You play a game with a coin. You may place a bet; if Heads is flipped then you receive your bet back plus the same in winnings. If Tails is flipped then you lose your bet.
You have \$10 and you want to turn this into \$100 by continuously betting \$1 at a time, walking away when you either have a total of \$100 or are bankrupt. What is the probability you will leave being bankrupt?
My idea is to create a set of stochastic equations and solve for the probability that you go bankrupt. However, is there a more simple way to solve this quickly? This looks similar to the Gambler's ruin problem. (My intuition says 10%)
This is a one dimensional random walk, and we can move either -1 or +1 with the same probability in each step. We wish to compute the probability of reaching zero (-10 from the starting point) before reaching 100 (+90 from the starting point). This is known to be $$ \frac{10}{10+90} = \frac{1}{10}.$$