HMMT 二月 2006 · GEN1 赛 · 第 7 题
HMMT February 2006 — GEN1 Round — Problem 7
题目详情
- The train schedule in Hummut is hopelessly unreliable. Train A will enter Intersection X from the west at a random time between 9:00 am and 2:30 pm; each moment in that interval is equally likely. Train B will enter the same intersection from the north at a random time between 9:30 am and 12:30 pm, independent of Train A; again, each moment in the interval is equally likely. If each train takes 45 minutes to clear the intersection, what is the probability of a collision today?
解析
- The train schedule in Hummut is hopelessly unreliable. Train A will enter Intersection X from the west at a random time between 9:00 am and 2:30 pm; each moment in that interval is equally likely. Train B will enter the same intersection from the north at a random time between 9:30 am and 12:30 pm, independent of Train A; again, each moment in the interval is equally likely. If each train takes 45 minutes to clear the intersection, what is the probability of a collision today? 13 Answer: 48 Solution: Suppose we fix the time at which Train B arrives at Intersection X; then call the interval during which Train A could arrive (given its schedule) and collide with Train B the “disaster window.” We consider two cases: 2 (i) Train B enters Intersection X between 9:30 and 9:45. If Train B arrives at 9:30, 1 the disaster window is from 9:00 to 10:15, an interval of 1 hours. If Train B 4 1 arrives at 9:45, the disaster window is 1 hours long. Thus, the disaster window 2 1 1 11 1 has an average length of (1 + 1 ) ÷ 2 = . From 9:00 to 2:30 is 5 hours. The 4 2 8 2 11 1 1 probability of a collision is thus ÷ 5 = . 8 2 4 (ii) Train B enters Intersection X between 9:45 and 12:30. Here the disaster window 1 1 1 3 is always 1 hours long, so the probability of a collision is 1 ÷ 5 = . 2 2 2 11 1 1 From 9:30 to 12:30 is 3 hours. Now case (i) occurs with probability ÷ 3 = , and 4 12 11 case (ii) occurs with probability . The overall probability of a collision is therefore 12 1 1 11 3 1 1 13 · + · = + = . 12 4 12 11 48 4 48