2011 AMC 10B 第 13 题

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}

乘积为正当且仅当两个数同号。

The product is positive when both numbers have the same sign

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

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 .

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