When playing poker holding two pairs means that you hold two

When playing poker, holding two pairs means that you hold two cards the same rank, another two cards of a second rank, and a third card of neither rank. For example 7744K is a two pair hand but 7777K and 77444 are not two pair hands. How many ways are there to get two pairs if you draw five cards from a standard deck at random? What is the probability of getting two pairs if you draw five cards from a standard deck at random? What is the probability that you have two pairs (in five cards drawn from a standard deck) given that you already know that you have one pair?

Solution

a) out of 5 cards, we must get 2 pairs

There are in total 52 cards with 4 cards of the same type for each.

Hence no of ways to get exactly 2 pairs = 13(4C2) 12(4C2) 44C1

=

(since first pair can be from any of the 13 and ii any of remaining 12 and last one from the remaining 44 excluding 2 pairs)

b) Prob = 13(4C2) 12(4C2) 44C1/52C5

= 247104/2598960

= 0.095078

c) Prob for 2 pairs given already one pair is there = 12(4C2) 44C1/48C5

= 3168/1712304 = 0.00185

247104