Ratio of two uniforms between one and two

Question

Let $X$ and $Y$ be independent and uniform on $[0,1]$ . Find the probability that $1<X/Y<2$ .

Short answer

$\tfrac14$ . The condition is $X/2<Y<X$ , a triangle in the unit square between the lines $y=x$ and $y=x/2$ . Its area is $\tfrac12-\tfrac14=\tfrac14$ , which is the probability.

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Cumulative Distribution Functions
Breadth-First and Depth-First Search

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