lean_workbook_plus_42806

Let, $p$ be the probability that the first archer wins. Take the first aim. The first archer wins on that aim with probability $\frac{1}{2}$ and with probability $\frac{1}{2},$ the game continues to the second shot. The moment that happens, it's just as if we were starting over, except that the roles are now reversed - and the first archer's overall winning probability, given that he didn't win on the first shot, is $1-p.$ So $p=\frac{1}{2}+\frac{1}{2}\cdot(1-p)\implies \frac 32p=1\implies p=\frac 23$