Team $A$ and team $B$, in a series of $7$ games, whoever wins $4$ games first wins. You want to bet $100$ that your team wins the series, in which case you receive $200$, or $0$ if they lose. However the broker only allows bets on individual games. You can bet $X$ on any individual game that day before it occurs to receive $2X$ if it wins and $0$ if it loses. How do you achieve the desired pay-out? In particular, what do you bet on the first match?
My initial thought was breaking this problem up in terms of combinatorics and probability by asking questions like: how many possible combinations are there for one of the two particular teams to win? What is the probability that $A$ wins?, what is the probability $A$ wins given $B$ wins the first game?, etc...
I was a bit stumped by this question so turning to the solution the author suggests that well this is just replicating a $4$-step symmetric binomial tree. Continuing on I could not really follow his solution, I was wondering if there were other ways of answering this problem. Any suggestions or guidance are greatly appreciated.