29th Putnam 1968

The random variables X, Y can each take a finite number of integer values. They are not necessarily independent. Express prob( min(X, Y) = k) in terms of p₁ = prob( X = k), p₂ = prob(Y = k) and p₃ = prob( max(X, Y) = k).

Solution

Let q₁ = prob(X = k, Y = k), q₂ = prob(X = k, Y > k), q₃ = prob(X > k, Y = k), q₄ = prob(X = k, Y < k), q₅ = prob(X < k, Y = k). These are all probabilities for disjoint events, so we can add them freely to get: prob( min(X, Y) = k) = q₁ + q₂ + q₃, p₁ = q₁ + q₂ + q₄, p₂ = q₁ + q₃ + q₅, p₃ = q₁ + q₄ + q₅. Hence prob( min(X, Y) = k) = p₁ + p₂ - p₃.

29th Putnam 1968

© John Scholes
jscholes@kalva.demon.co.uk
14 Jan 2002