Setting a coin bias to make a first-head race fair

Two players take turns tossing their own coins; whoever gets the first head wins. The friend's coin lands heads with probability $0.4$, and to compensate the friend tosses first. Steven, who tosses second, can choose any heads probability $p$ for his own coin. What $p$ makes the game fair, i.e. gives each player a winning chance of $\tfrac12$?

$p=2/3$. Each full round both tosses are tails with probability $0.6(1-p)$, so the friend wins with probability $\tfrac{0.4}{1-0.6(1-p)}$. Setting this to $\tfrac12$ gives $0.8=0.4+0.6p$, so $p=\tfrac23$.