Probability of Three of a kind losing (fraction)

This equation defines the probability that another hand can beat your three of a kind and this probability is expressed as a fraction.  

INPUTS

• Card Value of Three of a Kind - the card value of the three of a kind cards in your hand

NOTES

The Three of a Kind hand can be beaten by all the hands of greater value: straight, flush, full house, four of a kind, straight flush, and royal flush. So the probability that an opponent has a hand that can beat your hand is the sum of all the higher value hands'  probabilities, plus the probability that your opponent has a Three of a Kind hand with a higher value three of a kind.  See Notes Page for further explanation.

The probability that you can be beat is the sum of the probabilities of any hand that can beat your Three of a Kind hand. Each of those probabilities is based on the ratio of the combinations of ways to be dealt those winning hands to the number of five card hands possible,`((52),(5))`. The total number of five card hands possible is computed as `((52),(5))` (see the constant defining number of five card hands).

The probabilities that must be summed are:

• The probability of being dealt a straight
• The probability of being dealt a flush
• The probability of being dealt a full house
• The probability of being dealt four of a kind
• The probability of being dealt a straight flush
• The probability of being dealt a royal flush
• The probability of being dealt a higher three of a kind hand