Making a biased-coin race against a fair coin even

Abby and Ben take turns tossing coins, Abby first; the first person to toss a head wins. Ben's coin is fair, while Abby's comes up heads with probability $p$. For what value of $p$ do the two have the same probability of winning?

$p=\tfrac13$. A full round with no winner has probability $(1-p)\cdot\tfrac12$, so Abby wins with probability $\dfrac{p}{1-(1-p)/2}=\dfrac{2p}{1+p}$. Setting this equal to $\tfrac12$ gives $4p=1+p$, so $p=\tfrac13$.