An unfair coin comes up heads with probability $\frac{4}{7}$ and tails with probability $\frac{3}{7}$. Aerith and Bob take turns flipping the coin until one of them flips tails, with Aerith going first. What is the probability that Aerith wins the game?
Let $p$ be the probability Aerith wins. There is a $\frac{3}{7}$ chance Aerith wins right away, and if she doesn't win right away, there is a $1-p$ chance she will win after that (since at that point it's like starting the game over but with Bob going first. This gives

$$
p=\frac{3}{7}+\frac{4}{7}(1-p)=1-\frac{4}{7} p
$$

and solving this we get $p=\frac{7}{11}$.