What is the probability that the last passenger sat in their assigned seat?

Question

The 120 seats on an Alaska Airlines flight were completely booked, with each of the 120 passengers having different assigned seats. The passengers entered the plane one-by-one. Unfortunately, the first passenger, Joe Isclumsy, couldn't read his boarding pass because he spilled coffee on it and sat in a (uniformly) random seat. Each subsequent passenger sat in their assigned seat if it was available when they entered and sat in a (uniformly) random empty seat otherwise. What is the probability that the last passenger sat in their assigned seat?

Answer 1

The chance is 1/2. It really depends on whether Joe Isclumsy's seat is taken first, or if the last passenger's seat is taken first. The probabilities are equal for the two seats, so the overall probability is 1/2.

Let's call the seats A and B. Starting with Joe. If Joe picks seat A (his seat), then the chain finishes and everyone else can sit in their assigned seat (100% chance that the final passenger gets to sit in their seat B.)

On the other hand, if Joe picks seat B, everyone else takes their seat until the last passenger tries to board and is forced to take seat A (0% chance they will get their assigned seat.) These are equally likely, so the probability is 1/2.

Okay, but what if Joe sits in a random seat C? Well, things are all okay until passenger C arrives. Passenger C has to choose a new seat. He might pick seat A and end the chain. Then the last passenger will definitely get his seat. Or he might pick seat B and again the last passenger can't get his seat. The choice of A vs. B is equally likely so the probability is still 1/2.

And if the passenger takes seat D, we only extend the chain and transfer the choice to passenger D. But again it will be 1/2.

Interestingly the number of seats doesn't matter. It could be 2 seats or 20,000 seats. The chance is still 1/2.

Let's enumerate the possibilities for four seats. For the sake of simplicity, let's assume that each passenger is supposed to sit in seats 1, 2, 3, 4 in order.

Case 1:
Joe --> 1, #2 --> 2, #3 --> 3, #4 --> 4 (success)
Joe --> 2, #2 --> 1, #3 --> 3, #4 --> 4 (success)
Joe --> 2, #2 --> 3, #3 --> 1, #4 --> 4 (success)
Joe --> 2, #2 --> 3, #3 --> 4, #4 --> 1 (failure)
Joe --> 2, #2 --> 4, #3 --> 3, #4 --> 1 (failure)
Joe --> 3, #2 --> 2, #3 --> 1, #4 --> 4 (success)
Joe --> 3, #2 --> 2, #3 --> 4, #4 --> 1 (failure)
Joe --> 4, #2 --> 2, #3 --> 3, #4 --> 1 (failure)

4 successes out of 8 possible outcomes, probability is 50%. I'll leave it up to you to enumerate the cases for 120 seats. :-)