A bridge hand is found by taking 13 cards at random and with

A bridge hand is found by taking 13 cards at random and without replacement from a deck of 52 playing cards. Find the probability of drawing each of the following hands. a) One in which there are 5 spades, 4 hearts, 3 diamonds, and 1 club. b) One in which there are 5 spades, 4 hearts, 2 diamonds, and 2 clubs. c) One in which there are 5 spades, 4 hearts, 1 diamond, and 3 clubs. d) Suppose you are dealt 5 cards of one suit, 4 cards of another. Would the probability of having the other suits split 3 and 1 be greater than the probability of having them split 2 and 2?

Solution

a) In order to have 5 spades, 4 hearts, 3 diamonds and 1 club, the selection can be done in (13C5)(13C4)(13C3)(13C1)/(52C13) ways. The numerator shows the number of ways in each suit can be selected from a set of 13 cards each and the denominator shows the number of ways in which the overall selection of 13 cards out of a pack of 52 cards be done.

b) One in which there are 5 spades, 4 hearts, 2 diamonds, and 2 clubs:

13C5 . 13C4 . 13C2. 12C2 / (52C13) ways

c) One in which there are 5 spades, 4 hearts, 1 diamond, and 3 clubs. =

(13C5) (13C4) (13C1) (13C3) / (52C13) ways.