Dorsia 晚餐：相遇概率

Dinner at Dorsia

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

两位 quant 约在 Dorsia 吃晚饭。假设两人独立地在晚上 8:00 到 9:00 之间的某个均匀随机时刻到达，并且每人到达后只停留 10 分钟就离开。

问：他们能碰面并一起吃饭的概率是多少？

Two quants are planning for dinner at Dorsia. Assume that each independently arrives at some uniformly random time between 8:00pm and 9:00pm, for which they stay for exactly 10 minutes before leaving. What is the probability that they will meet each other and stay for dinner?

解析

设两人到达时刻（以 8:00 为 0）分别为 $X,Y\sim\mathrm{Unif}(0,60)$ 。

相遇当且仅当 $|X-Y|\le 10$ 。

在 $60\times 60$ 的正方形中，不相遇区域是两块边长 50 的直角三角形，总面积 $2\cdot\frac{50^2}{2}=2500$ 。

因此相遇概率为

1-\frac{2500}{3600}=\frac{1100}{3600}=\frac{11}{36}.