There is a fully-booked flight with 100 seats. The first person decides to ignore the seat assignments, and sits in a random seat. Each subsequent person sits in their assigned seats if available, or sits in a random unoccupied seat if not.
What is the probability that the last person happens to find their assigned seat unoccupied?
Solution
The probability is actually 50% regardless of the number of seats on the flight!
Good Thinking but No Luck
You can easily calculate the probabilities for small numbers of seats:
- 2 seats: 1/2 chance first person picks their own seat
- 3 seats: 1/3 chance first person picks their own seat, and 1/3 * 1/2 = 1/6 chance first person picks the 2nd person’s seat and 2nd person picks the 1st person’s seat, for a total of 1/2 (all other scenarios result in the last person’s seat being taken)
However, this doesn’t reveal an insight that allows us to apply this reasoning for larger numbers of passengers.
Key Insight
Only the first person’s seat and last persons’s seat matter! Why?
- If any person takes the first person’s seat, the last person will always get their own seat, because every subsequent person will find their assigned seat available. Thus no subsequent person will affect the probability.
- If any person takes the last person’s seat, the last person obviously cannot get their own seat anymore. Thus no subsequent person will affect the probability.
- If any person takes any other seat, it doesn’t affect the probabilities, it just delays the choice between scenarios 1 and 2 to a later person.
Since scenario 3 does not affect the probability, and since scenarios 1 and 2 are always equally likely (each person is equally likely to select the first person’s seat as the last person’s seat), the probability the last person sits in their assigned seat is 50%.