Question

在一个圆上随机且独立地选取三个点。三点两两之间的距离都小于该圆半径的概率是多少？

Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle?

$\dfrac{1}{36}$

$\dfrac{1}{24}$

$\dfrac{1}{18}$

$\dfrac{1}{12}$

$\dfrac{1}{9}$

答案：D

知识点：几何概率弧弦

难度评级：2270

小提示：

一条弦短于半径，当且仅当它对应的弧小于 $60^\circ$

A chord is shorter than the radius exactly when its arc measures less than $60^\circ$

大提示：

三条两两弦都短，恰好等价于三个点都落在某个 $60^\circ$ 弧内

All three pairwise chords are short precisely when the three points lie within some $60^\circ$ arc

解答：

一条弦的长度小于半径，当且仅当它所对的弧小于 $60^\circ$ ，因为 $60^\circ$ 弧对应的弦长正好等于半径。

三条两两弦都短于半径，恰好等价于三个点都位于某个 $60^\circ$ 弧内。

在任何成功的情形中，恰好有一个点是这样一段包含弧的逆时针端点（概率为零的边界情形除外）。这个端点有 $3$ 种选法；另外两个点各自独立地以概率 $\dfrac{1}{6}$ 落在其后的 $60^\circ$ 弧内。因此所求概率为 $3\left(\frac{1}{6}\right)^2 = \frac{1}{12}\text{。}$

因此，正确答案是 D。

A chord has length less than the radius exactly when the arc it subtends is less than $60^\circ,$ since a chord of a $60^\circ$ arc equals the radius.

All three pairwise chords are shorter than the radius precisely when the three points all lie within some arc of $60^\circ.$

For any successful configuration, exactly one of the three points is the counterclockwise endpoint of such a containing arc (apart from probability-zero boundary cases). Choose that endpoint in $3$ ways; each of the other two points independently has probability $\dfrac{1}{6}$ of lying in the next $60^\circ.$ Hence the probability is $3\left(\frac{1}{6}\right)^2 = \frac{1}{12}.$

Thus, the correct answer is D.

2003 AMC 12B 第 25 题

答案：D

难度评级：2270

小提示：

大提示：

解答：

其他年份的第 25 题