Equity vs a Hand vs Equity vs a Range
You almost never know the opponent's exact hand, so the equity that matters is your average against everything he can plausibly hold. This lesson defines ranges, walks the weighted-average math by hand, and shows why single-hand thinking gives wrong answers in both directions.
Assumptions: All examples use a 100bb-deep 6-max online cash game at $0.50/$1 with no rake unless a different setup is stated inline.
Every equity number so far has answered the question "how do I do against that exact hand?" Useful for building intuition — and almost never the question the table actually asks you. You don't get shown a hand. You get shown actions: a position, a raise size, a jam. The honest question is not "what does he have?" but "what hands can he have, and how do I do against all of them at once?"
That set of hands is his range: every holding consistent with what he's done, weighted by how likely he is to play each one this way. And your equity versus the range is the weighted average of your equity against each hand in it. That one number — not your equity against your fear, and not your equity against your hope — is what every correct decision is built on.
Where a range comes from: actions as filters
A range isn't a guess pulled from the air — it's the output of a filtering process that starts wide and narrows with every action. Before the cards are dealt, an opponent "has" all 1,326 two-card combos. Each thing he then does deletes the combos inconsistent with it.
The first and biggest filter is the preflop decision. A disciplined player who open-raises from UTG in a 6-max game is representing something like this site's baseline UTG opening range:
That chart contains 226 combos — 17% of all starting hands, roughly the top sixth of the deck. The moment he raises UTG, the other five-sixths of the combos stop existing for your purposes — he doesn't have 72o, and you should stop having nightmares about flopped two pairs that require it. Postflop actions keep filtering: a check-raise on a dry board deletes most air; a third barrel on a bricked river deletes most medium hands; a passive call deletes most monsters (good players usually raise them). Hand reading — the dedicated material later in the curriculum — is the craft of running these filters accurately. This lesson's job is the arithmetic that happens after the filtering, and for that we'll simply assert reasonable ranges and compute against them.
One situation, three numbers
You're on the button with T♠T♥. A tight, competent UTG player open-jams 100bb — an unusual, polarizing move. Watch the same decision produce three completely different equities depending on what you compare your hand to.
- 1.UTG jams $100
- 2.Folds to BTN, who must call $100 to win the $101.50 pot (jam + blinds)
Analysis
Versus AK alone TT is 57% — a happy call. Versus KK alone TT is 18% — a disaster. Versus the realistic jamming range {QQ+, AK}, 34 combos, TT averages out to 37%. The call needs 49.6%, so range thinking says fold — and neither single-hand number would have gotten you there: one is far too optimistic, the other too pessimistic, and both are answers to a question you can't actually ask.
Compute the pieces:
- Versus A♣K♦ alone: TT has 57%. If you could see ace-king, calling is automatic.
- Versus K♥K♠ alone: TT has 18%. If you could see kings, folding is automatic.
- Versus the realistic range {QQ+, AK} — that's QQ, KK, AA, AKs, AKo, 34 combos in all: TT averages 37%.
The price: you're calling $100 to win the $101.50 already out there, so you need 49.6% equity to break even. Against the range, TT is a clear fold at 37%. Notice that no single-hand comparison reaches that conclusion honestly. The player who thinks "he's probably on ace-king, I'm flipping ahead" talks himself into a call that torches money against the pairs. The player who thinks "what if he has kings" folds — right answer, wrong method, and the wrong method will betray him in the next spot, where the fold isn't right.
Why is the range number 37% and not midway between 57 and 18? Because the average is weighted by combos. There are 6 combos each of QQ, KK, and AA (18 combos of pairs — kings hold you to 18% and aces to 19%) but 16 combos of AK (4 suited, 12 offsuit). The offsuit AK combos are 57/43 dogs to you and the suited ones about 54/46 — computed, not assumed. More than half the range is AK by combo count — and you still can't call, because the pair half is so devastating. That's what averaging does: it lets the good and bad cases fight it out in correct proportion.
One more turn of the dial, because this is where range thinking starts paying rent: suppose the jammer is instead a frustrated short-tempered reg whose jamming range you read as {99+, AQ+}. Against that wider range — 63 live combos — TT computes to 44%. Still short of 49.6%, still a fold, but eight points closer, and a hand like JJ or AK would now play very differently. Same action, same price; the entire decision lives in the range you assign.
Doing the average by hand
The tool computed 37% in a second, but you should walk the arithmetic once yourself, because the mechanism — equity times weight, summed — is the engine under everything in this track.
The second example: you hold A♥8♥ on a 8♠6♦3♥ flop — top pair, top kicker, backdoor nut flush draw. You c-bet, and a sticky opponent check-raises. You give his check-raise a small, concrete range of five hand types: top set, middle set, an overpair, and two aggressive draws. Against each specific combo, computed individually:
| His hand | Your equity |
|---|---|
| 8♣8♦ (top set) | 4% |
| 6♠6♣ (middle set) | 5% |
| K♠K♣ (overpair) | 25% |
| 7♠5♠ (open-ended straight draw) | 65% |
| 5♥4♥ (OESD + backdoor hearts) | 67% |
First pass — pretend each hand is equally likely. The straight average: (4 + 5 + 25 + 65 + 67) ÷ 5 = 166 ÷ 5 ≈ 33%.
Second pass — weight by reality. Hands aren't equally likely, because the cards you can see remove combos. You hold an 8 and the board has another: only 1 combo of 88 is left. The 6♦ on board leaves 3 combos of 66. KK is untouched at 6 combos, and each suited draw has 4. Eighteen live combos, and the weighted sum:
(1 × 4 + 3 × 5 + 6 × 25 + 4 × 65 + 4 × 67) ÷ 18 = (4 + 15 + 150 + 260 + 268) ÷ 18 = 697 ÷ 18 ≈ 39%
(Using the unrounded per-hand equities, the tool gets the same answer: 39%.) Six points better than the naive average, and the difference is pure card removal: the monster you fear most — top set — is nearly impossible because you're holding one of its cards. This is why ranges are counted in combos and not in hand names. "He could have a set" is one sentence; "there are four set combos and eight draw combos" is an answer.
What does 39% mean for the decision? You're getting about 2.5-to-1 on the check-raise call (calling $7.50 into the $19 out there), needing about 28% — so against this range, calling with the intention of evaluating turns is comfortable, and folding top pair because "he could have a set" would be lighting money. A different read — say, this opponent only check-raises sets — collapses the range to the 4–5% rows and makes folding correct. The method doesn't tell you what his range is; it tells you exactly what to do once you've committed to one.
Weights: when a hand is only sometimes in the range
One refinement completes the definition. A range is a weighted set — and the weights aren't always 100%. Real players play the same hand different ways: maybe this opponent always check-raises his sets on 8♠6♦3♥, but check-raises his draws only about half the time (calling with them the other half). Then the draws belong in the check-raising range at half weight.
The averaging machinery doesn't change; the weights just stop being whole combos. Redo the A8 calculation with sets at full weight and draws at 50%:
(1 × 4 + 3 × 5 + 6 × 25 + 2 × 65 + 2 × 67) ÷ 14 = (4 + 15 + 150 + 130 + 134) ÷ 14 = 433 ÷ 14 ≈ 31%
Eight points worse than the 39% you had against the always-raise-draws version — because halving the draws shifted the range's center of gravity toward the hands that beat you. Against a passive opponent who only check-raises monsters, the draws drop to 0% weight and your equity collapses toward the 4–25% rows: now folding top pair is right. One player type calls for a comfortable continue, another for a disciplined fold, and the difference is nothing but weights.
You won't run decimal weights in your head at the table, and you don't need to. The working version is coarse: "his raising range here is mostly draws" (continue), "it's half draws, half monsters" (continue carefully, hate life on bad turns), "it's monsters only" (fold). What the exact arithmetic gives you is calibration for those judgments away from the table — run the numbers in review, and your in-game instincts inherit the precision.
Why the range number, and not the scariest hand
Single-hand thinking fails in two symmetrical ways, and you've now seen both:
The monster-under-the-bed fold. TT-versus-kings is 18%; A8-versus-top-set is 4%. If you make decisions against the worst hand in the range, you fold every time anyone raises, and observant opponents will raise you forever. The sets were in the A8 range above — all four combos of them — and calling was still right, because they were outnumbered by the hands you beat. Fear isn't a weighting scheme.
The hopeful hero call. TT-versus-AK is 57%. If you make decisions against the most convenient hand, every jam looks callable. The AK combos were in the jamming range too — sixteen of them, the biggest single block — and folding was still right. Hope isn't a weighting scheme either.
The range average prices both errors correctly and automatically. It is also, not coincidentally, how every solver, every equity calculator, and every strong player processes the game. When a coach says "you're doing fine against his range," this weighted average is the literal object being discussed.
Two practical habits to start now. First, assign the range before you look at your equity. Range first, from his position and actions; your hand second. If you peek at your own equity needs first, you'll unconsciously build him a range you can beat. Second, state ranges in combos when it matters. Especially postflop, where card removal does violent things — the A8 hand turned "five hand types" into an 18-combo range where the worst case was a 1-combo afterthought.
What this lesson deliberately left soft: where the ranges themselves come from. We asserted {QQ+, AK} for a tight UTG jam and a five-hand check-raising range by fiat. Building those assignments — from positions, sizings, player types, and the site's baseline charts — is its own skill, developed throughout the preflop and hand-reading material. The skill this lesson installs is what to do once the range is on the table: count the combos, average the equities, and let the number — not the scariest or friendliest face in the crowd — make the call.
Worked examples
- 1.UTG jams $100
- 2.Folds to BTN, who must call $100 to win the $101.50 pot (jam + blinds)
Analysis
Versus AK alone TT is 57% — a happy call. Versus KK alone TT is 18% — a disaster. Versus the realistic jamming range {QQ+, AK}, 34 combos, TT averages out to 37%. The call needs 49.6%, so range thinking says fold — and neither single-hand number would have gotten you there: one is far too optimistic, the other too pessimistic, and both are answers to a question you can't actually ask.