A and B play a series of games. Each game is independently won by A with probability $p$ and by B with probability $1-p$. They stop when the total number of wins of one of the players is $3$. Find the probability that A is the winner of the series given that A won the first game.

My attempt:

Let $E\_A$ be the event that A wins the series and $E\_1$ be the event that A wins the first game. Then I need to compute $$P[E\_A|E\_1]={P[E\_1|E\_A]P[E\_A]\over P[E\_1]}$$

$P[E\_1]=p$ and $P[E\_A]={\binom {5}{3}}p^3(1-p)^2$ this is because A and B can play a total of $5$ games: (for example A loses the first $2$ games but wins the last $3$ games) but we can do this in ${\binom {5}{3}}$ ways (becuase A needs to win 3 games out of 5)

But I don´t know how to compute $P[E\_1|E\_A]$

I would really appreciate if you can help me :)