A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled and n cards are dealt to $A$ and $B$. $A$ and $B$ alternately discard a card face up, starting with $A$. The game when the sum of the discards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$ wins if neither player makes a mistake?