Two numbers are chosen independently and uniformly from [0,1]. What is the probability that their product is less than 1/2?
Correct: A
The probability is the area under xy < 1/2 in the unit square. For x ≤ 1/2 the entire vertical strip satisfies xy < 1/2 (area 1/2). For x > 1/2 the height is 1/(2x), so integrate 1/(2x) from 1/2 to 1 giving (ln 2)/2. Total area = 1/2 + (ln 2)/2 = (1 + ln 2)/2.