Alternating tosses race to a second head

Alice and Bob alternate tossing one fair coin, with Alice going first. Heads are counted separately for each player, and the first player whose own total of heads reaches two (not necessarily in a row) wins. What is the probability that Alice wins?

$16/27$. Track (Alice's heads, Bob's heads) at the start of each full round of two tosses. Solving the first-step equations: from $(1,1)$ Alice wins with probability $\tfrac23$, from $(0,1)$ with $\tfrac29$, from $(1,0)$ with $\tfrac89$, and from $(0,0)$ the value is $\tfrac{16}{27}$.