咖啡店的两位女巫

Witches at the coffee shop

专题: Probability / 概率
难度: L2
来源: Brainstellar

题目详情

两位女巫每晚都会去一家 24 小时咖啡店。两人都在 0:00 到 1:00 之间独立均匀随机到达，并且各停留 30 分钟。

问：某晚两人相遇的概率是多少？

Two witches make a nightly visit to an all-night coffee shop. Each arrives at a random time between 0:00 and 1:00. Each one of them stays for exactly 30 minutes. On any one given night, what is the probability that the witches will meet at the coffee shop?

Hint

Graph Shading

解析

设两人到达时间（分钟）为 $X,Y\in[0,60]$ ，均匀独立。

两人相遇当且仅当

|X-Y|\le 30.

在 $60\times60$ 的正方形样本空间中，不相遇区域是两块直角三角形（边长 30），总面积

2\times \frac{1}{2}\times 30\times 30=900.

总面积是 $3600$ ，因此

P(\text{相遇})=1-\frac{900}{3600}=\frac{3}{4}.

Original Explanation

3/4

Solution

Plot the graph of X and Y, where X and Y denote the time each witch arrives. This will form a 60x60 square for total feasible region. The probability that they meet is |X-Y|<= 30. Draw this as favorable region, by joining line (30,0) with (60,30) and (0,30) with (30,60). Clearly the interior of this region has area 3/4 th of total. Hence the probability = 3/4 = 0.75