Pocket-hand counting
Count pairs and suited hands and understand why hand classes have different weights.
10 minutes · intermediate · Original Poker4Quant lessonWhat you will learn
Count physical hands in a rank category before assigning it a probability.
Prerequisites: Combinations versus permutations
A rank label is not one physical hand
For a pocket pair such as queens, choose two of the four queens: C(4,2)=6 combinations. Across thirteen ranks that makes 78 pairs among the 1326 possible two-card hands. Thus P(any pocket pair)=78/1326=1/17.
For two different ranks, there are 4×4=16 combinations. Four share a suit and twelve do not. These weights explain why the 169 common rank/suitedness labels cannot each receive probability 1/169 in a physical deal.
Known cards remove combinations
If one queen is visible outside your hole cards, only three queens remain, so a pocket-queen holding has C(3,2)=3 possible combinations. With two queens visible, only one pair combination remains.
Those are conditional counts, not the final probability by themselves. If the known cards are removed from a uniform deck, the denominator must also change. If the information was generated through a player action, a range model may weight the remaining combinations differently.
Worked example
Count ace-king starting hands, separating suited from offsuit.
- Choose one of 4 aces and one of 4 kings: 16 hands.
- The four same-suit pairs are suited.
- The other 12 use different suits.
AK has 16 combinations: 4 suited and 12 offsuit. Under uniform dealing, suited AK is 4/1326, not 1/169.
Common mistakes
- Treating each rank/suitedness class as equally likely.
- Using 4×4 for a pocket pair, which repeats cards and double-counts sets.
- Removing blocked favorable combinations without changing the sample-space denominator.
Keep these ideas
- Pocket pairs have six combinations before blockers.
- Different-rank classes have four suited and twelve offsuit combinations.
- Conditional information changes both weights and possible cards.