On a hundred-seat airplane, a hundred people are traveling, each with a pre-assigned seat. The first passenger, however, disregards this and randomly sits in one of the hundred seats. Thereafter, each passenger tries to sit in their assigned seat, or if it is occupied, randomly selects another one. What is the probability that the hundredth passenger will sit in their assigned seat, given that only one person takes a seat at a time?
Solution. When the hundredth passenger sits down, only one seat can be free (99 have already sat down), and this seat can only be the hundredth or the first seat. Indeed, suppose that the seat of the X-th passenger would be free in between, then X would have sat in their seat. Thus, the last remaining seat can only be the first or the hundredth passenger's seat.

Whether the first or the hundredth seat is free, it would have been free all along, even when the other one was taken. For example, if the "U" passenger sat in the first seat, either because they were the first passenger or because their seat was taken, they had an equal chance of sitting in the hundredth seat, and this would not have affected the seat selection of the subsequent passengers, as it made no difference to them whether the first or the hundredth seat was taken earlier.

Similarly, if "U" sat in the hundredth seat, they had an equal chance of sitting in the first seat, regardless of the passengers who had already taken seats, and their decision would not have affected the subsequent arriving passengers. The probability of the two scenarios is the same. Thus, until the hundredth passenger takes a seat, there has always been an equal chance that the hundredth seat or the first seat would be taken, so the hundredth passenger has a 50 percent chance of sitting in their seat. The sought probability is $\frac{1}{2}$.