---

Players A and B play a fair game for $20: the winner takes $20 from the loser.

They play until one of them has lost all of their money.

Player A starts with $100 and Player B starts with $60.

---

1.

What is the probability that A wins all of B's money?

This is simple:

A/(A+B) = 100/160 = 62.5%

What's proving difficult is the second part.

2.

It was described as a "fair game" because A and B had the same probability of winning each single game.

However, A has an advantage over B by having more money with which to play.

To make this really fair, so that A and B would have the same probability of winning all of the other player's money:

what should A's probability of winning for each single game be?

I really have no idea how to do this without building some sort of infinite probability tree.

My first thought was:

100(x) = 60(1-x)

x = .375

But this is wrong.

I have no idea how to do this, and I hope that someone here will be able to shed some light on it.