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
(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,