# 【Puzzle】To Begin or Not to begin

|

【对算法，数学，计算机感兴趣的同学，欢迎关注我哈，阅读更多原创文章】

# 问题描述

An urn contains k black balls and a single red ball. Peter and Paula draw
without replacement balls from this urn, alternating after each draw until the
red ball is drawn. The game is won by the player who happens to draw the
single red ball. Peter is a gentleman and offers Paula the choice of whether
she wants to start or not. Paula has a hunch that she might be better off if
she starts; after all, she might succeed in the first draw. On the other hand,
if her first draw yields a black ball, then Peter’s chances to draw the red ball
in his first draw are increased, because then one black ball is already removed
from the urn. How should Paula decide in order to maximize her probability
of winning?

Paula 可以决定是否要先抽取。

Paula 觉得她先抽取会更好，毕竟有可能第一抽就中红球。但是她又担心如果第一抽拿到黑球，Peter 抽到红球的概率就提高了。

Paula 为了获胜概率最大，应该先抽还是后抽。

# 思路参考

k 等于 0, 1, 2, 3 的情况具体分析如下:

• k = 0，则 $p(0) = 1$
• k = 1，则 $p(1) = \frac{1}{2}$
• k = 2，则 $p(2) = \frac{1}{3} + \frac{2}{3}(1 - p(1)) = \frac{2}{3}$
• k = 3，则 $p(3) = \frac{1}{4} + \frac{3}{4}(1 - p(2)) = \frac{1}{2}$

Paula 应该先抽取

Share