Question

从 $1$ 到 $2016$ 之间随机选取三个不同的整数。下列哪一句正确描述了这三个整数乘积为奇数的概率 $p$ ？

Three distinct integers are selected at random between $1$ and $2016,$ inclusive. Which of the following is a correct statement about the probability $p$ that the product of the three integers is odd?

$p \lt \dfrac{1}{8}$

$p = \dfrac{1}{8}$

$\dfrac{1}{8} \lt p \lt \dfrac{1}{3}$

$p = \dfrac{1}{3}$

$p \gt \dfrac{1}{3}$

答案：A

知识点：基本概率奇偶性无放回抽样

难度评级：1140

解答：

乘积为奇数，当且仅当选出的三个整数全是奇数。从 $1$ 到 $2016$ 中有 $1008$ 个奇数和 $1008$ 个偶数。

因为是不放回抽取，第一个因子为 $\frac12$ ，后两个因子都略小于 $\frac12$ ，所以 $p<\frac18$ 。 $p=\frac{1008}{2016}\cdot\frac{1007}{2015}\cdot\frac{1006}{2014}.$

所以正确答案是 A。

The product is odd exactly when all three selected integers are odd. There are $1008$ odd and $1008$ even integers from $1$ to $2016$ .

Because the integers are selected without replacement, $p=\frac{1008}{2016}\cdot\frac{1007}{2015}\cdot\frac{1006}{2014}.$ The first factor is $\frac12$ , and each of the next two factors is slightly less than $\frac12$ . Therefore $p<\frac18$ .

Thus, the correct answer is A.

2016 AMC 10A 第 12 题

答案：A

难度评级：1140

解答：

其他年份的第 12 题