PUMaC 2014 · 组合（B 组） · 第 1 题

PUMaC 2014 — Combinatorics (Division B) — Problem 1

[ 3 ] A girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will instantly fall in love and live happily ever after. But after 10 minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between 8 AM and 9 AM with equal probability. The guy will arrive between 7 AM and 8:30 p AM, also with equal probability. Let for p, q coprime be the probability that they fall in q love. Find p + q .

解析

[ 3 ] A girl and a guy are going to arrive at a train station. If they arrive within 10 minutes of each other, they will instantly fall in love and live happily ever after. But after 10 minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between 8 AM and 9 AM with equal probability. The guy will arrive between 7 AM and 8:30 p AM, also with equal probability. Let for p, q coprime be the probability that they fall in q love. Find p + q . Solution: If the guy arrive between 7 AM and 7 : 50 AM they will never meet. If the guy arrive at 7 : 5 x AM, the girl needs to arrive between 8 AM and 8 : 0 x AM. Hence x 1 probability of them meting is , and the average is . 60 12 If the guy arrive at 8 : 0 x AM, the girl can arrive between 8 AM and 8 : 1 x AM. Hence 10 + x 1 probability of them meting is , and the average is . 60 4 If the guy arrives past 8 : 10 AM, the girl can arrive in a 20 minutes range. Hence probability 1 of them metting is . 3 1 1 1 1 2 1 1 Thus the probabilty they will meet will be + + = , and p + q = 10 9 12 9 4 9 3 9