A gambler makes an “even money” bet on each toss of a coin. The amount he bets is half the amount of money he has in his possession. On a sequence of 10 bets, he wins 5 Times. Does the gambler have more, less or the same amount that he had at the beginning? Can you prove it?
Answer:
Each time the gambler wins, his money increases by a factor of 3/2 and each time he loses, his money is decreased by 1/2. Therefore, after 10 bets, if he wins w times, he is left with 3w/210 times the amount he started with no matter what the order in which his wins and losses occurred. Since he won 5 times, he ends up with 35/210 or 343/1024 times what he started with. Since 343/1024 is approximately 23.7%, the gambler has less than a quarter of what he started with.
The reason that the order of winning doesn’t change the result derives from the fact that the final amount, 3w/210 is the product of w factors of (3/2) and 10–w factors of (1/2) and the value of the product does not depend on the order of the factors.
