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. Including this factor ensures the calculation accurately reflects real-world conditions rather than idealized assumptions. Ignoring it can introduce systematic error that compounds with other inputs, potentially leading to incorrect sizing or planning decisions. The calculator exposes this parameter so your result is meaningful for practical use rather than purely theoretical estimation.
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.

Common Tip Percentages

ServiceTypical Tip RangeNotes
Restaurant (sit-down)15–20%20%+ for exceptional service
Food delivery10–20%Consider extra for long distances
Taxi / rideshare10–20%More for assistance with luggage
Hotel housekeeping$2–5/nightLeave daily, not just at checkout
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General Household Budget Ratios (50/30/20 Rule)

Category% of Net IncomeExamples
Needs50%Rent, groceries, utilities, insurance
Wants30%Dining out, entertainment, hobbies
Savings / Debt20%Emergency fund, retirement, extra payments

Common Measurement Conversions for Everyday Use

QuantityUS / ImperialMetric Equivalent
1 cup8 fl oz237 mL
1 pint16 fl oz473 mL
1 quart32 fl oz946 mL
1 gallon128 fl oz3.785 L
1 lb16 oz453.6 g
1 ft12 in30.48 cm

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