Capped and uncapped ranges

A capped range has an upper limit on the hand strength represented in a particular model. An uncapped range retains the strongest relevant possibilities. The useful part of either label is the explanation behind it: which holdings remain, which have been removed, and why that removal is justified on this board.

A check does not reveal a player's cards. If you remove every strong hand after a check, you have supplied an assumption about how that player acts. This lesson makes that assumption visible. Its exercise uses five exact river holdings, assigned retention weights and one optional known unavailable card. It reports a model ceiling, not a betting recommendation.

Ace: "They checked. Should I put a cover over the top shelf?"

Celeste: "Tell me why those hands leave the range first. The check did not remove any cards."

Name the ceiling and its reference

A ceiling describes the strongest holding still represented. It needs a reference point. A range might lack quads while retaining full houses, or lack full houses while retaining strong flushes. Calling either range capped says little until the board and the proposed upper limit are stated. A lower ceiling does not make every remaining hand weak.

The word uncapped also needs context. It can mean that the strongest relevant holdings remain possible, without saying they are common. One very lightly weighted strong holding can preserve a model's maximum strength while contributing almost nothing to its total weight. Presence and frequency answer different questions.

Poker discussion sometimes uses capped loosely for a range with very few strong hands. This exercise uses a stricter test: a listed candidate is represented whenever its legal retained weight is positive. Zero means absent. That convention lets you see the difference between removing a hand and merely reducing its frequency without arguing over an informal label.

Separate a known card from a behavioral read

If a specific card is visibly unavailable, an opponent cannot hold that card. This is a physical exclusion. If you believe a player always raises a certain hand, removing that hand from a calling range is a behavioral exclusion. The two claims require different evidence. A visible card establishes availability; an action alone does not establish an always-or-never strategy.

For example, a player may check a strong hand to induce a bet, control the betting sequence, or follow a mixed strategy. Those possibilities do not prove that any particular strong hand is present. They show why the step from checked to never strong needs support. The exercise's action dropdown deliberately leaves every weight unchanged.

A useful study note can say, "Assume this line retains no quads." You can then inspect the consequences and compare a version that retains some quads. That is more precise than presenting the same exclusion as a fact about an unidentified opponent. Keep the range model and the observation that motivated it on separate lines.

Fix the board before ranking the hands

The exercise uses this Hold'em river: king of spades, eight of diamonds, four of clubs, two of hearts and two of clubs. The five board cards are distinct. The paired twos matter to every candidate's best five-card hand. There is no future card inside this river exercise, and the component does not accept a different board.

The strongest hand available with no additional known cards is four twos, made with two of spades and two of diamonds. A full house loses to quads. Only the twos are paired on this board, so another pocket pair cannot make four of a kind with it. The board also cannot support a five-card flush or straight flush using just two hole cards.

If the board changes, re-evaluate the holdings before reusing a ceiling. A pocket pair that makes a full house here might make a set, two pair, or only one pair elsewhere. A range label recorded on the turn is not a permanent property of its hole cards. The river card can change both the strength order and the relevant comparison.

Inspect the entire toy inventory

There are exactly five listed alternatives: two of spades with two of diamonds; eight of clubs with eight of hearts; ace of spades with ace of hearts; king of hearts with queen of hearts; and ace of diamonds with queen of diamonds. Each is one exact two-card holding. None of these entries stands for every suit combination of that starting hand. They are alternatives for one player, not five simultaneous opponents.

On the fixed board, the first holding makes four twos. The second makes eights full of twos. Pocket aces makes two pair, aces and twos; king-queen makes two pair, kings and twos. Ace-queen has the board's pair of twos with ace, king and queen kickers. That is the strongest-to-weakest order used by the desk.

The inventory is deliberately incomplete. Pocket kings, for example, makes kings full of twos and beats the listed eights full of twos. It has been left out to keep the exercise small, not because a real player's line excludes it. The tool therefore cannot promote its strongest remaining example into a claim about every legal holding.

Read a retention weight as an assumption

A retention weight of 100% gives a listed holding one full unit of model mass. A weight of 50% gives it half a unit. Zero excludes it. Every candidate starts as one exact combination, so there is no hidden multiplier for suited, offsuit or paired groups. The weights describe how much of each candidate you choose to retain after the supposed line.

The five weights do not need to add to 100. They are separate retention percentages applied before normalization. Keeping all five at 100% produces five candidate-equivalents of mass, not an impossible 500% chance of being dealt a hand. The normalized shares divide each retained mass by the total retained mass.

Remove the top holding and compare the result

With no additional known card and every weight at 100%, the total mass is 5. Four twos has mass 1, so its normalized share is 1/5, or 20%. Every other listed candidate also has a 20% share. This equal weighting is a convenient starting assumption. It is not a default model of how opponents reach the river.

Select Exclude quads. Only the two-spades/two-diamonds retention changes to zero. The total mass becomes 4, and eights full of twos is the strongest listed holding. Its normalized share becomes 1/4, or 25%. The strongest-group share increased even though the represented ceiling became weaker. Those two changes are entirely consistent.

The statement supported by this result is narrow: under the supplied weights, the original inventory's quads candidate is absent. It does not prove that an opponent cannot have kings full, another full house, or any legal hand omitted from the inventory. A larger real range needs its own complete candidate list and justified exclusions.

Ace: "I left just a tiny piece of weight on the top holding. Is the shelf empty now?"

Celeste: "It still has a possible holding. Write down how little weight it has as well as how strong it is."

A small positive weight preserves possibility

Select Quads at 1%. The quads candidate now contributes 0.01 units while each of the other four contributes 1. The total is 4.01. Its normalized share is 0.01/4.01, about 0.249377%. The original inventory ceiling is represented again, although very little of this model's mass sits there.

At 0% the candidate is absent; at 1% it is present. The interface keeps that difference visible. It does not round the positive mass into an empty shelf. A one-percent retention weight is also different from a one-percent normalized share. The denominator includes the mass of every other retained legal candidate.

If an analysis depends on how often the strongest hands occur, carry the mass or normalized share into that analysis. The word uncapped cannot replace those numbers. Conversely, a high percentage in the strongest remaining bucket does not establish a high absolute ceiling. Both the identity of the bucket and its share need to stay in the record.

Test the stated model

Start with all five retained, exclude quads, then restore it at 1%. Compare the named ceiling with the normalized shares. Next, change only the action label to Checked the river or Called an earlier bet. The output will identify the label, but the arithmetic will remain the same. You have supplied no new retention weights by changing a description.

The weights accept whole percentages from 0 through 100. Blank fields are errors when you choose assigned weights. If every available candidate has zero retained weight, the model is empty and there is nothing to rank or normalize. The desk asks for a positive available candidate instead of dividing by zero or inventing a range ceiling.

Known cards can remove a candidate

Reset the inventory and choose two of spades as an additional known unavailable card. The quads holding contains that card, so it is removed regardless of its retention weight. The other four examples remain available. With their weights at 100%, the listed mass becomes 4 and the listed ceiling is eights full of twos.

That exclusion has a different basis from setting the quads weight to zero. One follows from the supplied card inventory; the other follows from a chosen strategy assumption. The same numerical result can have different evidential strength. Record which kind of exclusion produced it instead of treating all zeroes as interchangeable.

The full legal comparison also changes when two of spades is unavailable. Four twos is then impossible, and pocket kings can make the strongest available full house on this board. Pocket kings is still outside our five-candidate inventory. This is why even a correctly applied blocker does not make an incomplete model into a complete opponent range.

The dropdown permits only one additional known card at a time. It is not a full hero-hand entry or a general dead-card calculator. No hero hole cards are otherwise supplied. If you know more cards than the exercise accepts, use a fuller inventory before drawing conclusions about a real hand. The desk does not silently fill in the missing cards.

Unknown assumptions should stay unknown

Choose Unknown assumptions when you have not assigned a retention model. The desk can still report which listed holdings are physically available under the known-card selection. It withholds a ceiling conclusion and normalized shares because availability does not tell us which hands followed a particular line or how often they did so.

Unknown is different from assigning all weights to zero. An all-zero assigned model says no available candidate survives your exclusions. Unknown says the retention assumptions have not been established. Replacing uncertainty with zero would create an artificial cap. Replacing it with full weight would create an equally unsupported frequency model.

Ace: "This candidate disappeared when I marked a card unavailable. Did the river check do that?"

Celeste: "The known card did. Keep a physical exclusion separate from the story about the action."

Build a defensible range note

Write the game, current street and exact board first. Add every known unavailable card. Then list the candidate holdings or groups and the assumptions that retain them. If you use groups, specify their physical combination counts and whether different suits get different weights. A hand-category name on its own can hide substantial differences.

Attach each behavioral exclusion to its reason. For example, a study assumption might say that an exact strong holding always takes an earlier raising branch. Then test a second model where it sometimes calls instead. Compare the ceilings and the amount of top-end mass. If a conclusion depends entirely on an absolute zero, that assumption deserves a careful review.

Keep the ceiling separate from a decision

The desk supplies no hero hand, pot size, price, available bet sizes, opponent response model or expected-value calculation. Those missing inputs prevent it from selecting an action. A lower range ceiling alone does not show that a bluff succeeds, that a value bet gets called, or that any particular size earns more than checking.

For a real review, compare alternative range assumptions before considering an action. Keep the board, card exclusions, included holdings and retained mass visible. If the data do not support removing the strongest candidates, leave that uncertainty in the note. The purpose is to make a range claim inspectable, not to make an opponent's cards seem known.

Questions about capped ranges

Can a strong hand remain after a check or call?

Yes. Whether it remains in a particular model depends on the strategy assumptions and the evidence for them. A check or call does not physically prevent someone from holding a strong legal hand. The exercise requires you to change the retention weights explicitly rather than hiding that judgment inside the action label.

Is the strongest listed holding always the nuts?

No. The nuts require a comparison against all relevant legal hands on the current board with the known cards removed. The strongest listed holding is only the maximum inside the supplied inventory. In this exercise, eights full can be the listed maximum while a legal pocket-king combination outside the inventory would beat it.

Do these normalized shares estimate winning chances?

No. They divide the assigned retained mass among five candidate alternatives. No hero holding is being compared and no deal is simulated. The numerical output checks the model you entered. Its usefulness comes from showing exactly which assumptions create the ceiling and how that ceiling changes when those assumptions change.

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