Probability and One Pair
One pair is the lowest possible of the combination hands in poker. It is so frequent as to be common. Yet the greater hands are rare enough that it is often the winning hand. To find the chances of getting this hand we must first get an idea of the number of one-pair combinations possible in a poker hand:
((13 X 6) X (48 X 47 X 46)/6) - 247,104 - 3744 = 1,098,240
Here, as in two pairs 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 48, 47, 46 are the remaining cards available to choose from on the three subsequent cards, dividing this number by six results in the number of combinations possible from these three cards. Then we must subtract out the possible 2 pairs and full houses.
To get the odds on one pair divide by the number of combinations of 5 cards possible:
1,098,240/2,598,960 = 1 in 2.3
This means within every three hands you should get at least a pair dealt to you. Of course, this will not always happen. You can easily have a drought, so never count on getting a pair just because you haven't had one for three hands. Remember the odds of getting a hand never improve or decrease based on what happened in a previous hand.
A single pair can be improved any number of ways on a draw of three cards to two-pairs (1/5.5), three of a kind (1/8), full house (1/112), or even the lofty four of a kind (1/360).
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