Question

如图所示的转盘旋转两次，指针指向的随机数字被记录。第一个数字除以 $4$ ，第二个数字除以 $5$ 。第一个余数指定棋盘的一列，第二个余数指定一行。这个数对指定到阴影方格的概率是多少？

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by $4,$ and the second number is divided by $5.$ The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square?

$\dfrac{1}{3}$

$\dfrac{4}{9}$

$\dfrac{1}{2}$

$\dfrac{5}{9}$

$\dfrac{2}{3}$

答案：C

知识点：基本概率奇偶性模运算

难度评级：1490

解答：

阴影方格对应两个余数同奇或同偶。第一个余数为偶数（来自数字 $2$ 和 $6$ ）的概率为 $\dfrac13$ ，为奇数的概率为 $\dfrac23$ 。

第二个余数为偶数的概率为 $\dfrac12$ ，为奇数的概率为 $\dfrac12$ 。

它们奇偶性相同的概率为 $\dfrac13\cdot\dfrac12+\dfrac23\cdot\dfrac12=\dfrac12$ 。

所以正确答案是 C。

The shaded squares are those where the two remainders are both odd or both even. The first remainder is even (from the numbers $2$ and $6$ ) with probability $\dfrac13$ and odd with probability $\dfrac23.$

The second remainder is even with probability $\dfrac12$ and odd with probability $\dfrac12.$

The probability that they share parity is $\dfrac13\cdot\dfrac12+\dfrac23\cdot\dfrac12=\dfrac12.$

Thus, the correct answer is C.

2007 AMC 10B 第 19 题

答案：C

难度评级：1490

解答：

其他年份的第 19 题