Question

一只青蛙跳 $3$ 次，每次恰好跳 $1$ 米。每次跳跃的方向都是独立随机选择的。青蛙最终位置离起点不超过 $1$ 米的概率是多少？

A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently and at random. What is the probability that the frog's final position is no more than $1$ meter from its starting position?

$\dfrac{1}{6}$

$\dfrac{1}{5}$

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

答案：C

知识点：几何概率随机游走

难度评级：2030

解答：

这是一个连续的几何概率问题。把第二跳固定为从 $P=(0,0)$ 到 $Q=(1,0)$ ，令 $\alpha,\beta$ 为第一跳和第三跳的方向，则起点为 $A=(\cos\alpha,\sin\alpha)$ ，终点为 $B=(1+\cos\beta,\sin\beta)$ 。

取 $0\le\alpha\le\pi$ 和 $0\le\beta\le2\pi$ ，条件 $AB\le1$ 恰好等价于 $\alpha\le\beta\le\pi$ 。

在面积为 $2\pi^2$ 的 $\alpha\beta$ -矩形中，有利区域是面积 $\tfrac{\pi^2}{2}$ 的三角形，所以概率为 $\dfrac{\pi^2/2}{2\pi^2}=\dfrac14$ 。

因此，正确答案是 C。

This is a continuous (geometric) probability. Anchor the second jump from $P=(0,0)$ to $Q=(1,0),$ and let $\alpha,\beta$ be the directions of the first and third jumps, so the start is $A=(\cos\alpha,\sin\alpha)$ and the end is $B=(1+\cos\beta,\sin\beta).$

Taking $0\le\alpha\le\pi$ and $0\le\beta\le2\pi,$ the requirement $AB\le1$ holds exactly when $\alpha\le\beta\le\pi.$

In the $\alpha\beta$ -rectangle of area $2\pi^2,$ the favorable region is a triangle of area $\tfrac{\pi^2}{2},$ so the probability is $\dfrac{\pi^2/2}{2\pi^2}=\dfrac14.$

Thus, the correct answer is C.

2010 AMC 12B 第 18 题

答案：C

难度评级：2030

解答：

其他年份的第 18 题