You play a coin flipping game with 5 coins. On round 1 you flip all of them. On round 2, you pick up all the ones that came up tails (leaving all the heads alone) and flip them again. You continue to do this until all the coins are heads. For example:

Round 1:  H T T H T
Round 2:  - H T - H
Round 3:  - - T - -
Round 4:  - - T - -
Round 5:  - - H - -

Done in 5 Rounds.

What is the expected number of rounds you'll need to finish the game?
What is the probability you will finish the game in 3 rounds or less?

(In reply to solution (raw form) by Charlie)

In doing the summation, at any given step, one item (the last) has not been subtracted out at all but added in the full n times but the summation of the previous (negative one) values has taken place, so at a given stage the total is

n(1-1/2^n)^5 - Sigma{i=1 to n-1} (1-1/2^i)^5

The limit of this as n grows larger is presumably the 3.794... found empirically by adding the first few terms.

Posted by Charlie on 2004-10-13 19:21:42