Probability and Poker
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Probability and Two Pairs

There are many chances of getting dealt two pairs. You may even be shocked at how common it is. Even so, this does not mean that it is not a good hand to bet on, especially in the early rounds of a poker hand when it might be improved upon. To find the odds of getting dealt two pair, we first need to find the number of two-pair hands possible:

(((13 X 6) X (12 X 6))/2) X 44 = 123,552

Two pairs is made up of two like cards of one type, plus two like cards of another type plus a random card.The 13 X 6 is the thirteen card types times the number of ways to combine four cards to make 2 of a kind. The twelve times six is the number of ways to combine four cards to make 2 of a kind in the remaining twelve card types after the first type is selected. We then have to divide these by two in order to account for the diminishing number of combinations resulting from the fact that fewer combinations are possible as a limited number of pairs of cards are introduced. The 44 comes from the cards remaining to combine after the first two sets have been chosen.

To get the odds on two pairs divide by the number of combinations of 5 cards possible:

123,552/2,598,960 = 1 in 21

Two pair can easily be made into a better hand. The chance of drawing one card to make the full house are 4/47 (the four possible cards available to make up the full house, divided by the total number of possibilities).

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