AMC12 2011 B

AMC12 2011 B · Q9

AMC12 2011 B · Q9. It mainly tests Probability (basic), Geometric probability (basic).

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

从区间 $[-20,10]$ 中独立随机选取两个实数。这两个数的乘积大于零的概率是多少？

(A) \frac{1}{9} \frac{1}{9}

(B) \frac{1}{3} \frac{1}{3}

(D) \frac{5}{9} \frac{5}{9}

(E) \frac{2}{3} \frac{2}{3}

Answer

Correct choice: (D)

正确答案：（D）

Solution

For the product to be greater than zero, we must have either both numbers negative or both positive. Both numbers are negative with a $\frac{2}{3}*\frac{2}{3}=\frac{4}{9}$ chance. Both numbers are positive with a $\frac{1}{3}*\frac{1}{3}=\frac{1}{9}$ chance. Therefore, the total probability is $\frac{4}{9}+\frac{1}{9}=\frac{5}{9}$ and we are done. $\boxed{D}$

要使乘积大于零，必须两数同为负数或同为正数。两数都为负数的概率为 $\frac{2}{3}*\frac{2}{3}=\frac{4}{9}$。两数都为正数的概率为 $\frac{1}{3}*\frac{1}{3}=\frac{1}{9}$。因此总概率为 $\frac{4}{9}+\frac{1}{9}=\frac{5}{9}$。$\boxed{D}$

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