# Independent Bernoulli trials and the gambler's ruin

Suppose the independent Bernoulli trials with success probability 1/2 for each trial are performed, and keep performing until there are more than twice as many failures as successes, and then stop.

When I want to find the probability of ever having more than twice as many failures as successes with independent Bern(1/2) trials, the textbook says it is related to a gambler's ruin problem:

Two gamblers, A and B, make a series of bets, where each has probability 1/2 of winning a bet, but A gets \$2 for each win and loses \$1 for each loss. Assume that the gamblers are allowed to borrow money, so they can and do gamble forever. Let $$p_k$$ be the probability that A, starting with \$k, will ever reach \$0. And we can solve the original Bernoulli trials by finding $$p_k$$.

I can solve this gamber's ruin problem and find the $$p_k$$ by recurrence, but I can't find how it is related to the the probability of ever having more than twice as many failures as successes with independent Bern(1/2) trials. I'm very confuesd about that.

Did I misunderstand some key points in the problem?

Let us start the gambler'r ruin problem with \$1. Reaching \$0 means having $$n$$ wins and $$2n + 1$$ losses - which is exactly getting more than twice as many failures as successes in your initial problem.
So probability of quitting in your original problem is $$p_1$$.