Random Airplane Seats Puzzle

August 4, 2020

Random Airplane Seats Puzzle

People are waiting in line to board a 100-seat airplane. Steve is the first person in the line. He gets on the plane but suddenly can’t remember what his seat number is, so he picks a seat at random. After that, each person who gets on the plane sits in their assigned seat if it’s available, otherwise they will choose an open seat at random to sit in.
The flight is full and you are last in line. What is the probability that you get to sit in your assigned seat?

Solution

The probability is indeed 1/2. There are two things to realize:

  • The probability that Steve chooses his assigned seat is equal to the probability that he chooses your assigned seat.
  • In case that Steve would choose neither his own seat nor yours, then there are two alternatives: if somebody else would choose Steve’s seat at random, then you would get your assigned seat; otherwise you would be left with the Steve’s seat.

With that being said, we can go on to find out the probability. With every person choosing a seat at random (including Steve), there are there possible outcomes:

  • either he chooses your assigned seat, or
  • chooses the Steve’s seat, or
  • chooses someone else’s seat.

Notice, that the probability of choosing Steve’s seat is always equal to probability of taking your seat. That means that the probability of you getting your seat vs. not, is even. The case of a passenger choosing someone else’s seat doesn’t affect your final outcome in either way, it just passes that three possible alternatives to the next passenger.

Since the probability of someone taking your place is always equal to the probability of someone taking Steve’s place (and this also applies to the penultimate passenger with only two seats left), the probability of you getting your assigned seat is in the end 50%.