轮盘+对赌：最终期望收益

36 次押单号 13：一次期望为\n\n$$\n35\\cdot\\frac{1}{38}-1\\cdot\\frac{37}{38}=-\\frac{2}{38}.\n$$\n\n36 次期望为 $36\\cdot(-2/38)=-72/38\\approx -1.89$。\n\n若 36 次内从未命中 13，则总收益为 -36，概率为 $(37/38)^{36}\\approx 0.383$。朋友押“最终为负”，其对赌期望为\n\n$$\n20\\cdot(1-0.383)-20\\cdot 0.383\\approx +4.68.\n$$\n\n合并期望约为 $4.68-1.89\\approx \\boxed{+2.79}$。

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英文解析

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

is

$$35\left(\frac{1}{38}\right) - 1\left(\frac{37}{38}\right) = -\frac{2}{38} (\mathrm{dollars}),$$

and in 36 turns

$$-\frac{2(36)}{38}\approx -1.89 (\mathrm{dollars})$$

Against Kind Friend, Mr. Brown has an expectation of

$$+20(0.617) - 20(0.383)\approx +4.68 (\mathrm{dollars}).$$

And so all told Mr. Brown gains $+4.68 - 1.89 = +2.79$ 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.