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 re: New approach by Charlie)

Charlie - I wish I had a magic way to get the fraction.  All I did was carry out the calculation in excel to full decimal precision.  It is accurate to 13 or 14 decimals.  Consider it a guess.

btw, you get 1/2^5 + 2x211/2^10 + 3x9031/2^15 + 4x221551/2^20 + 5x4329151/2^25 + ...

Posted by bernie on 2004-10-16 22:21:11