### 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?

**Solution**

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