You are tossing a coin. If it is tails, your opponent gives you a dollar. If it is heads ($H$), you give your opponent a dollar. $P(H)=0.6$. Assume you both have unlimited money.
a) What is the probability $p(k,n)$ that you have exactly $k$ dollars more than your opponent after $n$ coin tosses? Express the answer as an expression.
b) You quit the game if you are ever losing by \$6. What is the probability you quit before turn $100$?
I have worked out some simple simulations and know that being down by \$6 is the same as having lost 3 more times than I've won. However, I can't figure out how to write a formula without double counting some probabilities. I've been using the negative binomial distribution, but do not know where to go next.
Think about this as a random walk with the probability of an up being 0.6, and with your having a lower bound. Then you can just apply the results from random walks (for example, see wikipedia).