The Two Decks of Cards and the Drunken Hat Check Girl

3 August 2009
  1. Imagine you have two decks of playing cards, each thoroughly shuffled.  You give one deck to your friend and keep the other.  Now each of you goes through your deck, one card at a time, flipping cards face up.  You compare your top cards, then you compare your next cards, and so on all through the deck.  If it ever happens that you both reveal the same card at the same time, you win; if you go through the whole deck without such a match, your friend wins.  Who is more likely to win?  You or your friend?
  2. Thirty-seven men attend a certain social event and check their hats as they enter.  However, the hat check girl has had a bit too much to drink, and when the time comes to leave, she gives back hats at random, with total disregard for which hat belongs to whom.  What are the chances that nobody ends up with his own hat?

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