Question: Independent Bernoulli trials are performed, with success probability 1/2 for each trial. An important question that often comes up in such settings is how many trials to perform. Many controversies have arisen in statistics over the issue of how to analyze data coming from an experiment where the number of trials can depend on the data collected so far. For example, if we can follow the rule "keep performing trials until there are more than twice as many failures as successes, and then stop", then naively looking at the ratio of failures to successes (if and when the process stops) will give more than 2:1 rather than the true theoretical 1:1 ratio; this could be a very misleading result! However, it might never happen that there are more than twice as many failures as successes; in this problem, you will find the probability of that happening.

(a) 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 (a very favorable game for A!). Assume that the gamblers are allowed to borrow money, so they can and do gamble forever. Let pk be the probability that A, starting with $k, will ever reach $0, for each k 0. Explain how this story relates to the original problem, and how the original problem can be solved if we can find pk.

(b) Find pk.

(c) Find the probability of ever having more than twice as many failures as successes with independent Bern(1/2) trials, as originally desired.