ILL 1974 Problem 16

A pack of $2n$ cards contains $n$ different pairs of cards. Each pair consists of two identical cards, either of which is called the twin of the other. A game is played between two players $A$ and $B$ . A third person called the dealer shuffles the pack and deals the cards one by one face upward onto the table. One of the players, called the receiver, takes the card dealt, provided he does not have already its twin. If he does already have the twin, his opponent takes the dealt card and becomes the receiver. $A$ is initially the receiver and takes the first card dealt. The player who first obtains a complete set of $n$ different cards wins the game. What fraction of all possible arrangements of the pack lead to $A$ winning? Prove the correctness of your answer.