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俄罗斯轮盘 IV

Russian Roulette IV

专题
Probability / 概率
难度
L4

题目详情

6 膛左轮里装有两发相邻的子弹（连续两个膛位），开始时转动弹巢一次并固定。

朋友先手扣扳机且存活。轮到你时可选择是否重转弹巢再扣扳机。

问：不重转与重转相比，你生存概率相差多少？（不重转生存概率减去重转生存概率）

You're playing a game of Russian Roulette with a friend. The six-chambered revolver is loaded with two consecutively placed bullets. Initially, the cylinder is spun to randomize the order of the chambers. Your friend goes first, and lives after the first trigger pull. You are then given the choice to either spin the barrel or not before pulling the trigger. What is the difference in probability of you surviving between not spinning and spinning the barrel?

解析

若重转：任意一膛等可能，2 膛致死，所以生存概率为 4/6=2/34/6=2/3。

若不重转：设两发子弹位于相邻两膛。朋友已存活，说明他击发的膛不在那两膛上，因此他击发的膛只可能是剩下的 4 个空膛之一。

在这 4 个等可能空膛中，只有 1 个空膛的“下一膛”是子弹，因此你生存概率为 3/43/4。

差值：

34−23=112.\frac{3}{4}-\frac{2}{3}=\frac{1}{12}.

Original Explanation

If you spin the barrel, the probability that you survive is 46\frac{4}{6}. If you don't spin the barrel, the probability that you survive is 34\frac{3}{4}. To understand this case, let us define the chambers 1-6 and further define chambers 5 and 6 to hold the two consecutive bullets, without loss of generality. Because our friend survives, we know that he shot either chambers 1, 2, 3, or 4. Of these 4 equal possibilities, only one of these leads to our loss (if he shot the fourth chamber, then the following is a bullet); hence, the probability of survival in this case is 34\frac{3}{4} and the difference between these probabilities is:

34−23=112\frac{3}{4} - \frac{2}{3} = \frac{1}{12}