轮盘+对赌：最终期望收益

Curing the Gambler

专题: Strategy / 策略
难度: L4
来源: QuantQuestion

题目详情

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

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 $4.68-1.89\\approx \\boxed{+2.79}$ 。