Suppose Bridget receives a uniformly drawn hand of 13 cards, in a uniformly random order. Because…

 

Suppose Bridget receives a uniformly drawn hand of 13 cards, in a uniformly random order. Because your ex-friend Peter was trying to cheat at poker with this deck, the A♣ card is marked. You observe that the card the fourth-from-the-right position in Bridget’s hand is A♣. What is the probability that Bridget also has the A♦ in her hand

Most casual bridge players sort their hands by suit (♠, ♥, ♣, ♦ from left to right), and decreasing from left to right by rank within each suit. (So one might have a hand like ♠AK4 ♥983 ♣AKQ ♦AJ98, reading from left to right.) Professional players are taught not to sort their hands, because doing so causes which card they play to leak information about the rest of their hand to the other players. Suppose Bridget receives a uniformly drawn hand of 13 cards, and sorts the cards in her hand. Peter’s card marking is still present, and you observe the A♣ in a particular position in Bridget’s hand. In the following scenarios, what is the probability that Bridget also has the A♦ in her hand? (That is: out of all hands for which A♣ is highest/lowest/etc. card, what fraction also have the A♦

 

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