A gambler believes that a strategy called "doubling up" is an effective way to gamble. The method requires the gambler to double the stake after each loss. Thus, if the initial bet is $1, after losing he will double the bet until he wins. After a win he resorts back to a $1 bet. The result is that he will net $1 for every win. The problem, however, is that he will eventually run out of money or bump up against the table limit. Suppose that for a certain game the probability of winning is .5 and that losing 6 bets in a row will result in bankrupting the gambler.
Find the probability of losing 6 times in a row.
Solution is provided.