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赶地铁

You have a train to catch!

专题
Probability / 概率
难度
L2

题目详情

蜘蛛侠每次任务后都会赶到地铁站：一条线去 Mary 家，另一条线去 Gwen 家。两条线的列车都每 10 分钟发车一次。

蜘蛛侠很“公平”，每次都上最先发车的那趟。

但他发现自己去 Mary 家的次数竟然是去 Gwen 家的 9 倍。为什么会这样？

提示：两条线的发车间隔不一定是均匀的 5 分钟对 5 分钟。

Spiderman has two close friends, Mary Jane & Gwen Stacy. After every mission, he rushes to the central subway. One line heads towards Mary's place, and another towards Stacy. Trains from each line leave every 10 minutes. Spiderman being impartial always boards the first train that leaves.

However, he observes that he ends up visiting Mary Jane nine times more often than Gwen Stacy. Can you decipher why?

Hint

The gap between the two trains may not be 5 minutes each.

解析

因为两条线虽然都“每 10 分钟一班”，但时刻表可能错位得很不均匀。

举例：

  • Mary 线在每个整 10 分钟发车：0,10,20,…0,10,20,\dots
  • Gwen 线在每个整 10 分钟后的第 1 分钟发车：1,11,21,…1,11,21,\dots

这样每个 10 分钟周期里：

  • 从 Mary 发车到 Gwen 发车的间隔只有 1 分钟（这 1 分钟到站的人会坐 Gwen）。
  • 从 Gwen 发车到下一班 Mary 发车的间隔有 9 分钟（这 9 分钟到站的人会坐 Mary）。

如果蜘蛛侠到站时间近似均匀随机，那么他坐 Mary 与 Gwen 的概率比就是 9:19:1。

Original Explanation

The gap from MM to GG is 1 minute, and the gap from GG to MM is 9 minutes

Solution

Since both trains leave at the same frequency of 10 minutes, probably the solution lies in the gap between the two consecutive trains of each line. Suppose the train MM (towards Mary) leaves at the minutes: 0,10,20,...0, 10, 20,...

And the train GG (towards Gwen) leaves at the minutes: x,(x+10),(x+20)...x, (x+ 10 ), (x+20)... and so on, where (0<x<10)(0 < x < 10)

Suppose Spiderman randomly reaches the station at some time t+10⋅nt + 10\cdot n minutes where nn is an arbitrary integer and tt is the fraction that determines his fate, 0<t≤100 < t \le 10.

Note that if 0<t≤x0 < t \le x means that he just missed MM and now has to wait for GG. While x<t≤10x <t \le 10 means the opposite.

Thus, the probability that he takes the train GG is:

P(G)=P(0<t≤x)P(G) = P(0 < t \le x)

0.1=x−010  ⟹  x=10.1 = \dfrac{x-0}{10} \implies x = 1

The train heading to Mary Jane's location leaves precisely 1 minute before the train heading to Gwen Stacy's location. Therefore, 9 out of 10 times, spiderman reaches the station when the next train is leaving towards Mary.