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: solution (raw form) by Charlie)

How about a transformation matrix?  Call it T.

1/32 5/32 10/32 10/32  5/32  1/32

0      1/16   4/16   6/16  4/16  1/16

0         0      1/8    3/8    3/8    1/8

0         0        0     1/4    2/4    1/4

0         0        0       0      1/2   1/2

0         0        0       0        0      1

 

Represents the probability of going from row to column.

Now use a vector whose elements are the probabilies of being in that same column.  Call this A.

Multiply A times T successively to get AT, AT^2, AT^3, AT^4, etc...  The far right element of each of these vectors is the cumulative probability of ending the game on the exponent turn.  The third one shows about .5129... or 16807/2^15.  Difference these values and multiply by the exponents will give you 3.79416... or 2470/651 for an expected value.

Edited on October 14, 2004, 7:02 pm

Edited on October 14, 2004, 7:02 pm

Posted by bernie on 2004-10-14 18:56:55