Abraham and Blaise each have $\$10$. They repeatedly flip a fair coin. If it comes up heads, Abraham gives Blaise $\$1$. If it comes up tails, Blaise gives Abraham $\$1$. What is the expected number of flips until one of them runs out of money?
I have no idea how to start this, I'm stuck. Solutions are highly appreciated. Thanks in advance!
Set up a recursive relationship by conditioning on the first toss. The probability that one of them, say Abraham WLOG, loses is given by $$ P(A\;\text{loses game})\\ =P(A\;\text{loses game}|A\;\text{A lost first round})P(A\;\text{A loses first round})+P(A\;\text{loses game}|A\;\text{A won first round})P(A\;\text{A won first round}) $$ Using the above relationship, you should be able to come up with a probability of losing in terms of initial conditions for any endowment.