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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?

  Submitted by Brian Smith    
Rating: 3.7143 (7 votes)
Solution: (Hide)
Prob. of finishing in 3: 16807/32768 = (7/8)^5

Expected number of rounds: 2470/651
To calculate this, first we need the probability of fininshing in n rounds.

First, we find the following, similar to the other part of the problem:
P(1) = (1/2)^5
P(2) + P(1) = (3/4)^5
P(3) + P(2) + P(1) = (7/8)^5
P(4) + P(3) + P(2) + P(1) = (15/16)^5
P(5) + P(4) + P(3) + P(2) + P(1) = (31/32)^5
P(n) + P(n-1) + . . . + P(2) + P(1) = (1 - 1/2^n)^5

From that:
P(1) = (1/2)^5
P(2) = (3/4)^5 - (1/2)^5
P(3) = (7/8)^5 - (3/4)^5
P(4) = (15/16)^5 - (7/8)^5
P(5) = (31/32)^5 - (15/16)^5
P(n) = (1 - 1/2^n)^5 - (1 - 1/2^(n-1))^5

The expected number of turns can be calculated from
E = P(1) + 2*P(2) + 3*P(3) + 4*P(4) + 5*P(5) + . . .
The expression for E approaches 2470/651 as a limit.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
The other way...Ady TZIDON2015-01-20 05:24:35
SolutionAnswersMath Man2015-01-19 08:47:53
not to be pickyhumfoozneydannle2005-01-02 02:17:26
re: No SubjectMichael2004-11-21 17:00:50
No SubjectMichael2004-11-21 16:43:29
Some Thoughtshow to get 2470/651owl2004-10-24 01:53:19
re(3): About the expected valueBon2004-10-23 05:09:29
re(2): New approachbernie2004-10-16 22:21:11
re(2): About the expected valueCharlie2004-10-15 15:17:35
re: About the expected valuenikki2004-10-15 13:14:39
Questionre: New approachCharlie2004-10-15 03:58:53
About the expected valueCharlie2004-10-15 03:56:13
To bernienikki2004-10-14 20:46:18
To Nikkibernie2004-10-14 20:33:27
SolutionMy approach - matches penny'snikki2004-10-14 20:22:04
New approachbernie2004-10-14 18:56:55
re: solution (raw form)Charlie2004-10-13 19:46:51
re: solution (raw form)Charlie2004-10-13 19:21:42
Some Thoughtssimulation for part 1Charlie2004-10-13 18:58:19
Solutionsolution (raw form)Charlie2004-10-13 18:37:50
SolutionWho gives a flip ? (Solution)Penny2004-10-13 17:46:50
Solution to second partJer2004-10-13 17:46:16
re: Solution to First Question (high chance of error)David Shin2004-10-13 17:30:08
SolutionSolution to First Question (high chance of error)David Shin2004-10-13 17:26:05
re: Flip vs. TossRandyOrton2004-10-13 17:12:57
QuestionFlip vs. Tossnikki2004-10-13 16:56:56
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