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3-Card Poker Hand Frequency

The table below lists the frequency and probability by percentage for the different hand ranks in Three Card Poker.

Calculating Frequency of 3-Card Poker Hands

The following describes a method for calculating the frequency of 3-card poker hands. The calculations make extensive use of the combinations without repetition formula (combin formula in Excel) ⁿC_r = n! / (r! x (n - r)!)

Any Hand - Total number of 3-card poker hands drawn from a 52 card standard deck.

⁵²C₃ = 22100

Straight Flush - There are 12 differently ranked straight flushes from A-2-3 up to Q-K-A in each of the 4 different suits.

¹²C₁ x ⁴C₁ = 48

Three of a Kind - There are 13 differently ranked three of a kinds using 3 of the 4 suits.

¹³C₁ x ⁴C₃ = 52

Straight - The are 12 differently ranked straights from A-2-3 up to Q-K-A. Each of the cards can be 1 of the 4 suits with the straight flushes being excluded.

¹²C₁ x (⁴C₁)³ - 48 = 720

Flush - A flush contains 3 of the 13 ranks, each card belonging to 1 of the 4 suits. The straight flushes are excluded.

¹³C₃ x ⁴C₁ - 48 = 1096

One Pair - There are 13 differently ranked pairs using 2 of 4 suits, the third card being 1 of the 12 remaining ranks using 1 of the 4 suits.

¹³C₁ x ⁴C₂ x ¹²C₁ x ⁴C₁ = 3744

Nothing - Any hand not being one of the above type of hands.

⁵²C₃ - 48 - 52 - 720 - 1096 - 3744 = 16440

Related Pages

• Three Card Poker - Hand Ranking