Probability of Four of a Kind (fraction)

The Probability of Four of a Kind (fraction) constant defines the probability of being dealt four-of-a-kind and is represented as a fraction.  The Four of a Kind hand is a five card hand having four of five cards being the same value cards.  For example:

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

would be a Four of a Kind hand.

The number of possibilities of a Four of a Kind hand is the number of card values, `((13),(1))`; times the number of possible card values of the fifth card (the card that is not part of the four off a kind), `((12),(1))`; times the number of suits of that fifth card could be, `((4),(1))`.  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 Four 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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