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 β2. Thus the probability is (β-α)2/(β)2.
22nd Putnam 1961
© John Scholes
15 Feb 2002