Ratio of two uniforms landing between 1 and 3

Question

Let $a$ and $b$ be independent and uniformly distributed on $[0,1]$ . Find the probability that $1\le a/b\le 3$ .

Short answer

$1/3$ . The condition is $a/3\le b\le a$ , a triangle in the unit square. For fixed $a$ it has length $\tfrac{2a}{3}$ , and $\int_0^1\tfrac{2a}{3}\,da=\tfrac13$ .

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Cumulative Distribution Functions
Sample Spaces, Events, and the Probability Axioms

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