Sheldon Ross Chapter 2

Question

 If it is assumed that all (⁵² ₅) 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)- First select a suit: $4$ choices

      then, we can have to select 5 cards out of 13:  $C(13,5)$

      required probability: $\frac{4*C(13,5)}{C(52,5)}$ $=$ $0.001980$

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b)- We have to select two cards to make a pair: $C(4, 2)$ choices (we have to select 2 from 4 because a suit can't have 2 cards    of  same number, so we are selecting 2 suits from 4),

     then select a number for the pair: $13$ choices,

     now for remaining three cards: each card can be of same suit but number must be different, i.e. $4^3 * C(12,3)$

     required probability: $\frac{13*C(4,2) * C(12, 3) * 4^3}{C(52, 5)}$ $=$ $0.422569$

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c)- First, we have to select two numbers for two pairs: $C(13, 2)$ ,

     Select suits for both pairs: $C(4,2) * C(4,2)$ ,
     For the remaining one card: $11$ choices for the number and $4$ choices for the suit,

     required probability: $\frac{C(13,2)*C(4,2)*C(4,2)*11*4}{C(52, 5)}$ $=$ $0.04753$