Range-versus-range equity experiments

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Range equity depends on the hands and weights supplied. Change that composition and the result can change, even on a river where every showdown is settled. Here you can change one opposing group at a time and account for the difference.

Hero has every ace-king combination. The opposing range contains pocket nines, king-queen, ace-ten and ace-king. These small authored ranges describe an off-table experiment. They do not estimate a particular player's hands after a real betting sequence.

We hold the completed board fixed, exclude matchups that share a physical card and weight the legal pairings. The result is a share of that specified showdown distribution. No future betting decisions enter the calculation.

Ace: "I gave all four groups the same weight. Does each group happen a quarter of the time?"

Aurora: "Count the compatible hands first. Equal weight per combination can produce different group shares."

Keep the board and ranges fixed

The game is heads-up Texas Hold'em with a standard 52-card deck. The river is Qh Js 8c 4d 2s. Qh means queen of hearts; T means ten. The suit letters are c, d, h and s. Text tokens identify cards without relying on color.

At showdown, each player uses the strongest five cards available from the board and their two hole cards. Both hole cards need not play. With the river complete, every legal pairing has a settled win, loss or tie.

Hero's range is all 16 AK combinations, each with initial weight 1. For example, As Kd and Ac Kh are separate entries. The opposing candidates are all 99, KQ, AT and AK combinations that avoid the board. A group weight applies equally to every combination in that group.

The controls change only the four opposing weights. A different board or an added hand group would require a new outcome check.

Separate available hands from compatible pairs

Before considering Hero's exact cards, there are six possible pocket-nine hands, 12 KQ hands, 16 AT hands and 16 AK hands. The board has removed Qh, leaving three queens for KQ. None of the other ranks used by the ranges appears on the board.

The 50 opposing hands and 16 Hero hands produce 800 candidate pairings. Candidates that give the same ace or king to both players are impossible and are excluded. A shared rank is allowed; a shared physical card is not.

Take Hero's As Kd. All six 99 hands remain available. KQ has three remaining kings and three queens, for nine hands. AT has three remaining aces and four tens, for 12. Opposing AK has three aces and three kings, for nine. That leaves 36 compatible opponents for this Hero hand.

Every Hero AK combination has those same 36 compatible opponents here. Across all 16, that gives 576 joint matchups. The other 224 candidates overlap a hole card. These are physical counts before zero or positive weights are assigned.

Check the showdown before changing weights

On Qh Js 8c 4d 2s, Hero's AK plays ace, king, queen, jack and eight. It is ace-high. There is no ten to complete an ace-high straight. The board also has only two spades and one card in each other suit; two hole cards cannot supply the five cards needed for a flush.

Every compatible 99 makes one pair and beats Hero. Every KQ makes a pair of queens and also beats Hero. AT remains ace-high, but its best five ranks are ace, queen, jack, ten and eight. Hero's king kicker wins that comparison. Opposing AK has the same five ranks as Hero, so those matchups tie.

The worksheet uses this fixed outcome matrix after checking the physical cards. An independent seven-card evaluator checked every compatible pairing. The controls accept weights only; arbitrary hands and boards are outside its scope.

Adding mass to AT adds Hero wins; adding it to 99 or KQ adds losses. AK adds ties. Changing a weight leaves every existing pairing's showdown outcome unchanged.

Build the equal-weight receipt

Start with all four controls at 100. Every opposing physical combination then has the same initial weight. For counting the baseline, divide all those weights by 100 and treat each legal matchup as one unit. That common rescaling leaves every normalized share unchanged.

The baseline has 192 Hero wins against AT, 240 losses against 99 and KQ combined, and 144 ties against AK. Their sum is 576. The outright win share is 192/576, or one third. The tie share is 144/576, or one quarter. Those are different outcome categories.

For this heads-up fractional pot model, a tied matchup contributes half a pot to each player. Hero's equity is therefore (192 + 144/2)/576 = 264/576 = 11/24. The display rounds that value to 45.833333%. It is larger than the outright win share because split pots also contribute value.

At the actual control value of 100, the raw masses are 19,200 wins, 24,000 losses and 14,400 ties, totaling 57,600. The receipt shows these larger integers so the weight convention remains visible. Dividing the numerator and denominator by the same factor recovers the unit-weight calculation.

Opposing pot-share equity is 13/24, about 54.166667%. The two shares sum to one pot before display rounding. This exercise measures fractional shares and includes no rake, side pots, odd-chip allocation or previous investment. It does not calculate either player's net profit.

A group weight is not its final share

A weight is an input multiplier on each hand in a group. A normalized group share is an output after compatibility and all weights are accounted for. Setting AT to 50 means that each AT combination has half the input weight of a combination in a group set to 100.

It does not mean AT occupies half of the opposing distribution. At baseline, AT contributes 192 of the 576 compatible pairings, so its group share is one third. If AT alone drops to 50, its mass becomes 9,600 while the other groups together contribute 38,400. Its final share is 9,600/48,000, or 20%.

The baseline shares are one sixth for 99, one quarter for KQ, one third for AT and one quarter for AK. These differences come from the number of compatible physical pairings. Equal control values do not flatten those counts into four equal groups.

Averaging the four group equities of 0%, 0%, 100% and 50% gives 37.5%. That assigns each group a quarter of the distribution, a different weighting from our baseline. Using compatible matchup masses gives the requested 45.833333%.

Keep the weight convention with your note. The label AT 50 alone cannot distinguish a per-combination multiplier from a final group probability. Here it always means a whole-number multiplier from zero through 100.

Ace: "AT is at 50, so I wrote a fifty-percent AT range."

Aurora: "Write the multiplier and the final share separately. In this experiment the final AT share is twenty percent."

Change one group and keep the denominator

First set AT to zero and leave the other three groups at 100. You removed all of Hero's winning mass. There are now 384 compatible pairings with positive weight, including 144 ties. Hero receives half of those tied pots, giving 72/384 = 18.75% equity.

Restore the baseline, then set opposing AK to zero. The 144 tied pairings disappear. The remaining 432 contain 192 Hero wins, so equity is 192/432 = 4/9, about 44.444444%. The denominator changes along with the outcome mass you removed.

Removing the tying group lowers equity slightly because that group contributed a 50% pot share, above the baseline average of about 45.83%. A label such as "I removed a hand I do not beat" misses this comparison. Ask how the removed group's share compares with the original whole.

Now set AT to 50 while keeping the others at 100. In unit terms, winning mass is 96, losing mass is 240 and tying mass is 144. Total mass is 480. Hero's share is (96 + 72)/480 = 35%. Half the AT multiplier does not halve the whole result because the distribution is normalized again.

For another check, remove KQ alone. Wins and ties stay at 192 and 144, while losses drop to 96. Total mass becomes 432, and equity is 264/432 = 11/18, about 61.111111%. You changed the relative frequency of losing matchups without changing any showdown winner.

Use endpoints to test the interpretation

Give positive weight only to AT. Hero wins every compatible pairing, so the normalized equity is 100%. Give positive weight only to opposing AK and every remaining matchup ties, so each player receives a 50% pot share. Give positive weight only to 99 or KQ and Hero's equity is zero.

These endpoints describe restricted, authored ranges. They supply no evidence about a real opponent's composition or the profit from a betting action.

All four weights at zero is a different case. It leaves no positive joint mass, so there is no probability distribution to normalize. The tool reports an error and keeps the result empty. Zero equity would mean a valid distribution in which Hero receives no share; that is not what an empty range describes.

A zero weight removes a group's contribution without changing the underlying physical inventory. The receipt therefore distinguishes 576 compatible pairings before weights from the number of pairings with positive mass now. With only AT positive, those counts are 576 and 192 respectively.

Run the comparison and read its receipt

Enter the four whole-number weights and compare the ranges. The result separates outright wins, losses and ties, then gives pot-share equity. It includes each group's compatible count, input weight, weighted mass and normalized share so you can reconstruct the answer.

The receipt prints winning and tying masses over total mass. It also gives twice winning mass plus tying mass over twice total mass, keeping half-tie arithmetic exact before the percentage is rounded.

The change from baseline is in percentage points. If Hero moves from about 45.833333% to 35%, the change is about minus 10.833333 percentage points. A relative percentage change uses a different denominator and is not the quantity shown by this receipt.

Changing a field clears old feedback. Submit again for the new result. Reset restores all weights to 100 and focuses the first field. Blank, fractional and out-of-range values require correction.

Scale the weights without changing the experiment

Try all four weights at 50, then all four at 100. In the second run every legal pairing has twice the mass. Winning, losing, tying and total masses all double, so the normalized outcome shares stay the same. This is a useful check that the controls represent relative weights.

You can make the same comparison with unequal weights, such as 10, 20, 30 and 40 against 20, 40, 60 and 80 in the order 99, KQ, AT and AK. Keep the common multiplier the same for every group. Changing only one weight changes composition, which is the sensitivity experiment this lesson asks you to explain.

Keep conditioning visible

The model starts with independent per-hand weights for the two ranges, then conditions their product on physical compatibility. For a legal Hero hand h and opposing hand v, joint mass is weight(h) times weight(v). An illegal pairing has zero mass. Sum all legal mass before dividing any outcome total by it.

A different construction could choose a Hero hand first, independently renormalize the opposing range for that hand, and then average Hero hands equally. That procedure requires a stated sampling model. It can differ from normalizing one joint distribution when Hero hands face different amounts of opposing mass.

Our fixture has a convenient symmetry: every Hero AK combination faces six 99, nine KQ, 12 AT and nine AK combinations. With weights shared within each opposing group, every Hero hand therefore faces the same total mass. The two averaging approaches happen to agree here. Do not assume that agreement survives a different board or suit-specific weights.

This is why a range label and a headline percentage are incomplete study records. The board, exact combinations, weight convention and compatibility rule determine which joint distribution was used. Preserve them when comparing two experiments so a change can be traced to its actual input.

Ace: "The new result is higher. Have we found a better play?"

Aurora: "We changed the assumed range. Record which mass moved before asking a separate betting question."

Write the result the experiment supports

A useful note might read: "On Qh Js 8c 4d 2s, Hero's equal-weight AK range has 35% pot-share equity against the stated four-group range when AT has weight 50 and the others have weight 100. The change comes from reducing winning mass, then renormalizing all legal pairings." That sentence names the assumption and the consequence.

An actual opponent's composition remains unresolved, as does betting behavior. A call or bet question needs the relevant price, responses, investments and other assumptions. Exact showdown arithmetic cannot supply those inputs.

Use the exercise away from live decisions. Any use of software in a poker environment depends on that environment's current rules; this page does not determine permission. For adults of legal playing age. Keep gambling within affordable limits.

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