Five cards are dealt from a well shuffled poker deck of 52 c

Five cards are dealt from a well shuffled poker deck of 52 cards. What is the probability that we get exactly two pairs? By this we mean that we have two cads of one rank, two card of another rank, and one card of a third rank.

Solution

First we need to find number of poker hands that have exactly two pairs :

First, assume you choose the cards in order of pair 1, pair 2, non pair card.

The first card in pair 1 can be any of 52 cards. The second card in pair 2 can be one of 3 cards (the other cards of the same number). The first card of pair 2 can be one of the 48 cards which are not of the number in pair 1. The second card of pair 2 is one of the 3 cards of the same number as the first card in the pair. The last card in the hand is not of the number in the first pair or second pair, so there are 52-8 = 44 cards which you can select from.

The division by 2\'s in the numerator account for ignoring order in each pair and ignoring the order the pairs were selected in.

= ((52*3)/2 * (48*3)/2 )/2 * 44 = 123552

Total number of possible hands :

= 52C5

= 2598960

Therefore, Probability that we get exactly two pairs :

= 123552 / 2598960

= 0.0475 Answer