掷飞镖：期望半径与连续命中

Throwing a dart

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Suppose you are throwing a dart at a circular board. What is your expected distance from the center? Make any necessary assumptions. Suppose you win a dollar if you hit 10 times in a row inside a radius of $R / 2$ , where $R$ is the radius of the board. You have to pay 10c for every try. If you try 100 times, how much money would you have lost/made in expectation? Does your answer change if you are a professional and your probability of hitting inside $R / 2$ is double of hitting outside $R / 2$ ?

解析

假设落点在圆盘内面积均匀。半径 $R$ 的圆盘中，半径分布密度为 $f(r)=2r/R^2$ 。

期望距离：

\mathbb{E}[r]=\int_0^R r\cdot \frac{2r}{R^2}\,dr=\frac{2R}{3}.

命中 $R/2$ 的概率为 $(1/2)^2=1/4$ 。连续 10 次概率为 $(1/4)^{10}=2^{-20}$ 。

若尝试 100 次，期望奖金为 $100\cdot 2^{-20}\approx 9.54\times 10^{-5}$ 美元，成本 $100\times 0.1=10$ 美元，期望净亏约 $-9.9999$ 。

若高手使 $P(\text{命中 }R/2)=2/3$ （外圈 1/3），则 10 连中概率 $(2/3)^{10}\approx 0.0173$ ，100 次期望奖金约 $1.73$ 美元，净亏约 $-8.27$ ，仍不划算。