2017 AMC 10A Problem 18

Let $x$ be the probability that Amelia wins. There is a $\frac{1}{3}$ chance Amelia wins off her first flip. If she gets tails, Blaine must also get tails for the game to return to the same state; this happens with probability $\frac{3}{5}.$ The total probability of this case is $$\dfrac{2}{3} \cdot \dfrac{3}{5} = \dfrac{2}{5}.$$ The game then goes back to Amelia, who then again has a $x$ chance of winning. Therefore, we get the following equation. $$x = \dfrac{1}{3} + \dfrac{2}{5}x$$$$\dfrac{3}{5}x = \dfrac{1}{3}$$$$x = \dfrac{5}{9}.$$ The difference between the denominator and numerator is $9 - 5 = 4.$ Thus, D is the correct answer.