Question

A line of 100 passengers is waiting to board a plane. They each hold a ticket to one of the 100 seats on that flight. (For convenience, let's say that the nth passenger in line has a ticket for the seat number n.) Unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. All of the other passengers are quite normal, and will go to their proper seat unless it is already occupied. If it is occupied, they will then find a free seat to sit in, at random. What is the probability that the last (100th) person to board the plane will sit in their proper sea?

Answer 1

The probability that the last passenger to board the plane will sit in their proper chair is 0.5

Explanation:

First, let's analyze the same exercise but with two seats and two passengers.

Seat 1: crazy passenger

Seat 2: passenger 2

Scenario 1: The crazy passenger sits in their respective chair, therefore passenger 2 also sits in their proper chair.

Scenario 2: The crazy passenger sits in chair 2, therefore passenger 2 must occupy chair 1.

According to this exercise, the probability that passenger 2 sits in his chair is 1/2 (0.5).

Let's analyze the exercise with 3 passengers and 3 seats

Seat 1: crazy passenger

Seat 2: passenger 2

Seat 3: passenger 3

Scenario 1: The crazy passenger sits in chair 1, therefore passenger 2 and 3 sit in their proper chairs.

Scenario 2: The crazy passenger sits in chair 2 and passenger 2 sits in chair 1. This leaves chair 3 free for passenger 3.

Scenario 3: The crazy passenger picks chair 2 and passenger 2 sits in chair 3. Leaving passenger 3 with the only option of taking chair 1. Everyone occupies wrong chairs

Scenario 4: The crazy passenger occupies chair 3, passenger 2 will occupy their respective chair and passenger 3 will again occupy chair 1.

Again, there are 50% chances that the last passenger will sit in his seat.

This means that in an event with n number of passengers, the probability of the passenger n will always be 1/2 (0.5). Since the number of scenarios where the last seat is available is the same as the scenarios where that seat is not available.