Question

玩家从 $1$ 到 $4$ 中选择一个数字。选好后，掷两个普通四面骰，每个骰子的面标有 $1$ 到 $4$ 。如果选中的数字恰好出现在一个骰子的底面，玩家赢得一美元；如果它出现在两个骰子的底面，玩家赢得两美元；如果它没有出现在任何骰子的底面，玩家损失一美元。每次掷骰玩家的期望收益是多少美元？

A player chooses one of the numbers $1$ through $4.$ After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered $1$ through $4.$ If the number chosen appears on the bottom of exactly one die after it is rolled, then the player wins $1. If the number chosen appears on the bottom of both of the dice, then the player wins $2. If the number chosen does not appear on the bottom of either of the dice, the player loses $1. What is the expected return to the player, in dollars, for one roll of the dice?

$-\dfrac{1}{8}$

$-\dfrac{1}{16}$

$0$

$\dfrac{1}{16}$

$\dfrac{1}{8}$

答案：B

知识点：期望值二项概率骰子（概率）

难度评级：1780

解答：

每个骰子底面出现所选数字的概率为 $\dfrac14$ 。所以该数字出现 $0,1$ 和 $2$ 次的概率分别为 $P(0)=\dfrac{9}{16},\$ $P(1)=\dfrac{6}{16},\$ $P(2)=\dfrac{1}{16}$ 。

期望收益为 $(-1)\cdot\dfrac{9}{16}$ $+(1)\cdot\dfrac{6}{16}$ $+(2)\cdot\dfrac{1}{16}$ $=\dfrac{-9+6+2}{16}$ $=-\dfrac{1}{16}$ 。

所以正确答案是 B。

Each die shows the chosen number on the bottom with probability $\dfrac14.$ So the number appears $0,1,$ or $2$ times with probabilities $P(0)=\dfrac{9}{16},\$ $P(1)=\dfrac{6}{16},\$ $P(2)=\dfrac{1}{16}.$

The expected return is $(-1)\cdot\dfrac{9}{16}$ $+(1)\cdot\dfrac{6}{16}$ $+(2)\cdot\dfrac{1}{16}$ $=\dfrac{-9+6+2}{16}$ $=-\dfrac{1}{16}.$

Thus, the correct answer is B.

2007 AMC 10B 第 22 题

答案：B

难度评级：1780

解答：

其他年份的第 22 题