猜对方抛硬币

Guess the Toss

专题: Probability / 概率
难度: L4
来源: Brainstellar

题目详情

A 和 B 组队对抗 C。若 AB 赢则赢得 3 卢比，否则 0。

游戏：A 和 B 被分在两个无法通信的房间，各自抛一次硬币。A 需要猜 B 的结果，B 需要猜 A 的结果。若两人都猜对，则 AB 获得 3 卢比，否则 0。

如果开局需要先付 1 卢比入场费，他们是否应该参加？

A and B are in a team called AB, playing against C. If AB team wins they win Rs 3, nothing otherwise.

The Game: A and B are placed in 2 separate rooms far away. A will toss a coin and B will also toss a coin; A will have to guess the outcome of B's toss and B will guess A's. If both guesses are right, team AB wins Rs 3, nothing otherwise. Should they play the game, by paying Rs 1 at the start?

Hint

The winning probability is not 1/4. They can make a strategy before the game.

解析

应该参加。

他们可以事先约定策略：两人都把“自己抛到的结果”作为对对方的猜测。

这样当两人抛到相同结果（HH 或 TT）时，两人都猜对，概率为 $1/2$ 。

期望收益： $3\times\frac{1}{2}-1=\frac{1}{2}>0$ 。

Original Explanation

1/2

Solution

Let's list down all the possibilities.

A	B
H	H
H	T
T	H
T	T

Observe that $2$ out of $4$ times, their coins have the same face, i.e. either both heads or both tails.

They can speak their own coin's face as their guess. They win the game with a probability of $1/2$ .

The pay-off will be positive ( $1/2 \cdot 3 - 1 = 0.5$ ), and hence they should play this game.