随机比值

Random Ratio

专题: Probability / 概率
难度: L4
来源: Brainstellar

题目详情

在区间 $(0,1)$ 上独立均匀随机取两点 $p,q$ 。

求比值 $\frac{p}{q}$ 落在 $[1,2]$ 之间的概率。

p and q are two points chosen at random between 0 & 1. What is the probability that the ratio p/q lies between 1 & 2?

Hint

Graph Shading. (Whenever we see two uniform random variables, we graph them up!)

解析

答案是 $0.25$ 。

在单位正方形上令 $(q,p)$ 为均匀点。条件 $1\le p/q \le 2$ 等价于

q \le p \le 2q,

并且 $0<q<1,\ 0<p<1$ 。

画出直线 $p=q$ 与 $p=2q$ ，在 $[0,1]^2$ 内满足条件的区域面积为 $1/4$ ，因此概率为 $1/4$ 。

Original Explanation

0.25

Solution

Assume that the points are $x$ & $y$ respectively. We want to know if $x/y$ is between 1 and 2. If we plot this, the limits are $x/y=1$ and $x/y = 2$ . Thus, the desired region is the area between lines $y=x$ & $y=x/2$ .

plot

This region is $1/4$ th of the rest.