## Problem of the Day #52: A Coin-Flipping Game
*May 10, 2011*

*Posted by Billy in : potd , trackback*

Albert is playing a game with himself. Albert has one weighted coin, with the probability of landing heads $p=\frac{7}{8}$ and the probability of landing tails $\frac{1}{8}$. A round in the game consists of Albert flipping the coin some number of times. Albert’s goal is to get as many heads as possible in a given round. For each heads that Albert flips, he gives himself one point. A round ends when either Albert voluntarily ends the round or Albert flips a tails. If Albert chooses to end the round, he retains however many points he got during that round. However, if Albert flips a tails, the round ends and Albert loses any points he gained during that round. At the end of a round, the coin magically morphs itself such that the new probability of getting heads is $p_{new} = \frac{2 p_{old}+1}{4}$ and the new probability of getting tails is $1-p_{new}$. If Albert plays 100 rounds and ends each round at the optimal time (if he can), what is the expected value of the total number of points Albert will have gained in the 100 rounds?

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