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俄罗斯轮盘赌系列题

Russian Roulette Series

专题
Strategy / 策略
难度
L4

题目详情

  1. 传统版本（不重新旋转）：6 发左轮随机装 1 发子弹，两人轮流扣扳机直到中弹。若可选先手/后手，是否有差别？输的概率是多少？

  2. 每次扣扳机前都重新旋转弹巢：若你先手，你输的概率是多少？

  3. 两发子弹随机放置：对手先手且幸存，你可以选择旋转或不旋转，应该怎么选？

  4. 两发子弹相邻放置：对手刚幸存第一枪，你应该旋转吗？

  1. Traditional version (no re-spin)
    • A 6-chamber revolver has 1 bullet placed randomly. Two players alternate pulling the trigger on themselves until the bullet fires. If you could choose to be the first or second player, does it matter and what is your probability of losing?
  2. Spin every time
    • Now the barrel is spun afresh before each pull. If you go first, what is your probability of losing?
  3. Two bullets, random placement
    • Your opponent goes first and survives. You may choose to spin or not. Should you spin?
  4. Two bullets, consecutive placement
    • Now the two bullets are in consecutive chambers. Your opponent just survived the first pull. Should you spin?
解析

  1. 不旋转：子弹位置固定。先手会在弹巢位置为 1/3/5 时中弹，概率 1/21/2；后手同理。先后手无差别，输的概率均为 1/21/2。

  2. 每次旋转：设先手最终输的概率为 pp。

p=16+56(1−p)⇒p=611.p=\frac16+\frac56(1-p)\Rightarrow p=\frac{6}{11}.

因此先手输 6/116/11，后手输 5/115/11，应选后手更优。

  1. 两发随机：若旋转，下一枪中弹概率 2/6=1/32/6=1/3；若不旋转，在已知对手第一枪未中弹条件下，相当于 5 个未知位置里 2 发子弹，概率 2/5=0.42/5=0.4。因此应旋转（1/3<0.41/3<0.4）。

  2. 两发相邻：若不旋转，已知对手未中弹后，你立即中弹的概率为 1/41/4；若旋转，中弹概率 2/6=1/32/6=1/3。因此不应旋转（1/4<1/31/4<1/3）。

Original Explanation

  1. Once the bullet’s position is fixed, if you go first, you lose if it’s in chambers 1, 3, or 5. That happens with probability 1/21/2. By symmetry, both have the same 1/21/2 chance of losing. So it does not matter.
  2. Let pp be the probability the first player eventually loses. Immediately there is a 1/61/6 chance. If not (5/65/6 chance), roles switch, so the first player’s chance is 1−p1 - p. Thus p=16+56(1−p)⟹p=611.p = \frac{1}{6} + \frac{5}{6}(1-p) \quad \Longrightarrow \quad p = \frac{6}{11}. So the first player has a 611\frac{6}{11} chance of losing, and the second player has 511\frac{5}{11}. You should choose to be the second player.
    • If you spin: probability of losing is 26=13.\frac{2}{6}=\frac{1}{3}.
    • If you do not spin: given your opponent survived the first pull, effectively 2 bullets remain among 5 unknown positions, so 25=0.4.\frac{2}{5}=0.4.
      Thus spinning is better (13<0.4\frac{1}{3} < 0.4).
    • If you do not spin: the chance of facing the next bullet right away is 14.\frac{1}{4}.
    • If you spin: the two consecutive bullets are among the 6 chambers, so probability is 26=13.\frac{2}{6}=\frac{1}{3}.
      Hence you should not spin because 14<13.\frac{1}{4} < \frac{1}{3}.