Probability and Three of a Kind
Now we are getting into fairly common hands. You will seldom be dealt many of the previously mentioned hands. However the three of a kind is seen fairly frequently by the average poker player. Even so, it is a good hand to bet on and can be improved as a hand progresses. To find out the chances of getting 3 of a kind, we first must find out how many possible ways there are of making up such a hand:
((13 X 4) X (48 X 47 / 2))  3744 = 54,912
In this case the (13 X 4) gives us the 3 like cards as explained in the section on the full house. The second section of the equation is the combinations that can be made from the remaining two card positions. At the end we must subtract off the possible combinations for full house, which is a glorified 3 of a kind we have already accounted for. (The possibilities for four of a kind have already been excuded earlier in the formula. They were taken out when we decided that we were only going to accept 3 of the 4 cards in any card type (the four possible combinations of four distinct cards in a group of three represented by the four in our formula). Note also that the 48 and 47 in our formula are one less than you might expect because we have thrown out the possibilty of picking up the final card of this group.)
To get the odds on three of a kind can then be done by dividing by the number of combinations possible:
54,912/2,598,960 = 1 in 47
The numerator is the number of 3 of a kinds possible, while the denominator is the number of all possible hands. Interestingly enough, your chances of getting dealt three of a kind are roughly equal to filling an inside straightflush. Three of a kind can easily build into better hands, 4 of a kind and full houses. It can be gotten to easily from two of a kind with about a 1 in 8 chance on a three card draw:
47 X 46 X 45 / 3 X 2 = 16,215
2 X (45 X 44 / 2) = 1980
1980/16,215 = 1 in 8
Quickly, the first equation gives us the total combinations possible with the next three cards. The second equation gives us the total combinations where one of the two remaining (from the previous 2 of a kind) are included. In the final equation we divide the number of possibilites that include one of the two cards needed by the to total number of possibile combination of draws. The result is the probability of making three of a kind from a pair drawing three cards.
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