This is an article I posted earlier on a previous version of my blog. It’s about an interesting puzzle that goes like this:

Imagine a plane with a hundred seats. There are also a hundred passengers for this fully booked plane. Each passenger has a ticket indicating his or her seat number. However, the first passenger lost his ticket and takes a random seat. The other passengers enter the plane one by one and they either take their legitimate seat (if it’s not taken) or a random seat (if their seat is taken). The question is: what are the odds for the last person to be able to take the seat matching his seat number?

For the first person there are two choices of particular interest: the person can take his own seat (by chance) – or he can take the seat of the last passenger. In the latter case, the game is over and the last person will NOT be able to take his own seat. In the first case, all passengers (including the last one) take their own seats as none is randomly taken by someone else. So, in the first case the last person WILL be able to take his own seat. Both cases have the same odds (1/100) and they average 50%. But what about the other 98 possibilities? Let’s see…

In the other cases one of the seats is taken. Passengers enter the plane and take their legitimate seats one by one – until some unlucky passenger finds his seat taken. Out of the remaining seats again two choices are conclusive. If he takes his own seat by chance, all further passengers can take their own seat, including the last passenger. If he takes the seat of the last passenger, then the last passenger can no longer hope to find his seat to be empty. The odds for both cases are unknown (as we do not know how many seats are left) but we do know that they are equal and therefore average 50%. What about the other cases? Let’s see..

You can repeat the above until the end. Each time a passenger is about to take a random seat, he can take his own seat by chance or he can take the seat of the last passenger. These cases average 50% and the rest of the cases are handled by the next iteration where a passenger takes a random seat.