This Probability of Four of a Kind losing (fraction) equation defines the probability that another hand can beat your Four of a Kind and this probability is expressed as a fraction. The Four of a Kind hand can be beaten by all the hands of greater value: 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 Four of a Kind hand with a higher Four of a Kind card value. 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 Four of a Kind. 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: