Question

在一个圆形区域内随机选取一点。该点到圆心的距离小于它到圆边界的距离的概率是多少？

A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

$\dfrac{2}{3}$

$\dfrac{3}{4}$

答案：A

知识点：几何概率面积比

难度评级：1260

解答：

设圆半径为 $1$ 。距圆心 $r$ 的点到圆心比到边界更近，当且仅当 $r \lt 1 - r$ ，即 $r \lt \tfrac12$ 。

有利区域是半径为 $\tfrac12$ 的圆，面积为 $\pi(\tfrac12)^2 = \tfrac{\pi}{4}$ ，整个圆面积为 $\pi$ 。概率为 $\dfrac{\pi/4}{\pi} = \dfrac14$ 。

所以正确答案是 A。

Take the radius to be $1$ . A point at distance $r$ from the center is closer to the center than to the boundary when $r \lt 1 - r$ , i.e. $r \lt \tfrac12$ .

The favorable region is a circle of radius $\tfrac12$ , with area $\pi(\tfrac12)^2 = \tfrac{\pi}{4}$ , out of the total area $\pi$ . The probability is $\dfrac{\pi/4}{\pi} = \dfrac14$ .

Thus, the correct answer is A .

1996 AMC 8 第 25 题

答案：A

难度评级：1260

解答：

其他年份的第 25 题