Mr. Smith goes to Atlantic City to gamble for a weekend. To guard against bad luck, he sets a policy at the start: In every game he plays, he’ll bet exactly half the money he has at the time, and he’ll make all his bets at even odds, so he’ll have an equal chance of winning and of losing this amount. In the end he wins the same number of games that he loses. Does he break even?
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No. Every win multiplies Smith’s holdings by 1.5, and every loss multiplies them by 0.5. One win and one loss (the order is immaterial) leaves him with 0.75 times his initial holdings, and n wins and n losses leaves him with 0.75n his initial assets. Ten such blows will leave him with only about 5 percent of his initial stake.
From David L. Book, Problems for Puzzlebusters, 1992.
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