Sheldon Ross Chapter 2

Question

 If it is assumed that all (⁵²₅) poker hands are equally likely, what is the probability of being dealt (a) a flush? (A hand is said to be a flush if all 5 cards are of the same suit.)

(b) one pair? (This occurs when the cards have denominations a, a, b, c, d, wherea, b, c, and d are all distinct.)

(c) two pairs? (This occurs when the cards have denominations a, a, b, b, c, wherea, b, andc are all distinct.) 

Answer 1

(A) choosing a suit out of 4=4C1, choosing any five cards from the same suit=13C5

choosing cards from the same suit=$\frac{4C1*13C5}{52C5}$

(B) choose any 2 suits from 4(two same numbers cannot be from the same suit) and choosing a pair=4C2*13C1

the remaining three numbers has to be from the same suit. Also one number has already been selected for the pair so we can't select the same number in the remaining three cards. Hence, choosing the remaining three numbers=12C3*4C1*4C1*4C1

choosing one pair=$\frac{4C2*13C1*12C3*4C1*4C1*4C1}{52C5}$

(C)two pairs can be selected in 4C2*4C2*13C2 ways. remaining one number can be chosen in 11C1 ways from any one suit(4C1)

choosing two pairs=$\frac{4C2*4C2*13C2*11C1*4C1}{52C5}$