From possible hands to weighted ranges.

After-play study. All hands, ranges and action frequencies below are fictional teaching inputs, not population data or solver output.

Count the hands. Then ask how they got here.

“They can have ace-king or a missed draw” is a start, but it is not yet a range model. How many physical combinations remain? Which of them took the earlier actions? How often does each bet this river? A list of possible hands becomes useful only when those questions are kept separate.

This workshop extends the viewer's introduction to ranges. The aim is not to assign a stranger a perfectly known strategy. It is to make a hand-review claim specific enough to examine and change.

A grid square is not a physical hand.

A hold'em starting hand contains two of 52 cards. There are 52 × 51 / 2 = 1,326 unordered combinations. The familiar 13-by-13 matrix groups them into 169 labels, but the labels do not have equal numbers of combinations.

A pair such as QQ has six combinations: choose two of four queens. A suited unpaired label such as KQs has four, one per suit. Its offsuit counterpart KQo has twelve. Across the full deck, that is 78 pair combinations, 312 suited combinations and 936 offsuit combinations. Their total is 1,326.

Counting selected matrix squares therefore does not give a range's percentage of dealt hands. Ten pair labels and ten suited labels cover different numbers of physical combinations. Once the board and your hand are known, count again rather than carrying the preflop totals forward unchanged.

Use a physical-card example.

You hold ace of spades and queen of clubs. The river board is ace of hearts, seven of diamonds, two of clubs, nine of spades and three of hearts. Your hand is a pair of aces with a queen kicker. Seven specific cards are known, so an opponent's two cards must come from the remaining 45, giving 45 × 44 / 2 = 990 possible combinations before considering their actions.

Combinations remaining after removing those seven known cards

Hand label	Before any removal	In this river example
AA	6	1
AK, all suits	16	8
AKs only	4	2
AKo only	12	6
KQs only	4	3

Only the ace of clubs and ace of diamonds remain, so AA has one combination. Each remaining ace can pair with four kings, giving eight AK combinations. Two are suited. The queen of clubs removes the clubs version of KQs, leaving three suited versions.

These are availability counts. They do not prove that all eight AK combinations would have played the hand this way or that any KQs combination would bluff. The board also does not reveal a folded opponent's cards. Remove only genuinely known cards, not a convenient guess about what someone probably folded.

A possible bluff is not a full-weight bluff.

For a deliberately narrow river model, suppose the opponent's only value bets are the one AA and eight AK combinations. Suppose the only candidate bluffs are the three remaining KQs combinations. Your hand loses to every value combination and beats every bluff, with no ties.

If all twelve combinations arrive and bet with equal weight, three out of twelve are bluffs: 25%. Facing a half-pot river bet, the call would be exactly break-even in the simple chip model because its threshold is also 25%.

More generally, assign a weight to each surviving combination and normalize by the total weight. The weight must represent how likely that combination is to reach the decision and take the observed action. If it already includes the earlier action history, do not multiply that history in again.

Blockers can remove good news as well as bad news.

Your ace removes value combinations. Your queen also removes one candidate KQs bluff. Calling a card “a blocker” says that it changes availability; it does not, by itself, say that the net change favors a call or a bluff.

Check the hands it removes from both sides of the proposed range. Then ask whether those hands actually take the relevant line. A card that removes three hypothetical bluffs matters little if those hands never bluff in the first place. Equally, an attractive narrative about blocking the nuts does not establish the opponent's remaining frequencies.

Three versions are more informative than one confident version.

Keep a conservative, central and more bluff-heavy version of the same clearly defined model. “Conservative” should describe an assumption, not be mistaken for a statistically proven bound. Record which hands or frequencies changed and whether your conclusion survived.

If a call is profitable only when every candidate missed draw bets, that dependence belongs in the review. If the conclusion survives several reasonable constructions, explain the common mechanism rather than displaying an artificially precise single estimate.

Make the next range claim auditable.

Write the known cards, value candidates, bluff candidates and the reasons each could reach this street. Cross out physically impossible combinations. Give the remaining candidates explicit frequencies, including uncertainty. Only then compare the weighted range with the call price.

A useful narration might say: “I can name three suited combinations that miss, but the important question is whether they actually bet. With half-weight bluffs, my simplified model is not close to the required frequency.” That is more honest and more interesting than announcing that an opponent “always has bluffs.”