1996 AMC 12 第 22 题

Question

选取四个不同的点

A

、

B

、

C

和

D

，它们来自圆周上均匀分布的

1996

个点，每个四点组被选中的可能性相同。弦

\overline{AB}

与弦

\overline{CD}

相交的概率是多少？

Four distinct points,

A,

B,

C,

and

D,

are to be selected from

1996

points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord

\overline{AB}

intersects the chord

\overline{CD}?

\frac14

\frac13

\frac12

\frac23

\frac34

答案：B

知识点：几何概率对称性

难度评级：1520

小提示：

固定任意四个选出的点，考察把它们配成两条弦的三种方式

Fix any four selected points and examine the three ways to pair them into two chords

大提示：

恰有一种配对会连接圆周上交错排列的点

Exactly one pairing joins alternating points around the circle

解答：

对于任意固定的四个点，把它们的标号分成两组无序弦对共有三种方式。恰有一种配对连接圆周上交错排列的点，因此两弦相交。指定的弦对

\overline{AB},\overline{CD}

等可能地对应这三种配对中的任意一种，所以概率为

\frac{1}{3}

。正确答案是 B。

For any fixed four points, there are three ways to partition their labels into two unordered chord pairs. Exactly one pairing joins alternating points around the circle and therefore crosses. The named pair

\overline{AB},\overline{CD}

is equally likely to be any of these three pairings, so the probability is

\frac{1}{3}.

The correct answer is B.

1996 AMC 12 第 22 题

答案：B

难度评级：1520

小提示：

大提示：

解答：

其他年份的第 22 题