Answer to Question #344592 in Statistics and Probability for tori

Question

in the game of bridge each of 4 players is dealt 13 cars from an ordinary well-shuffled deck of 52 cards. Find the probability that one of the players gets a) 7 diamonds, 2 clubs, 3 hearts, and one spade, b) a complete suit, c) 9 cards are of one suit

Answer 1

a)

$$P(7D\ \& \ 2C\ \&\ 3H \ \& \ 1S)=\dfrac{\dbinom{13}{7}\dbinom{13}{2}\dbinom{13}{3}\dbinom{13}{1}}{\dbinom{52}{13}}$$

$$=\dfrac{1716(286)(78)(13)}{635013559600}=0.00078368$$

b)

$$P(complete\ suit)=\dfrac{4\dbinom{13}{13}\dbinom{39}{0}}{\dbinom{52}{13}}$$

$$=\dfrac{4}{635013559600}=6.3\times10^{-12}$$

c)

$$P(9\ of\ one\ \ suit)=\dfrac{4\dbinom{13}{9}\dbinom{39}{4}}{\dbinom{52}{13}}$$

$$=\dfrac{4(715)(82251)}{635013559600}=0.00037045$$