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

小提示：

乘积为正当且仅当两个数同号。

The product is positive when both numbers have the same sign

大提示：

区间中负数部分长度为 $20$ ，正数部分长度为 $10$ 。

The interval has $20$ negative units and $10$ positive units

解答：

选中的数为负数的概率是 $\frac{20}{30}=\frac{2}{3}$ ，为正数的概率是 $\frac{10}{30}=\frac{1}{3}$ 。（恰好选中 $0$ 的概率为 $0$ 。）乘积为正当且仅当两个数同号，所以所求概率为 $\left(\dfrac23\right)^2+\left(\dfrac13\right)^2=\dfrac49+\dfrac19=\dfrac59\text{。}$

所以正确答案是 D。

A selected number is negative with probability $\frac{20}{30}=\frac{2}{3}$ and positive with probability $\frac{10}{30}=\frac{1}{3}.$ (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

小提示：

大提示：

解答：

其他年份的第 13 题