Question

从区间 $[-20, 10]$ 中独立随机选择两个实数。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：几何概率分类讨论

难度评级：1310

解答：

取到 $0$ 的概率为 $0$ ，所以只需考虑乘积是正还是负；乘积不是小于 $20/30=2/3$ 且等于 $10/30=1/3$ 的概率也是。 $\left(\dfrac23\right)^2+\left(\dfrac13\right)^2=\dfrac49+\dfrac19=\dfrac59.$

乘积为负当且仅当一个数小于、另一个数大于。有种顺序，概率为和。由于乘积等于或边界为的情况概率为零，乘积为负的概率为所以乘积为正的概率为。所以正确答案是 D。

A selected number is negative with probability $20/30=2/3$ and positive with probability $10/30=1/3.$ (Selecting exactly $0$ has probability $0.$ ) The product is positive exactly when both numbers have the same sign, so the probability is $\left(\dfrac23\right)^2+\left(\dfrac13\right)^2=\dfrac49+\dfrac19=\dfrac59.$

Thus, the correct answer is D .

2011 AMC 10B 第 13 题

答案：D

难度评级：1310

解答：

其他年份的第 13 题