Awnik repeatedly plays a game that has a probability of winning of . The outcomes are independent. What is the expected value of the number of games he will play until he has both won and lost at least once?
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After the first game, there is a chance he has won a game and a chance he has lost a game. If he has won, then he plays until he loses a game, so in expectation, he will play more games. If he has lost, then he plays until he wins a game, so in expectation, he will play more games. Hence, the expected value of the number of additional games he will play is given by:
Hence, including the first gmae, Awnik will play games on average, which corresponds to .
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