Two players, A and B, alternatively toss a fair coin (A tosses the coin first, then B tosses the coin, then A, then B...). If there is a head followed by a tail, the game ends and the person who tosses the tail wins. What is the probability that A wins the game?
From the solution, it claims: Let $P(A)$ be the probability that A wins, $P(A|T)$ represent the probability that A wins given that the first toss is a tail, then: \begin{equation} P(A|T) = P(B) \end{equation} It says in the solution that "if A's first toss is T, then B is essentially the first to toss", I don't see why this gives the equation. I can calculate the value of $P(A|T)$ explicitly by conditional probability, but I really want to understand this approach, help is much appreciated.
Let $p$ be the probability that the second player wins. Clearly, $P(B) = p$, since $B$ is the second player. And clearly, $P(A|T) = p$, since after an initial $T$, $A$ is the second player.