred seats, one at a time. They are in no particular order. The first person is drunk and has lost their boarding pass, so sits in a random seat. The second person does the following:

Goes to their seat (the one it says to go to on the boarding pass).
If unoccupied, sits in it.
If occupied, find a random seat to sit in.
Each subsequent passenger follows the same process.

What is the probability that the last person sits in their correct seat?

Try and solve the puzzle and use the below simulation to see a visual representation of the puzzle playing out.

100 Passengers Simulation

Simulation Controls

Seats:

Simulation Key

The drunk passenger when he has randomly selected their allocated seat.

The drunk passenger when he has NOT randomly selected their allocated seat.

Any passenger boarding after the drunk passenger who is able to sit in their allocated seat.

Any passenger boarding after the drunk passenger who is NOT able to sit in their allocated seat.

The last passenger when he is able to sit in his allocated seat

The last passenger when he is NOT able to sit in his allocated seat.

n

Allocated seat number in order of people boarding the plane.

Explanation

The above visual representation demonstrates how the puzzle plays out. Here's the best explanation for the solution:

There are two key stages to consider where randomness comes into play:

The first (drunk) passenger randomly picks a seat.
Subsequent passengers find their assigned seat occupied and must randomly pick a different seat.

Stage 1: The First Passenger

The first passenger is equally likely to choose any seat on the plane. This means the probability of them choosing their own seat or the 100th passenger's seat is the same.

Stage 2: Subsequent Passengers

For the second stage, it’s important to note that passengers only pick a random seat if their assigned seat is already occupied. As soon as someone randomly selects the first seat, the randomness resolves, and the puzzle is effectively determined: the 100th passenger will end up in their assigned seat.

However, if no one randomly picks the first seat, the randomness continues. For any passenger picking a seat randomly, they are equally likely to choose between the first seat and the 100th seat.

The Outcome

At every point in the puzzle, the probabilities remain equal for the first and the 100th seat being selected. If the puzzle continues all the way to the end, the last passenger will face a 50/50 chance of sitting in the first seat or their assigned seat.

Help me out and share this with your friends!