Buses, Bullies, and Bijections

In times past in a country far away, passengers on intercity buses were all assigned seat numbers, and the buses were always full. There were many casual people, and there were a few people who were real sticklers for the rules. This is where our problem begins. Adam is the first person to enter an n seat intercity bus, and, since he is a casual person, he takes a seat at random. Subsequently, n − 2 more casual people enter the bus, and they also happily take seats at random. Of course, the inevitable occurs, and the last person on the bus is one of the aforementioned sticklers for the rules. If luck prevails, the stickler’s assigned seat is free; he sits down, and the bus drives away. On the other hand, if his seat is occupied, the stickler insists on having his assigned seat, and he even insists that the person whom he displaces must go to his or her officially assigned seat. Moreover, the stickler continues to oversee the commotion he has created. He insists that each successively displaced person must go to his or her assigned seat until there are no more displaced persons. The n − 1 casual people are not happy about this; but, eventually, the brouhaha dies down, and the bus hits the road. Here is the question: “What is the probability that mellow Adam, the first person on the bus, will be forced to move from his randomly chosen seat?” If n = 2, it is easy to see that the answer is 1/2. If Adam happens to sit in his assigned seat, then everything is fine; otherwise, he will have to move. Next, we can consider a bus with n = 3 seats, but things become more complicated. For a three seat bus, there are 6 = 3! possibilities that one needs to consider. To work through these possibilities in a systematic way, one needs some tools like the notation that we develop in the next section. Still, after an examination of the six cases, one comes to a noteworthy observation. In exactly half of these cases, Adam is forced to move. This coincidence suggests a bold speculation: “Can it be true that the probability that Adam will be forced to move is always one-half whatever the size of the bus may be?”