Why every face on a die is equally likely
A fair die with d faces is what statisticians call a discrete uniform distribution: each of the d outcomes has exactly the same 1/d chance, with no face favored over another. For that distribution, the mean is (d + 1) / 2 and the variance is (d² − 1) / 12 (Weisstein, "Discrete Uniform Distribution," Wolfram MathWorld). A d6 averages 3.5 (there is no face showing 3.5, but that is where repeated rolls center), a d20 averages 10.5, and a d100 averages 50.5.
That single-die formula is the building block for everything else on this page. Roll more than one die and add them up, and the total's average and spread follow directly from it: for n dice of d faces, the expected total is n × (d + 1) / 2 and its variance is n × (d² − 1) / 12, because expectation and variance both add across independent draws.
Summing dice: why 7 is the most common roll on two d6s
A single d6 roll is uniform, every face equally likely, but the sum of two d6 rolls is not, because there are more ways to add up to some totals than others. There is only one combination that makes 2 (a 1 and a 1) and only one that makes 12 (a 6 and a 6), but six different combinations make 7 (1-6, 2-5, 3-4, 4-3, 5-2, 6-1). Out of the 36 equally likely combinations two dice can land on, that puts 7 at 16.67% and pushes 2 and 12 down to 2.78% each.
This tool computes that kind of distribution exactly, for any number of dice and any die size, by convolution: start with one die's uniform distribution, then repeatedly combine it with another copy of itself, once per additional die. Each combination step is exact because every die is an independent draw from the same distribution, so the result is the real probability, not a simulated approximation from sampling.
Advantage and disadvantage: what rolling twice actually buys you
D&D 5th Edition's System Reference Document 5.1 states the rule directly: "you roll a second d20 when you make the roll. Use the higher of the two rolls if you have advantage, and use the lower roll if you have disadvantage" (p.76). It is specific to a single d20 check, save, or attack roll, not to dice pools in general, which is why this tool only offers it when the dice type is set to d20.
The effect on your odds grows as the target number gets harder to hit. Needing a 15 or better on a plain d20 is a 30% shot (six faces out of twenty clear it). Roll with advantage and the odds rise to 51%, because the only way to miss is for both d20s to come up under 15: squaring the 70% miss chance gives 49%, so the hit chance is 1 − 0.49 = 51%, not the 60% a naive "just double it" guess would give. Disadvantage cuts the other way: squaring the 30% hit chance itself drops it to 9%, since now both rolls need to clear the bar.
Worked example: a +3 attack against DC (or AC) 15
A d20 roll with a +3 modifier needs a raw roll of 12 or higher to reach a total of 15, since 12 + 3 = 15. On a plain roll, that's 9 faces out of 20 that clear 12, for 45%. With advantage, the chance of both rolls landing under 12 is (11/20)², so the chance of at least one clearing it is 1 − 0.3025 = 69.75%. With disadvantage, both rolls need to clear 12, which is (9/20)² = 20.25%.
The same math works without a d20 or a fixed +3: entering any dice count, die size, modifier, and target into the probability calculator below runs the identical convolution and reads off the exact chance for that combination, for example a 3d6 damage roll with a +2 modifier reaching 15 total comes out to 7/27, about 25.93%, a number the calculator will confirm instantly rather than requiring you to work the fractions by hand.