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Dice probability: the odds for 2d6, 3d6 and d20 rolls

By TabBenchHow we check our guides

A single fair die is easy: every face is equally likely. The moment you add dice together, the odds change shape, and the middle totals become far more common than the ends. That is why 7 dominates board games with two dice, why ability scores cluster around 10 and 11, and why advantage on a d20 is worth so much.

This guide gives the exact probabilities for the most common rolls. Each figure is computed by counting every possible outcome, so you can use them to plan a game, set a difficulty or simply settle an argument.

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Step by step

  1. Two six-sided dice (2d6)

    There are 36 equally likely outcomes. A total of 7 can be made six ways (1+6, 2+5, 3+4 and their reverses), so it comes up 6 in 36 (16.7%). 6 and 8 each come up 5 in 36 (13.9%), 5 and 9 four in 36 (11.1%), 4 and 10 three in 36 (8.3%), 3 and 11 two in 36 (5.6%), and 2 and 12 only once in 36 (2.8%). The chance of rolling doubles is 1 in 6, and of at least one six is 11 in 36 (30.6%).

  2. Three six-sided dice (3d6)

    There are 216 outcomes. The most likely totals are 10 and 11, each 27 in 216 (12.5%). The extremes, 3 and 18, are 1 in 216 (0.46%). Rolling 15 or higher happens in 20 of 216 rolls (9.3%), and 16 or higher in 10 of 216 (4.6%). The curve is close to a bell shape centred on 10.5.

  3. A single d20 and advantage

    With one d20 each number has a 5% chance, so a target of 11 or higher succeeds half the time. Rolling two d20 and keeping the higher (advantage) gives 1 − (1 − p)² where p is the single-roll chance: 50% becomes 75%, 30% becomes 51%, and 5% becomes 9.75%. Disadvantage keeps the lower and gives p²: 50% becomes 25%, 30% becomes 9%, and 5% becomes 0.25%. The biggest swing, 25 percentage points either way, is for a target of 11.

  4. Rolling ability scores: 4d6 drop the lowest

    Rolling four six-sided dice and discarding the lowest raises the average of the three kept dice from 10.5 to about 12.24. The chance that such a score is 15 or higher is about 23.1%, compared with 9.3% for straight 3d6.

  5. Use the roller to check your own odds

    In the Dice Roller, roll any combination and the chart shows the exact probability of each total for ordinary dice rolls. For notation with keep-highest, keep-lowest or exploding dice, roll it many times and watch how the results cluster, or work the odds out with the same counting approach: list the equally likely outcomes and count the ones you care about.

Things worth knowing

  • Dice have no memory: a number that has not come up for ten rolls is no more likely next time.
  • Adding dice narrows the spread. The standard deviation of nd6 grows with the square root of n, so the average is more predictable than a single die.
  • A modifier shifts every outcome equally. It does not change the shape of the distribution.
  • Exploding dice have no maximum: a six is rolled again and added, so totals above the nominal range are possible.

Frequently asked questions

Why is 7 the most common roll with two dice?

More combinations add up to 7 than to any other total: six of the 36 equally likely pairs. Totals near the ends have few combinations, so they are rare.

What is the probability of rolling a 6 on at least one of two dice?

11 in 36, about 30.6%. It is easier to count the complement: 25 of 36 outcomes have no six, so 1 − 25/36 = 11/36.

Is advantage or +5 better on a d20?

It depends on the target. Advantage is worth up to about +5 on the die at the midpoint and much less near the ends. A flat +5 is worth exactly 25 percentage points until the target leaves the die's range.

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