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串关投注：10:1 是否划算

Parlay card

专题
Strategy / 策略
难度
L4

题目详情

If you like gambling and you like betting on the outcome of sports matches, then you may like the "parlay card." A parlay card lets you bet on the outcomes of more than one match. In order to win a parlay bet, you must be correct on each of the matches you bet upon. Parlay cards offer big payoffs if you are right on every match (some even offer a payoff for "almost wins"). Suppose that your bookie will give you 10- to- 1 odds for a parlay bet that covers four sports matches (with no almost wins). Should you take the bet? 12

解析

若对 4 场比赛都无信息、且每场胜率为 1/21/2 且独立，则串关全中的概率为

(12)4=116.\left(\frac12\right)^4=\frac{1}{16}.

若下注 1 单位，给 10:1 的赔率通常理解为“赢时净赚 10、输时亏 1”。则期望净收益为

116⋅10−1516⋅1=10−1516=−516<0.\frac{1}{16}\cdot 10-\frac{15}{16}\cdot 1=\frac{10-15}{16}=-\frac{5}{16}<0.

公平赔率应约为 15:1 才能期望为 0。

因此在风险中性/无信息假设下，不该下注\boxed{不该下注}。