Ability Score Odds Calculator

Four six-sided dice keeping the best three is how a character has been rolled for fifty years, and almost everything said about the odds is guesswork. The exact distribution is not hard to compute — you enumerate each distinct set of faces once and weight it by how many orders produce it — and it settles the arguments. An eighteen is one roll in sixty-two, but across six scores it turns up about one character in ten.

Dice rolled

How many dice go into one score. Four is the usual number for a character's abilities.

Sides per die

Six for the classic roll. Anything up to twenty works, so house rules using other dice are covered.

Lowest dice dropped

How many of the worst dice are discarded before adding up. One is the standard rule; nought means every die counts.

Scores in the set

How many of these you roll for one character. Six is standard, and it is what makes an extreme score so much likelier than it looks.

Your result will appear here.

What this generator does

Enumerates every distinct set of faces the dice could show, weights each by the number of orders that produce it, keeps the highest dice and adds them up. That gives the exact probability of each score, and the set-level figures follow from it.

How to use this tool

Set how many dice you roll and how many sides they have.
Set how many of the lowest are dropped.
Set how many scores make up a set.
Read the exact chance of every score, and of a maximum somewhere in the set.

Understanding the controls

Dice rolled: How many dice go into a single score. Four is the usual number.
Sides per die: Six for the classic roll, though anything up to twenty is allowed for house rules.
Lowest dice dropped: How many of the worst dice are discarded before adding up. One is standard; nought means every die counts and gives the plain sum.
Scores in the set: How many of these rolls make up one character. Six is standard, and it is what makes an extreme score so much likelier than its single-roll odds suggest.

Common use cases

Settling how likely an eighteen really is
Comparing 4d6 drop lowest against 3d6 straight
Checking a house rule before adopting it, such as dropping two dice
Working out how good a rolled character is against the average
Explaining why extreme scores appear more often than the single-roll odds suggest

How this generator works

Rolling four dice gives 1,296 ordered outcomes, but only 126 distinct sets of faces — and each set has a known number of orders that produce it, given by the multinomial coefficient. Enumerating the sets rather than the orders makes the calculation exact and quick even for larger dice. The set-level figures follow: the chance of at least one maximum is one minus the chance of missing it every time, and the expected best and worst come from the cumulative curve raised to the number of scores. Before display the probabilities are summed and must come to one, the range is confirmed to span exactly the achievable scores, and for a small enough roll the whole distribution is checked again against a direct enumeration of every ordered outcome.

Randomness and fairness

Nothing here is random. This computes what dice would do rather than rolling them — the dice rollers are the ones that produce an actual result.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

Only dropping the lowest dice is modelled. Rules that reroll ones, or that let you swap dice between scores, change the distribution and are not covered.
Scores in a set are treated as independent, which they are when rolled honestly and are not under point-buy or array rules.
This gives the odds of a score, not what it is worth in play — a low score is only a problem for the abilities the character actually uses.
Very large dice and counts are capped so the enumeration stays quick in a browser.

Privacy and your data

The calculation runs in your browser and the settings you choose are never transmitted, stored or included in analytics.