Probability of Three of a Kind (fraction)

The Probability of a Three of a Kind (fraction) constant defines the probability of being dealt three-of-a-kind and the probability is represented as a fraction.  The Three of a Kind hand is a five card hand having three of five cards being the same value cards and the remaining two cards not being the same value as each other.  For example:

• 4 of spades
• 4 of hearts
• 4 of diamonds
• Jack of clubs, and
• an 8 of spades

would be a Three of a Kind hand.

The number of possibilities of a Three of a Kind hand is the number of card values, `((13),(1))`; times the number of combinations of three suits being chose from four suits, `((4),(3))`; times the number of combinations of choosing two cards that are not the same value as the three of the same value, `((12),(2))`; times the number of choices of the two un-like cards done twice, `((4),(1))^2`.  The total number of five card hands possible is computed as `((52),(5))` (see the constant defining number of five card hands), so this constant represent the ratio of number of possible Three of a Kind hands to the number of five card hands possible,`((52),(5))`).

Additional Resources:

The Poker Calculator provides all the equations, the calculations, to determine the probabilities of success and failure when playing Poker with a standard five card deck.

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