PUMaC 2013 · 团队赛 · 第 10 题

PUMaC 2013 — Team Round — Problem 10

On a plane, there are 7 seats. Each is assigned to a passenger. The passengers walk on the plane one at a time. The first passenger sits in the wrong seat (someone else’s). For all the following people, they either sit in their assigned seat, or if it is full, randomly pick another. You are the last person to board the plane. What is the probability that you sit in your own seat?’

解析

On a plane, there are 7 seats. Each is assigned to a passenger. The passengers walk on the plane one at a time. The first passenger sits in the wrong seat (someone else’s). For all the following people, they either sit in their assigned seat, or if it is full, randomly pick another. You are the last person to board the plane. What is the probability that you sit in your own seat?’ SOLUTION: Label the correct seats for passengers 1 , 2 , · · · , 7 as S , S , · · · , S . 1 2 7 If 1 takes S , then P = 0. 7 1 If 1 takes S , then 2 , 3 , 4 , 5 take the correct seats, so that P = . 6 2 If 1 takes S , then 2 , 3 , 4 take the correct seats. If 5 takes S , then everything is fine. If 5 takes 5 1 1 1 1 1 1 S , then there is still a chance of for 7 to take S . This yields P = + · = . 6 7 2 3 3 2 2 3 1 Similarly, one can show that P = for any choice of 1 (except taking S ) by induction. Then 7 2 5 1 5 the answer is P = · = . 6 2 12 5 ANSWER: 12