Mock AIME 2 2006-2007 Problems/Problem 13

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In his spare time, Richard Rusczyk shuffles a standard deck of $\displaystyle 52$ playing cards. He then turns the cards up one by one from the top of the deck until the third ace appears. If the expected (average) number of cards Richard will turn up is $\displaystyle \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $\displaystyle m+n.$

Problem Source

4everwise enjoys playing and watching card games. In fact, he thought of this problem when watching Round 4 of the Professional Poker Tour. (Guy caught an Ace on the river.)

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