转动弹巢

Rolling the bullet

专题: Probability / 概率
难度: L2
来源: Brainstellar

题目详情

两发子弹被连续装入一个 6 发容量的左轮弹巢。开枪一次但没有击发。

在下一次开枪前把弹巢转一下（相当于重新随机位置），会不会提高击发子弹的概率？

Two bullets are loaded into a gun's round barrel consecutively. The barrel has a capacity of 6. The gun is fired once, but no bullet is shot. Does rolling the barrel (shuffling) before next shot increase the probability of firing a bullet?

barrel

Hint

Since the bullets are loaded consecutively, the next shot is also constrained.

解析

会提高。

不转弹巢：第一次未击发意味着击针落在 4 个空仓之一，而两发子弹仍然相邻。此时下一枪击发的概率为 $\frac{1}{4}=25\%$ 。
转动弹巢：等价于把“2 发子弹在 6 个仓位中随机分布”的状态重置，下一枪击发概率为 $\frac{2}{6}=\frac{1}{3}\approx 33.33\%$ 。

因此转动弹巢会把概率从 $25\%$ 提高到约 $33.33\%$ 。

Original Explanation

Yes, shuffling increases the probability of firing a bullet from $25\%$ to $33.\bar{3}\%$ )

Solution

Initial Misstep: If the two bullets are randomly put instead of consecutively, then, after firing one empty shot, there are $2$ bullets and $5$ total slots. The probability would be $2/5 = 40\%$ , but that's not the case here.

Correct step: The probability of firing a bullet without a shuffle is $1/4 = 25\%$ . To understand this, imagine that the firing pin was on one of the empty slots $(3, 4, 5, 6)$ , and the first shot was taken, but no bullet was fired. Now assumming that the barrel rotates clockwise, the pin will move to one of these slots: $(2, 3, 4, 5)$ . Out of these four slots, only the slot $(1)$ has a bullet. Hence probability of firing a bullet is $1/4 = 25\%$ . Note that the same is true in anti-clockwise direction.

barrel

After the shuffle, the state is reset. There are $6$ total slots with $2$ bullets, the probabilty of firing a bullet after a shuffle is $2/6 = 1/3 \approx 33\%$ .

Thus, shuffling does increase the probability of firing a bullet (from $25\%$ to $33\%$ )