[math-fun] Propp's birthday problem

Propp: If N denotes the number of times you toss a k=2-sided coin until one side has come up Q=3 times, then the expected value of N-choose-3 is (2/8)(1)+(6/16)(4)+(12/32)(10)=11/2, not k^(Q-1)=4, so Warren's formula is incorrect, and the reasoning that he applied to the case Q=2 is inadequate. --WDS: I agree something must be wrong... But what? Apparently my reasoning was valid (anyhow, yielded the right answer!) when Q=2, but not for Q>2. Perhaps my reasoning still works when Q>2 to yield a lower bound?