Methodology

A standard 52-card deck, four suits, ranks two through ace. A card can appear once; the engine refuses any scenario that uses a card twice. When we say “unseen cards”, we mean the 52 minus the cards you can see — your own and the board. An opponent’s hidden cards are unseen and therefore still counted, which is the convention behind every outs and draw-odds figure (47 unseen on the flop, 46 on the turn).

Standard high-hand rankings; the best five of seven cards; the ace plays high and low in straights (A-2-3-4-5 is the lowest straight); suits never break ties. The evaluator is verified by enumerating all 2,598,960 five-card hands on every test run and all 133,784,560 seven-card hands in the extended suite, and matching the published frequency of each category exactly.

When a question can be answered by counting, we count. For a hand against a known hand on the flop, that is all 990 turn-and-river combinations; preflop, all 1,712,304 boards. For a hand against a range, every live combination in the range times every board. Results are exact fractions held in double precision and rounded only when displayed — the engine itself never rounds.

Some questions have too many outcomes to count in a browser — a hand against an unknown hand before the flop is around two billion evaluations. Then we deal random run-outs from a seeded generator and report the number of trials, the seed, and the standard error. A simulated result is always labelled as one. The two methods are never silently swapped.

With no opponent specified, a card is an out if it raises your hand to a better category and your hole cards are part of what defines the new hand; a card that merely pairs the board is not your improvement. With an opponent specified, an out is a card that moves you from behind to strictly ahead — and on the flop we also count how many rivers put the opponent back ahead, which is the honest measure of a clean versus dirty out.

Draw probabilities use the hypergeometric formula. Nine outs on the flop: 19.15% on the turn (9 ÷ 47), 34.97% by the river (1 − C(38, 2) ÷ C(47, 2)). The Rule of 2 and 4 is shown beside the exact figure wherever it applies, with its signed error.

Pot odds are (pot + bet) : call. Break-even equity is call ÷ (pot + bet + call). The expected value of a call is equity × (pot + bet) − (1 − equity) × call, measured against folding and assuming the call closes the action. Implied odds are taught as a separate concept and never folded into these numbers unannounced.

A range is a weighting over the 169 starting-hand classes. Combinations blocked by known cards are removed before weights are applied. Hand-versus-range equity is the combination-weighted average of hand-versus-each-holding — a property the test suite checks directly.

We do not present approximations as exact, or simulations as enumerations. We do not label a card “good” or “bad” unless an equity change supports it. We do not present sample ranges as strategy. And we do not design tools for real-time assistance during play; everything here is for study, review and practice.

Details for developers, including the engine layout and test plan, live in the project documentation. Questions or a number you cannot reproduce? See About.