This Probability of High Card losing (percentage) equation defines the probability that another High Card hand can beat your High Card and this probability is expressed as a percentage.  The High Card hand can be beaten by all the hands of greater value: one pair, two pair, three of a kind, straight, flush, full house, four of a kind, straight flush, and royal flush but in this equation we are just addressing the case where both you and an opponent have a High Card hand. So, the probability that an opponent has a hand that can beat your hand is the sum of all the  High Card hands that can beat one of your cards. See the Notes Page for further explanation.

This equation assumes you have not input two two cards of the same value; ie., for example, you have not input two cards that both have value 10.  If you enter a pair of identical values for any two of your cards, or enter any card hand higher than a high card hand you make the calculation invalid.

Notes

The High Card Hand is any hand that doesn't qualify as a hand containing one pair, two pair, three of a kind, a straight, a flush, a full house, four of a kind, or a straight flush.

The probability that you can be beat is the sum of the probabilities of any hand that can beat your High Card hand, and 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).

This particular equation computes only the probability that your High Card hand can be beat by another High Card hand.

Thus this probability is the sum of the following possibilities divided by the total number of five card hands:

• Your opponent's highest card beats your highest card
• Your highest card ties you opponents highest card and your opponent's second highest card beats your second highest card
• Your highest and second highest card ties you opponents highest and second highest card and your opponent's third highest card beats your third highest card
• Your highest, second highest, and third highest cards ties you opponents highest, second highest, and third highest cards and your opponent's fourth highest card beats your fourth highest card
• Your highest, second highest, third highest, and fourth highest cards ties you opponents highest, second highest,  third highest, and fourth highest cards and your opponent's lowest card beats your lowest card