Solution
A Product Below One Half
Show the problem again
X and Y are independent and uniform on [0, 1]. What is the probability that XY < 1/2? Give an exact closed form, or a decimal rounded to 4 places.
Worked solution
The answer is (1 + ln 2)/2 ≈ 0.8466. If X ≤ 1/2 the product is automatically below 1/2; that contributes 1/2. If X = x > 1/2, we need Y < 1/(2x), which has probability 1/(2x). So P(XY < 1/2) = 1/2 + ∫_{1/2}^1 dx/(2x) = 1/2 + (1/2) ln 2. Geometrically, the complement is the region above the hyperbola xy = 1/2 in the unit square, whose area is 1/2 − (ln 2)/2 ≈ 0.153.
Source: Classic geometric-probability exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.