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面值 vs 固定奖金

Face Value

专题
Probability / 概率
难度
L4

题目详情

桌上有 10 张牌，面值分别为 1 到 10，背面朝上。你随机抽一张并看见面值。

你可以选择拿固定奖金 3.50，或拿这张牌的面值。

问：该游戏的公平价值（期望收益）是多少？

Ten cards with values 1−101-10 are face down in front of you. You select one card at random and look at it. You can either choose the payout of $3.50 or the face value of the card. What is the fair value of this game?

解析

当面值为 1、2、3 时选择 3.50 更好；当面值为 4..10 时选择拿面值。

因此期望为

310⋅3.5+710⋅4+5+6+7+8+9+107=5.95.\frac{3}{10}\cdot 3.5+\frac{7}{10}\cdot \frac{4+5+6+7+8+9+10}{7}=5.95.

Original Explanation

There is a 310\dfrac{3}{10} probability that the card is worth less than the payout, so you should choose the payout of 3.53.5. Otherwise, you should take the face value of the card. Therefore, the expected value is:

310⋅72+710⋅4+5+6+7+8+9+107=$5.95\dfrac{3}{10} \cdot \dfrac{7}{2} + \dfrac{7}{10} \cdot \dfrac{4 + 5 + 6 + 7 + 8 + 9 + 10}{7} = \$5.95