Product of two uniforms above a half

Let $X, Y \sim \text{Uniform}(0,1)$ be independent. Compute $\mathbb{P}\!\left[XY > \tfrac12\right]$ , rounded to the nearest hundredth.

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For a fixed $x$ , the condition $xy>\tfrac12$ means $y>\tfrac{1}{2x}$ - feasible only when $x>\tfrac12$ .
Integrate the admissible $y$ -length $1-\tfrac{1}{2x}$ over $x\in(\tfrac12,1)$ ; you'll meet a $\ln$ term.

Answer

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0.15 (± 0.01)

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Asked at: IMC, SIG