2003 AMC 12B 第 25 题

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：

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