29th Putnam 1968 Problem B1 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 p 1 = prob( X = k), p 2 = prob(Y = k) and p 3 = prob( max(X, Y) = k).
Let q 1 = prob(X = k, Y = k), q 2 = prob(X = k, Y > k), q 3 = prob(X > k, Y = k), q 4 = prob(X = k, Y < k), q 5 = 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) = q 1 + q 2 + q 3 , p 1 = q 1 + q 2 + q 4 , p 2 = q 1 + q 3 + q 5 , p 3 = q 1 + q 4 + q 5 . Hence prob( min(X, Y) = k) = p 1 + p 2 - p 3 . 29th Putnam 1968 © John Scholes jscholes@kalva.demon.co.uk 14 Jan 2002