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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?

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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 $p_1$.

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