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 to First Question (high chance of error) by David Shin)

Doh. Not thinking about it properly. f(n) is the expected number of rounds to finish when starting with n **heads**, not coins. Assuming I computed the values of f correctly, the solution becomes

p(0,5)*f(0)+p(1,5)*f(1)+...+p(5,5)*f(5).

I don't want to do this out...I wouldn't be surprised if it came out to something nice.