John L. Weatherwax∗ September 4, 2007
Special thanks to Vincent Frost and Andrew Jones for helping ﬁnd and correct various typos in these solutions.
The Crazy Passenger Problem
The following is known as the “crazy passenger problem” and isstated as follows. A line of 100 airline passengers is waiting to board the plane. They each hold a ticket to one of the 100 seats on that ﬂight. (For convenience, let’s say that the k-th passenger in line has a ticket for the seat number k.) Unfortunately, the ﬁrst person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. All the other passengers arequite normal, and will go to their proper seat unless it is already occupied. If it is occupied, they will then ﬁnd a free seat to sit in, at random. What is the probability that the last (100th) person to board the plane will sit in their proper seat (#100)? If one tries to solve this problem with conditional probability it becomes very diﬃcult. We begin by considering the following cases if theﬁrst passenger sits in seat number 1, then all
the remaining passengers will be in their correct seats and certainly the #100’th will also. If he sits in the last seat #100, then certainly the last passenger cannot sit there (in fact he will end up in seat #1). If he sits in any of the 98 seats between seats #1 and #100, say seat k, then all the passengers with seatnumbers 2, 3, . . . , k − 1 will have empty seats and be able to sit in their respective seats. When the passenger with seat number k enters he will have as possible seating choices seat #1, one of the seats k + 1, k + 2, . . . , 99, or seat #100. Thus the options available to this passenger are the same options available to the ﬁrst passenger. That is if he sits in seat #1 the remaining passengerswith seat labels k +1, k +2, . . . , 100 can sit in their assigned seats and passenger #100 can sit in his seat, or he can sit in seat #100 in which case the passenger #100 is blocked, or ﬁnally he can sit in one of the seats between seat k and seat #99. The only diﬀerence is that this k-th passenger has fewer choices for the “middle” seats. This k passenger eﬀectively becomes a new “crazy”passenger. From this argument we begin to see a recursive structure. To fully specify this recursive structure lets generalize this problem a bit an assume that there are N total seats (rather than just 100). Thus at each stage of placing a k-th crazy passenger we can choose from • seat #1 and the last or N-th passenger will then be able to sit in their assigned seat, since all intermediate passenger’sseats are unoccupied. • seat # N and the last or N-th passenger will be unable to sit in their assigned seat. • any seat before the N-th and after the k-th. Where the k-th passenger’s seat is taken by a crazy passenger from the previous step. In this case there are N −1 −(k + 1) + 1 = N − k − 1 “middle” seat choices. If we let p(n, 1) be the probability that given one crazy passenger and n totalseats to select from that the last passenger sits in his seat. From the argument above we have a recursive structure give by 1 1 1 p(N, 1) = (1) + (0) + N N N = 1 1 + N N
N −1 k=2 N −1 k=2
p(N − k, 1)
p(N − k, 1) .
where the ﬁrst term is where the ﬁrst passenger picks the ﬁrst seat (where the N will sit correctly with probability one), the second term is when the ﬁrst passenger sits in theN-th seat (where the N will sit correctly with probability zero), and the remaining terms represent the ﬁrst passenger sitting at position k, which will then require repeating this problem with the k-th passenger choosing among N − k + 1 seats. To solve this recursion relation we consider some special cases and then apply the principle of mathematical induction to prove it. Lets take N = 2....