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# Likelihood of getting a Yarborough

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In the card game Bridge (and its predecessor Whist), a cooloquial term often used for a very bad hand is called a Yarborough.  It is defined as any hand of 13 cards with no card greater than a 9 (i.e. no 10's, Jacks, Queens, Kings, or Aces).  If a standard deck of 52 cards is dealt randomly to four players in a game of Bridge, what is the probability of:

(1) One player being dealt a Yarborough

(2) Two players being dealt a Yarborough

Jul 30, 2021

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Solutions:

(1) Calculate the total number of 13-card hands when face-cards, aces, and tens are removed.

Divide this by the total number of 13-card hands from the complete deck

nCr(32, 13) / nCr(52, 13) = 0.0005470333581834

This corresponds to odds of about (1) in (1829) hands.

(2)  This is a conditional probability. Given that one player has a Yarborough, what is the probability that a second player also has a Yarborough?

Calculate the total number of 13-card hands after an additional 13 cards are removed from the reduced deck. Divide this by the total number of 13-card hands from the complete deck.

nCr(32, 13) / nCr(52, 13) = 0.0000000427266467.

The over all probability of two players being dealt a Yarborough in a single round is

(0.0005470333581834) * (0.0000000427266467) = 0.00000000002337290102821668560478

This corresponds to odds of about (1) in (42 784 590 531) rounds,

GA

Jul 31, 2021