20, 27, 29, 39, 48, 52, 60, 61, 64, 73

20.

Alice flips a fair coin \(n\) times and Bob flips another fair coin \(n+1\) times, resulting in independent \(X\sim Bin(n,.5)\) and \(Y\sim Bin(n+1, .5)\).

  1. Let \(V = min(X,Y)\) be the smaller of \(X\) and \(Y\) and let \(W = max(X,Y)\) be the larger of \(X\) and \(Y\). Find \(E(V) + E(W)\).

  2. Show that \(P(X<Y) = P(n- X < n + 1 - Y)\)

Solution

  1. \[E(V) + E(W) = E(V + W) = E(X + Y) = E(X) + E(Y) = n(1/2) + (n+1)(1/2) = n + (1/2)\]