AMC10 2007 A

AMC10 2007 A · Q16

AMC10 2007 A · Q16. It mainly tests Probability (basic), Parity (odd/even).

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?

整数 $a, b, c,$ 和 $d$（不一定互异）独立且随机地从 $0$ 到 $2007$（包含端点）中选取。$ad - bc$ 为偶数的概率是多少？

(A) $\frac{3}{8}$ $\frac{3}{8}$

(B) $\frac{7}{16}$ $\frac{7}{16}$

(D) $\frac{9}{16}$ $\frac{9}{16}$

(E) $\frac{5}{8}$ $\frac{5}{8}$

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): The number $ad-bc$ is even if and only if $ad$ and $bc$ are both odd or are both even. Each of $ad$ and $bc$ is odd if both of its factors are odd, and even otherwise. Exactly half of the integers from 0 to 2007 are odd, so each of $ad$ and $bc$ is odd with probability $(1/2)\cdot(1/2)=1/4$ and are even with probability $3/4$. Hence the probability that $ad-bc$ is even is $$ \frac14\cdot\frac14+\frac34\cdot\frac34=\frac58. $$

答案（E）：数 $ad-bc$ 为偶数，当且仅当 $ad$ 和 $bc$ 同为奇数或同为偶数。$ad$ 与 $bc$ 中的每一个在其两个因子都为奇数时为奇数，否则为偶数。从 0 到 2007 的整数中恰好有一半是奇数，因此 $ad$ 和 $bc$ 各自为奇数的概率为 $(1/2)\cdot(1/2)=1/4$，为偶数的概率为 $3/4$。因此 $ad-bc$ 为偶数的概率为 $$ \frac14\cdot\frac14+\frac34\cdot\frac34=\frac58. $$

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