Question

A pack of playing cards is being used by four players for a game of bridge, so each is dealt 13 cards. The King, Queen and Jack are referred to as picture cards. Find the probability that a bridge hand (13 cards) contains (a) 3 spades, 4 diamonds, 1 heart and 5 clubs; (b) 3 aces and 4 picture cards. 

Answer 1

a) The number of possible distinct 13-card hands is

$$\dbinom{52}{13}=\dfrac{52!}{13!(52-13)!}$$

$$=\dfrac{52(51)(50)(49)(48)(47)(46)(45)(44)(43)(42)(41)(40)}{1(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)}$$

$$=635013559600$$

We need to have 3 spades, 4 diamonds, 1 heart and 5 clubs

$$\dbinom{13}{3}\dbinom{13}{4}\dbinom{13}{1}\dbinom{13}{5}=\dfrac{13!}{3!(13-3)!}$$

$$\times\dfrac{13!}{4!(13-4)!}\times\dfrac{13!}{1!(13-1)!}\times \dfrac{13!}{5!(13-5)!}$$

$$=286\times715\times13\times1287=3421322190$$

$$P(A)=\dfrac{3421322190}{635013559600}\approx0.005388$$

b) We need to have 3 aces, 4 picture cards and 6 numerals

$$\dbinom{4}{3}\dbinom{12}{4}\dbinom{36}{6}=$$

$$=\dfrac{4!}{3!(4-3)!}\times\dfrac{12!}{4!(12-4)!}\times \dfrac{36!}{6!(36-6)!}$$

$$=4\times495\times1947792=3856628160$$

$$P(B)=\dfrac{3856628160}{635013559600}\approx0.006073$$