Poker Equity Calculator

Leave Outs at 9, Unknown Cards at 2 and Known Board Cards at 3 — the factory defaults, and the exact shape of a nut-flush draw on the flop. Two hearts in your hand, two on the board, nine hearts still in the stub, and both the turn and the river still to come. The engine rebuilds the unseen deck first: fifty-two minus three flop cards minus two hole cards leaves forty-seven. It then asks how often both remaining streets miss you entirely: C(47 − 9, 2) divided by C(47, 2), which is C(38, 2) / C(47, 2) = 703 / 1081 ≈ 65.03 percent. Equity is the complement, (1 − 0.6503) × 100 ≈ 34.97 percent. That is the chance at least one of those nine hearts lands among the two unknown cards if you see both streets.

Read 34.97 percent as a frequency, not a verdict. Over a thousand identical flops you complete the flush roughly three hundred and fifty times, provided the nine outs are live and the villain is not already drawing to a better heart or a full house. If the board pairs on a heart, some of those nine cards also fill a set; if an opponent holds a made flush with a higher heart, every out is dead. The calculator does not know suits or kickers. It only knows the counts you typed.

The street-counting error is the one that wrecks prices. Leaving Unknown Cards at 2 after the turn has already fallen double-counts a card that is no longer unknown. Once you have seen fourth street, set Known Board Cards to 4 and Unknown Cards to 1. Nine clean outs then become 9 / 46 ≈ 19.57 percent, which is the real turn-to-river number and the origin of the “rule of two.” The “rule of four” (outs × 4 on the flop) would have printed 36 percent here — close, but systematically high relative to the combinatorial 34.97 percent this page actually computes.

Compare the 34.97 percent with the price of the pot before you call. If a flop bet asks you to win more than 35 percent of the time and you will not see the river for free, the raw flush draw is a fold unless future bets (implied odds) close the gap. If the price is 25 or 30 percent and the outs look clean, calling is the mathematical default. Next, drop Outs to 8 for an open-ended straight, raise it to 12 for a combo draw, or set Unknown Cards to 1 to isolate the turn. Do not treat thirty-five percent as a license to stack off into a raise that prices the draw out.

Two more traps sit in the Outs box. Overcounting — treating a gutshot plus two overcards as twelve live outs when two of those overcards crash into a better two-pair — inflates equity by several points. Undercounting the runner-runner redraw is usually the safer bias. Backdoor flush and straight combinations are not in this model at all; the formula only cares about hitting at least one of the outs you declared among the unknown cards. If you need runner-runner, this is the wrong tool.

This page prices a drawing hand from outs, remaining board cards and how many streets are still unknown. It is the combinatorial core of hold’em decision-making on the flop and turn: you name the cards that win, the engine names the frequency they appear. Cash regulars use it between orbits to check whether a “standard” call is still standard after the bet sizes moved. Tournament players use it on the bubble when a 35 percent draw is no longer worth the chips because ICM has warped the value of a double-up. Coaches walk students through it so the rule-of-four shortcut is no longer a superstition.

The outs framework is older than online poker. Doyle Brunson’s Super/System already talked about counting winners; David Sklansky’s The Theory of Poker made the comparison with pot odds explicit; later solvers replaced the count with full range-versus-range equity. This calculator sits in the middle generation. It is more precise than “nine outs times four,” and far less precise than a Monte-Carlo equity engine that knows both hole-card combinations. It assumes a fifty-two-card deck, two hole cards for you, and a board of the length you typed. It does not know the opponent’s hand, blockers, or the chance you are already drawing dead.

Who it is not for: stud players who need seventh-street runouts with folded-up cards, Omaha players who must count wraps and redraws from four hole cards, or anyone trying to compute all-in equity against a range. Changing Known Board Cards does not insert specific ranks; it only changes how many cards have been removed from the stub. There is no input for “villain has a set,” no input for “one heart is in the muck,” and no input for dead money already in the middle. Those belong in a pot-odds or implied-odds tool, not here.

Limitations follow from the combinatorics. Multiway pots share outs. A flush draw against two opponents is not 35 percent to win; it is 35 percent to make a flush, which may still lose to a bigger flush or a boat. Short-deck and mix games change the remaining-card count; this engine never leaves fifty-two. Reverse implied odds — the times you hit and still pay off a better hand — are invisible. The 34.97 percent default is a clean nut-flush-draw illustration, not a claim that hearts always win.

What it will not do is tell you to call, fold or raise. Equity is one input into expected value. Stack depth, position, ICM, and the chance of being raised off the draw all sit outside the formula. Treat the number as a frequency you can take to a pot-odds comparison, then stop. It is not a prediction of the next street and it is not financial advice.

Hypergeometric combinatorics, not a named procedure, is the correct language for pricing a drawing hand on a finite pack. Two private cards together with a known board of three cards leave a stub of remaining cards R = 47. Nine outs and two unknown cards still to come is the classical nut-flush illustration: the turn and the river will be taken without replacement from those forty-seven unseen cards. Equity is the frequency that at least one of the nine winners appears among the two unknown cards, obtained most cleanly by complementing a total miss.

The miss event counts the ways both remaining cards can be chosen from the thirty-eight non-outs. After that counting sentence, the ratio of combinations is C(38, 2) / C(47, 2) = 703 / 1081. Expanding the binomials yields 38 × 37 / (47 × 46) = 1406 / 2162, which reduces by two to the same 703 / 1081. One minus that fraction is 378 / 1081 ≈ 0.349676, reported as equity of approximately 34.97 percent. In symbols the same hypergeometric statement is miss = C(R − O, U) / C(R, U) with O = 9 and U = 2, provided enough dead cards remain to fill the two streets.

Worked flop arithmetic should be distinguished from the linear shortcut called the rule of 2 and 4. Multiplying nine outs by four prints 36 percent on the flop; multiplying by two on the turn prints 18 percent. Both figures sit about one point above the combinatorial values 34.97 percent flop-to-river and 9 / 46 ≈ 19.57 percent turn-to-river. The shortcut is an approximation of the exact ratio, useful at speed, systematically slightly high for this out-count. Order of the two streets does not matter because combinations enumerate sets rather than sequences.

Assumptions are severe and belong in the lecture as first-class hypotheses. Sampling without replacement from a well-shuffled fifty-two-card deck is taken as given. Outs are treated as live and as full wins: a chopped pot on a shared straight is invisible, and a heart that completes a better flush for an opponent is still counted if the out-count was left at nine. Other players’ hole cards appear only insofar as the out-count was already reduced. Multiway dilution, blockers, and runner-runner paths sit outside this model.

Historical table primers of the 1970s and 1980s tabulated these draw frequencies before David Sklansky made the comparison with pot odds a doctrine; later exhaustive enumerators replaced outs with range-versus-range matching. The closed-form outs calculation remains the object of this lecture: remaining R = 47, miss 703 / 1081, equity about 34.97 percent, hypergeometric, sampling without replacement. It is a hit frequency under those counting hypotheses, not a long-run profit rate and not a claim that hearts always take the pot. Tainted outs cut realised win rate below the printed 34.97 percent without changing the combination count itself.