A pack of n cards, including three aces, is well shuffled. Cards are turned over in turn. Show that the expected number of cards that must be turned over to reach the second ace is (n+1)/2.
Solution
For each arrangement A of the cards, let A' be the reflection about the middle of the pack, so that if a card is in position m in A, then it is in position (n+1-m) in A'. Then all possible arrangements can be grouped into pairs (A, A') (note that A cannot equal A'). If the position of the second ace in A is m, then it is n+1-m in A', so the average over A and A' is (n+1)/2. Hence that is also the average over all the arrangements.
© John Scholes
jscholes@kalva.demon.co.uk
11 May 2002