22nd Putnam 1961 Problem B2 Two points are selected independently and at random from a segment length β. What is the probability that they are at least a distance α (< β) apart?
Consider which points (x, y) of the square x = 0 to β, y = 0 to β have |x - y| ≥ α. Evidently the acceptable points lie in the two right-angled triangles: (0, α), (0, β), (β-α, β) and (α, 0), (β, 0), (β, β-α). These fit together to give a square side β-α, so the area of the acceptable points is (β-α) 2 out of a total area of β 2 . Thus the probability is (β-α) 2 /(β) 2 . 22nd Putnam 1961 © John Scholes jscholes@kalva.demon.co.uk 15 Feb 2002