Flop runout count
After four known hole cards and three board cards, 45 cards remain and C(45,2)=990 unordered runouts.
Poker Probability Explained
Learn how combinations, conditional probability, best-five comparison, split-pot equity, and seeded simulation fit together. This poker probability guide follows the same transparent definitions used by the browser calculator rather than treating an odds result as a black box.
equity(player) = Σ outcome share ÷ legal outcomes × 100%
How the result is built
Poker Probabilities Depend on the Sample Space
Concrete unordered card combinations and abstract starting-hand classes answer different counting questions. They must not share a denominator.
- Physical deck52 cards
- Unordered two-card combinationsC(52,2) = 1,326
- Abstract starting-hand classes169
Five-card combinations2,598,960C(52,5)
Two-card combinations1,326physical deals
Starting-hand classes169abstract labels
Combinations remove irrelevant order
A flop with two missing board cards needs unordered turn-river sets for showdown equity. Counting both A-then-B and B-then-A as separate runouts would double the same final board.
Condition on every known card
Hole cards and board cards leave the sample space before enumeration or sampling. Range calculations also remove blocked concrete combinations before selecting hands.
Equity divides tied outcomes
A sole winner receives one unit of an outcome. Two tied winners receive one half each; three receive one third. Across players, equity sums to 100% within floating-point tolerance.
Simulation estimates the same target
Monte Carlo samples legal runouts without replacement. More iterations reduce sampling error, while the seed makes the selected pseudo-random sequence reproducible.
Worked examples
Three-way chop
When the board is the best hand for all three players, each receives 33.333…% equity.
Frequently asked questions
Why not use Math.random sorting?
It is biased, hard to reproduce, and unnecessary; partial Fisher–Yates sampling is used.
Does exact mean future certainty?
No. It means every legal modeled outcome was counted.
How does a seed help?
It reproduces the same finite simulation for testing and comparison.
Why can percentages have long decimals?
Split shares and finite sample counts often do not terminate in base ten.