From a hand to a range
Beginners often try to put an opponent on one exact hand. A more useful idea is a range: the whole set of hands that opponent could plausibly hold given what they have done so far, each with its own weight. Thinking in ranges is a probability exercise. You are not guessing a card; you are describing a distribution over possible holdings and updating it as new information arrives.
This article covers the arithmetic only. It is not advice on how to play, and it cannot promise any result.
Counting hand combinations
In Texas hold’em each player starts with two cards, and there are C(52, 2) = 1,326 possible two-card holdings. Ranges are usually described by hand type (such as a pair of kings or ace-king), so the first skill is to count how many specific card combinations, or combos, each type contains.
- Pocket pairs: choose 2 of the 4 cards of a rank, which is 6 combos for each pair.
- Suited hands with two different ranks: 4 combos (one for each suit).
- Offsuit hands with two different ranks: 4 × 3 = 12 combos.
As a check: there are 13 pairs, giving 13 × 6 = 78 combos. There are 78 pairs of different ranks (13 × 12 / 2), giving 78 × 4 = 312 suited and 78 × 12 = 936 offsuit combos. The total is 78 + 312 + 936 = 1,326, which matches the direct count. That agreement is a useful sanity check whenever you build a table of combos.
Combos matter because hand types are not equally common. Ace-king in any suits has 16 combos (4 suited and 12 offsuit), while a specific pair like aces has only 6. A range that contains many pairs and few unpaired hands has a different shape from one that is the reverse, even if both are written as “strong hands.”
Blockers: your cards change their combos
The cards in your own hand cannot also be in your opponent’s. This is the idea of blockers, and it is an application of counting without replacement.
Suppose you hold one ace. Then 3 aces remain unseen, so the number of ways an opponent can hold a pair of aces falls from 6 to C(3, 2) = 3. Their ace-king combos fall from 16 to 3 aces × 4 kings = 12. Holding an ace does not reveal their hand, but it quietly shifts the weights within their range. Reasoning about hidden information this way is mostly careful bookkeeping.
If you hold both a king and an ace, the effect is stronger still. The general point is that you should count the combos that remain after removing the cards you can see, not the combos from a full deck.
Weighting a range
Once combos are counted, a range is a list of hand types with weights. One simple way to describe the chance that an opponent holds a given type is the combos of that type divided by all combos in the range.
Illustration (numbers invented for the example): a range with 6 combos of one pair, 12 of one offsuit hand and 6 of another pair has 24 combos in total. The first pair makes up 6/24 = 25% of the range, the offsuit hand 12/24 = 50%, the second pair 25%. Each number comes from counting, not from feeling.
Bayesian updating
A range is not fixed. Each action an opponent takes is evidence, and the right way to revise the range is Bayes theorem, which is built on conditional probability. The prior is the chance a hand belongs to a category before the action; the likelihood is how often that category would produce the action; the posterior is the revised estimate.
Illustrative example with invented numbers: suppose you think an opponent’s range is 25% strong hands and 75% weak hands. You further assume strong hands take an aggressive action 80% of the time and weak hands 20% of the time.
- Strong and aggressive: 0.25 × 0.80 = 0.20
- Weak and aggressive: 0.75 × 0.20 = 0.15
- Probability of the aggressive action overall, adding the two cases as the law of total probability does: 0.20 + 0.15 = 0.35
- Posterior chance of a strong hand given the action: 0.20 / 0.35 = 4/7, about 57.1%
The strong share rose from 25% to about 57%, but notice it did not become certain. Half of the work is in the inputs: if the assumed percentages are wrong, the output is wrong. In real situations nobody knows these likelihoods exactly, so the figures are estimates, and a precise-looking posterior can hide a very uncertain starting point.
The common error here is base rate neglect: forgetting how common each category was before the action. If strong hands are rare, even a very aggressive action may be more likely to come from the many weak hands.
Limits of range maths
- Inputs are assumptions. Combos can be counted exactly, but how often someone takes an action with each hand is an estimate.
- Players change their behaviour. A distribution you assumed in one situation may not describe the next.
- Probability does not remove luck. Even a correct estimate describes long-run frequencies. A single hand can go either way.
- Results stay uncertain. Understanding ranges helps avoid arithmetic mistakes and does not guarantee winnings. Playing for money carries real risk of loss, and anyone who is struggling with gambling should contact a local support service.
Used carefully, combo counting and Bayesian updating are a way to keep your reasoning consistent, and they make clear how much of any conclusion depends on assumptions.