Question

整数 $a, b, c$ 和 $d$ 不要求互不相同，分别独立地从 $0$ 到 $2007$ （含端点）中随机选择。 $ad - bc$ 为偶数的概率是多少？

Integers $a, b, c,$ and $d,$ not necessarily distinct, are chosen independently and at random from $0$ to $2007,$ inclusive. What is the probability that $ad - bc$ is even?

$\dfrac{3}{8}$

$\dfrac{7}{16}$

$\dfrac{1}{2}$

$\dfrac{9}{16}$

$\dfrac{5}{8}$

答案：E

知识点：奇偶性基本概率独立事件

难度评级：1540

解答：

从 $0$ 到 $2007$ 的整数中一半为奇数，所以 $ad$ 为奇数的概率为 $\tfrac12 \cdot \tfrac12 = \tfrac14$ ，为偶数的概率为 $\tfrac34$ 。 $bc$ 也是如此。

$ad - bc$ 为偶数当两个乘积奇偶性相同： $\tfrac14 \cdot \tfrac14 + \tfrac34 \cdot \tfrac34 = \tfrac{1}{16} + \tfrac{9}{16} = \tfrac58.$

所以正确答案是 E。

Half the integers from $0$ to $2007$ are odd, so each of $ad$ and $bc$ is odd with probability $\tfrac12 \cdot \tfrac12 = \tfrac14$ and even with probability $\tfrac34.$

The difference $ad - bc$ is even when both products have the same parity: $\tfrac14 \cdot \tfrac14 + \tfrac34 \cdot \tfrac34 = \tfrac{1}{16} + \tfrac{9}{16} = \tfrac58.$

Thus, the correct answer is E.

2007 AMC 10A 第 16 题

答案：E

难度评级：1540

解答：

其他年份的第 16 题