6 Sided Dice Roller Calculator

Enter how many six-sided dice you roll and a target total. Get the exact probability of hitting that total, of rolling it or less, or it or more — plus the mean and standard deviation of the sum.

Quick Facts

Method
Exact enumeration of all 6ᴺ outcomes
Each die is independent and uniform on 1–6; mean = 3.5N, variance = 35N/12.

Your Results

Calculated
P(sum = target)
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Exactly this total
P(sum ≤ target)
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This total or less
P(sum ≥ target)
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This total or more
Mean ± SD of sum
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Expected total and spread

Ready

Enter a dice count and target sum, then calculate.

Understanding the 6-sided dice roller

A standard six-sided die (a "d6") lands on each face 1 through 6 with equal probability 1/6. When you roll several dice and add them up, the individual uniform distributions combine into a bell-shaped distribution over the possible totals. This calculator enumerates every possible outcome exactly to tell you how likely any target sum is — no simulation, no approximation.

How the probability is computed

Rolling N dice produces 6N equally likely ordered outcomes (36 for two dice, 216 for three, and so on). The probability of a given total s is simply the number of outcomes that add up to s, divided by 6N:

  • P(sum = s) = (ways to make s) / 6N
  • P(sum ≤ s) and P(sum ≥ s) add up the ways for every total on that side of s.

The "ways to make s" are found by convolution: start with one die, then repeatedly combine the running distribution with another die's 1–6 faces. For two dice the counts form a symmetric triangle — 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 for totals 2 through 12 — so a 7 (six ways) is exactly six times as likely as a 2 or a 12 (one way each).

Mean and spread of the total

  • Mean: one die averages (1+2+3+4+5+6)/6 = 3.5, so N dice average 3.5 × N.
  • Variance: one die has variance 35/12 ≈ 2.9167, so N dice have variance 35N/12 and standard deviation √(35N/12) ≈ 1.708 × √N.

So two dice average 7 with an SD of about 2.42; three dice average 10.5 (SD ≈ 2.96); four dice average 14 (SD ≈ 3.42).

Common reference values (two dice)

  • P(2) = P(12) = 1/36 ≈ 2.78%
  • P(3) = P(11) = 2/36 ≈ 5.56%
  • P(7) = 6/36 ≈ 16.67% (the most likely total)
  • P(sum ≥ 10) = 6/36 ≈ 16.67%; P(sum ≤ 4) = 6/36 ≈ 16.67%

Frequently Asked Questions

What is the probability of rolling a 7 with two dice?
Six of the 36 equally likely outcomes total 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so P(7) = 6/36 = 1/6 ≈ 16.67%. Seven is the single most likely total with two dice because it has more combinations than any other sum.
Why is 7 more likely than 2 or 12?
There is only one way to roll a 2 (1+1) and one way to roll a 12 (6+6), but six different ways to roll a 7. Since every ordered outcome is equally likely, totals with more combinations are more probable, which is why the distribution peaks in the middle.
What is the average total when rolling N dice?
Each fair d6 averages 3.5, so the sum of N dice averages 3.5 × N: 7 for two dice, 10.5 for three, 14 for four. The standard deviation of the total is √(35N/12) ≈ 1.708 × √N.
Does this calculator simulate rolls or compute exact odds?
It computes exact probabilities by enumerating all 6N outcomes via convolution — it does not use random simulation. The results are the true theoretical odds, not a sample estimate, so they are identical every time you run them.

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