Casino Table
Optiver
At a casino table, Maria begins with $\$1$ and James with $\$2$. They play with a special coin that has a $\frac{2}{3}$ chance of landing heads. When heads appears, James pays Maria $\$1$; when tails shows up, Maria pays James $\$1$. The game continues until someone loses all their money. What's the probability of Maria winning?
Answer
Draw a tree. First flip can be H or T. T leads to Maria's loss, whereas H means James pays her 1. If second flip is T, Maria gives 1 back and her chance of winning from then on is the same as at the start - as if the game resets. If second flip is a H, Maria gets 1 more and wins. So Maria loses if T is first, wins if we have HH and game resets if we have HT. Her chance of winning is then: $P = \frac{2}{3} \cdot \frac{2}{3} + \frac{2}{3} \cdot \frac{1}{3} \cdot P$. Solving for P, we get $\frac{4}{7}$.