Question

从区间 $[-20, 10]$ 中独立随机选取两个实数。它们的乘积大于零的概率是多少？

Two real numbers are selected independently at random from the interval $[-20, 10].$ What is the probability that the product of those numbers is greater than zero?

$\dfrac{1}{9}$

$\dfrac{1}{3}$

$\dfrac{4}{9}$

$\dfrac{5}{9}$

$\dfrac{2}{3}$

答案：D

知识点：几何概率独立事件

难度评级：1390

解答：

区间长度为 $30$ ，其中负数部分长度为 $20$ ，正数部分长度为 $10$ 。因此每个数为正的概率为 $\dfrac13$ ，为负的概率为 $\dfrac23$ 。

乘积为正发生在两数都正或都负： $\left(\dfrac13\right)^2+\left(\dfrac23\right)^2=\dfrac19+\dfrac49=\dfrac59.$

所以正确答案是 D。

The interval has length $30,$ with $20$ of it negative and $10$ of it positive. So each number is positive with probability $\dfrac13$ and negative with probability $\dfrac23.$

The product is positive when both are positive or both are negative: $\left(\dfrac13\right)^2+\left(\dfrac23\right)^2=\dfrac19+\dfrac49=\dfrac59.$

Thus, the correct answer is D.

2011 AMC 12B 第 9 题

答案：D

难度评级：1390

解答：

其他年份的第 9 题