2011 AMC 12B 第 9 题

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}

乘积为正当且仅当两个数同号。

The product is positive exactly when both numbers share the same sign

每个数为负的概率是

\dfrac{20}{30}

，为正的概率是

\dfrac{10}{30}

Each number is negative with probability

\dfrac{20}{30}

and positive with probability

\dfrac{10}{30}

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\text{。}

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.

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