Reading dice notation: NdS+M
Every combination of dice can be written the same short way: NdS+M. N is how many dice, S is how many sides each one has, and M is a flat modifier added on afterwards. "3d6+2" means roll three six-sided dice, add them together, then add 2. Drop the modifier entirely and "2d20" just means two twenty-sided dice, added together with nothing else applied.
The expected value formula comes from a single die first. A fair die numbered 1 to S lands on each face equally often, so its long-run average is just the midpoint of its range, (S+1)÷2. A six-sided die averages 3.5 over enough rolls, a twenty-sided die averages 10.5. Add several dice together and the averages add too, so N dice average N times that midpoint, and the modifier simply shifts the whole thing up or down by a fixed amount.
Why 7 turns up more than anything else on two dice
Roll one die and every face from 1 to 6 is equally likely, a flat line. Roll two dice and add them, and the flat line disappears: 7 becomes the single most common total, and the further a total sits from 7, the rarer it gets. This is the whole reason Catan places its best resource-producing tiles on the numbers around 7, and why a come-out roll of 7 in craps is both the most likely single number and the instant win for a pass-line bet.
The shape is simple combinatorics, not luck. There is only one way to make 2 (a 1 and a 1) and only one way to make 12 (a 6 and a 6), but there are six different ways to make 7: 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1. Order matters here because the two dice are separate objects, a red one and a blue one, even if they look identical, so "1 then 6" and "6 then 1" are two distinct outcomes out of the 36 equally likely pairs two six-sided dice can produce. Six of those 36 pairs sum to 7, giving a probability of 6/36, one in six, comfortably ahead of any other total. The distribution chart below draws this out for whatever number of dice and sides you have set above, whenever the pool is small enough to show clearly.
Physical dice bias, and why a good RNG beats them
A real die is a small manufactured object, and manufacturing is never perfect. Rounded corners, drilled-and-filled pips that leave one face lighter than its opposite, and tiny asymmetries in an injection mould all nudge a physical die away from a true 1-in-S chance per face, usually only by a fraction of a percent, but casinos still weigh and inspect their dice for exactly this reason. A cheap die bought in bulk carries more of this bias than a precision-cut casino die, and no ordinary player can tell the difference by eye or by feel.
This calculator has no manufacturing tolerances to worry about. Each die is drawn with the browser's cryptographic random number generator rather than the more familiar everyday one, and a rejection-sampling step discards any raw value that would make one face even fractionally more likely than another through the classic modulo-bias trap, so every side genuinely gets an equal share, not just an approximately equal one.
Advantage and disadvantage, explained with the numbers
Advantage, the mechanic used across modern tabletop games, means rolling two d20s and keeping the higher one; disadvantage means the same two rolls but keeping the lower. Neither changes what a single d20 can land on, still 1 to 20, but both shift the average a long way from the ordinary 10.5, because keeping "the better of two" pulls the result toward the top of the range, and keeping "the worse of two" pulls it toward the bottom.
Worked out exactly rather than estimated, the average result with advantage is 13.825, and with disadvantage it is 7.175, the two figures sitting the same distance either side of 10.5. That is a swing of over 3 points in either direction from a single ordinary roll, purely from choosing which of two rolls to keep, which is why advantage and disadvantage are treated as a meaningfully large edge at the table rather than a cosmetic flourish.
Expected value is not a prediction
The expected value shown for your current settings is a long-run average, not a forecast for the next roll. Set 2d6 and the expected value is exactly 7, yet 7 comes up on only 6 of the 36 equally likely pairs, meaning roughly five rolls out of six land somewhere else entirely. Expected value only becomes a good description of what actually happens after enough rolls to smooth out the individual ups and downs, which is also why the session log below tracks your running average across the rolls you have actually made, alongside the theoretical figure above it.
Questions people ask
Is this dice roller actually fair?
As fair as software gets. Each die uses the browser's cryptographic random number generator rather than a simple pseudo-random function, and a rejection-sampling step throws away any draw that would otherwise make one face slightly more likely than another. No physical manufacturing tolerances apply, unlike an actual die, so every side has an equal share of the outcomes.
Why does the number of dice max out at 20?
Twenty covers every common tabletop use, from a single ability check to a big pool of damage dice, while keeping the results readable as a row of individual chips rather than an unreadable wall of numbers.
Why does the probability chart sometimes disappear?
An exact probability distribution needs one bar per possible total, and that count grows fast: two d6 need 11 bars, but two d100 would need 199. Past a reasonable number of bars the chart stops being readable, so it is hidden rather than drawn too small to read, and the note above it says so.
Can the modifier be negative?
Yes. A negative modifier is exactly how a penalty is written in dice notation, such as a d20−1 attack roll with a cursed weapon, and it shifts the minimum, maximum and expected value down by the same amount it would otherwise shift them up.
What happens if I type a custom number of sides below 2?
It gets held at 2, the smallest number of sides a die can meaningfully have. A one-sided die cannot land on anything else, so there is nothing left to roll.
Does the session log save between visits?
It saves in this browser only, in local storage on this device, up to the most recent 20 rolls. Clearing it, clearing your browser data, or opening the page in a different browser or device starts a fresh log.
Why does the total not update while I am still setting up the dice?
Rolling is the one genuinely random step on this whole site, so it only happens when you actually ask for it by pressing Roll. The expected value, minimum, maximum and probability chart above the log update immediately as you change the dice, since those are worked out from the settings alone and involve no randomness at all.
This calculator generates dice rolls for games, teaching probability and general curiosity. It has no connection to any particular game's rules and is not intended for wagering.