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Flipping Coins (Posted on 2004-10-13) Difficulty: 4 of 5
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?

See The Solution Submitted by Brian Smith    
Rating: 3.7143 (7 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution to First Question (high chance of error) | Comment 3 of 26 |

Please see my next comment - there is an error in this reasoning.

Let f(n) be the expected number of rounds to finish the game when starting with n coins.  We seek f(5).  Note that

f(n)=1 + (p(0,n)*f(n)+p(1,n)*f(n-1)+...+p(n-1,n)*f(1)+p(n,n)*f(0)),

where p(i,j) is the probability of getting i heads out of j tosses.  Using p(i,j) = (j choose i)/(2^j), and f(0)=0, we can recursively compute f(1),f(2),...

I probably made a mistake (doing it by hand), but here's what I got:

f(1)=2
f(2)=8/3
f(3)=22/7
f(4)=368/105
f(5)=2270/651

Edited on October 13, 2004, 5:30 pm
  Posted by David Shin on 2004-10-13 17:26:05

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