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戒掉强迫性赌徒

Curing the Compulsive Gambler

专题
Probability / 概率
难度
L3

题目详情

布朗先生总是不听好友 Kind Friend 的劝告，在轮盘赌中每次都押 1 美元买 13 号。为了帮布朗戒赌，Kind Friend 总是和他额外对赌 20 美元、赔率为 1:1，赌布朗在玩完 36 局后会处于亏损状态。这个“戒赌方案”效果如何？

（大多数美式轮盘有 38 个等可能结果。如果玩家押中的号码开出，则按 35 倍赔率赔付，并返还本金；否则输掉本金。）

Mr. Brown always bets a dollar on the number 13 at roulette against the advice of Kind Friend. To help cure Mr. Brown of playing roulette, Kind Friend always bets Brown $20 at even money that Brown will be behind at the end of 36 plays. How is the cure working?

(Most American roulette wheels have 38 equally likely numbers. If the player's number comes up, he is paid 35 times his stake and gets his original stake back; otherwise he loses his stake.)

解析

如果 布朗 在 36 局中恰好中一次，他对赌场正好不亏不赚。因此他在 36 局结束后仍然落后的唯一情况，是 36 局一局都没中，其概率为

(3738)36≈0.383\left(\frac{37}{38}\right)^{36} \approx 0.383

单局对赌场的期望收益为 35(138)−1(3738)=−238 美元35\left(\frac{1}{38}\right)-1\left(\frac{37}{38}\right) = -\frac{2}{38}\ \text{美元}

所以 36 局合计的期望是 −2⋅3638≈−1.89 美元-\frac{2\cdot 36}{38} \approx -1.89\ \text{美元}

而和 Kind Friend 的附加赌注中，布朗 以概率 0.6170.617 赢 20 美元，以概率 0.3830.383 输 20 美元，因此其期望为

20(0.617)−20(0.383)≈4.68 美元20(0.617)-20(0.383) \approx 4.68\ \text{美元}

综合起来，布朗 每 36 局的总期望收益变为

4.68−1.89=2.79 美元4.68 - 1.89 = 2.79\ \text{美元}

也就是说，这个“戒赌疗法”反而让他在轮盘赌上有了正期望；也许先被治好的会是 Kind Friend。当然，如果 布朗 36 局全输，他会一次性亏掉 56 美元，这倒可能给他一点震撼。

Original Explanation

If Mr. Brown wins once in 36 turns, he is even with the casino. His probability of losing all 36 time is (3738)36≈0.383(\frac{37}{38})^{36} \approx 0.383. In a single turn his expectation is:

35(138)−1(3738)=−238(dollars)35 (\frac{1}{38}) - 1(\frac{37}{38}) = -\frac{2}{38} (\text{dollars})

and in 36 turns

−2(36)38≈−1.89(dollars)- \frac{2(36)}{38} \approx -1.89 (\text{dollars})

Against Kind Friend, Mr.Brown has an expectation of

20(0.617)−20(0.383)≈4.68(dollars)20(0.617) - 20(0.383) \approx 4.68 (\text{dollars})

And so all told Mr.Brown gains 4.68−1.89=2.794.68 - 1.89 = 2.79 dollars per 36 trials; he is finally making money at roulette. Possibly Kind Friend will be cured first. Of course, when Brown loses all 36, he is out $56, which may jolt him a bit.