Dice Roller

What each roll actually does

Computed when this page was built, using the same engine the roller runs on. Every figure here is exact rather than sampled.

• 1d20 1 to 20 · average 10.50 · most likely 1 or 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9 or 10 or 11 or 12 or 13 or 14 or 15 or 16 or 17 or 18 or 19 or 20 A single twenty-sided roll. The highest possible total comes up about 5.0% of the time.
• 2d20kh1 1 to 20 · average 13.83 · most likely 20 Advantage: two d20, keep the higher. The highest possible total comes up about 9.8% of the time.
• 2d20kl1 1 to 20 · average 7.18 · most likely 1 Disadvantage: two d20, keep the lower. The highest possible total comes up about 1 in 400 of the time.
• 4d6dl1 3 to 18 · average 12.24 · most likely 13 Ability score: four d6, drop the lowest. The highest possible total comes up about 1.6% of the time.
• 3d6 3 to 18 · average 10.50 · most likely 10 or 11 Three six-sided dice added. The highest possible total comes up about 1 in 216 of the time.
• 2d6 2 to 12 · average 7.00 · most likely 7 The classic board-game roll. The highest possible total comes up about 2.8% of the time.
• 8d6 8 to 48 · average 28.00 · most likely 28 A big damage roll. The highest possible total comes up about 1 in 1,679,616 of the time.
• 1d8+1d6+3 5 to 17 · average 11.00 · most likely 10 or 11 or 12 Mixed dice with a modifier. The highest possible total comes up about 2.1% of the time.

Reading dice notation

Dice notation looks cryptic and takes about a minute to learn. It is worth learning, because once you can write a roll down you can type it here in one go instead of adding numbers by hand.

The core of it is XdY: X dice with Y sides each. So 3d6 is three six-sided dice, 1d20 is a single twenty-sided die, and d20 means the same thing because the 1 is assumed. Add a modifier with plus or minus: 1d8+3 rolls a d8 and adds three. Mix dice freely, as in 1d8+1d6+3. Percentile dice are written d%, which is just a hundred-sided die.

The interesting part is keep and drop, and it is where most online rollers give up:

• kh — keep highest. 2d20kh1 rolls two d20 and keeps the better one. This is advantage.
• kl — keep lowest. 2d20kl1 is disadvantage.
• dl — drop lowest. 4d6dl1 rolls four d6 and throws away the worst.
• dh — drop highest. Rarer, but it appears in some systems as a penalty.

Keep and drop are two ways of saying the same thing: 4d6dl1 and 4d6kh3 produce identical results. Use whichever you find easier to read. The number after the letters is how many dice are affected, and it defaults to one, so 2d20kh and 2d20kh1 are the same roll.

Why 3d6 is nothing like a single d18

Both range from 3 to 18. That is where the similarity ends, and the difference is the single most useful thing to understand about dice.

A single die is flat. Every face is equally likely, so on a d18 each of the eighteen outcomes turns up about 5.6% of the time. Rolling a 1 is exactly as likely as rolling a 10.

Add dice together and that stops being true, because totals in the middle have more ways of happening. On 3d6, a total of 3 requires all three dice to show 1 — one combination out of 216. A total of 10 can be made dozens of ways. The result is that 10 and 11 each come up 12.5% of the time while 3 and 18 each come up under half a percent. Put differently: on 3d6 you are twenty-five times more likely to roll a 10 than a 3, and on a d18 those two are equally likely.

This is why the choice of dice is a design decision rather than a formality. Flat dice produce swingy, unpredictable results where anything can happen on any roll. Multiple dice produce reliable results that cluster around the average and only rarely go to extremes. A game built on d20 rolls feels dramatically different from one built on 3d6, and the difference is entirely in the shape of the curve.

The chart above the fold shows this shape for whatever you have rolled. Two dice give a triangle. Three or more give something close to a bell. One die gives a flat line.

What advantage is really worth

Roll two d20 and keep the higher. The average goes from 10.5 to 13.825, an improvement of about 3.3 — and quoting that number is the most common way to misunderstand what advantage does.

The average is not what you care about. What you care about is whether you hit the number you need, and advantage helps enormously in the middle of the range and barely at all at the edges:

• You need a 5 or better. 80% becomes 96%. Useful, but you were probably going to make it.
• You need an 11 or better. 50% becomes 75%. This is the maximum benefit advantage can ever give you.
• You need a 15 or better. 30% becomes 51%.
• You need a natural 20. 5% becomes 9.75%. Nearly double, but still rare.

The pattern is a rule of thumb worth carrying: advantage is worth most when the roll is close to even, and worth progressively less as the task becomes very easy or very hard. If you can choose when to spend an effect that grants advantage, spend it on the roll you have roughly a coin-flip chance of making.

Disadvantage is the mirror image, and it is brutal. Needing a 15 with disadvantage drops you from 30% to 9% — you will fail nine times out of ten. Advantage and disadvantage are symmetric in average (13.825 and 7.175 sum to 21, exactly twice the ordinary 10.5), which is a neat way to remember that they cancel out.

Rolling ability scores: 4d6 drop lowest

The traditional method is 3d6, which averages 10.5 and produces a lot of mediocre characters. Rolling four dice and discarding the worst — 4d6dl1 — is the modern standard, and the difference is larger than it looks:

• The average rises from 10.50 to 12.24.
• The most likely single result moves from 10 or 11 to 13.
• The chance of a score of 15 or better goes from about 9% to about 23%.
• The chance of an 18 goes from 1 in 216 to roughly 1 in 62.
• The chance of a 3 collapses from 1 in 216 to 1 in 1,296, because now all four dice have to fail.

Both methods share the same 3-to-18 range, so the extremes are unchanged. What changes is everything in between. Drop-lowest does not raise the ceiling, it raises the floor and the middle, which is exactly what most tables want: fewer characters who are bad at everything, without producing superheroes.

If you are rolling a full set of six scores, remember that each is an independent roll. The chance of at least one 18 across six scores is far higher than the chance on any single one — which is why an 18 somewhere in the array is common even though an 18 on a given roll is not.

Where the probability curve comes from

Most dice tools that show odds at all get them by rolling a very large number of times and counting the results. That is easy to build and always slightly wrong, and it gives a different answer every time you reload.

The figures here are exact. For ordinary dice the tool combines the distributions mathematically, one die at a time, which gives the true probability of every possible total. For keep-and-drop rolls the calculation is harder — you cannot simply add distributions together when the dice affect each other — so it works through the dice in rank order, tracking how much of the kept total has accumulated. Both approaches produce the true answer rather than an estimate.

That is why the average for 4d6dl1 shows as a stable 12.24 and not a number that shifts on each visit. The exact value is 15869/1296.

There is a limit. Above a few thousand possible totals the computation stops being instant, so very large rolls skip the curve and show the total, the dice and the range instead. Everything anyone actually rolls sits comfortably inside that limit.

Digital dice versus the ones on your table

A fair physical die is harder to make than it sounds. Ordinary dice have rounded corners for rolling, pips drilled into the faces and then filled, and small density variations from moulding — all of which introduce bias. Casino dice are made with sharp edges, flush pips and tight tolerances precisely because ordinary dice are not good enough for money.

Digital dice have none of those problems, but they have a different one: the random number generator has to be genuinely uniform. The usual shortcut is to take a random number and use its remainder after division, which subtly favours the lower faces. This tool draws from your browser’s cryptographic generator and discards the small range of values that would cause that skew, so every face is exactly equally likely.

None of which means you should stop using your dice. Physical dice are part of the pleasure of playing, and their bias is far too small to notice. Use this when you have not got them with you, when you need to roll a hundred dice at once, or when you want to know the odds.

What people use this for

Tabletop games

The main use. Type your attack, keep the notation in the box and press Enter to roll it again. Advantage and disadvantage are one click from the common rolls list, and the individual dice are always shown, with dropped dice struck through, so you can see what happened rather than trusting a total.

Board games with lost dice

The d6 button is the fastest path to a replacement die. Set the notation to 2d6 for anything using a standard pair.

Teaching probability

The curve makes the central limit theorem visible in about ten seconds. Roll 1d6 and the chart is flat; roll 2d6 and it is a triangle; roll 5d6 and it is a bell. Very few ideas in statistics can be demonstrated that quickly, and the target-number readout turns it into a question students can answer.

Settling things

A d100 is a quick way to pick a number nobody can argue with. For choosing between named people or options, the Name Wheel is a better fit, and for a straight two-way call the Coin Flip is simpler.

Getting more out of it

• Press Enter to re-roll. The notation stays in the box, so repeated rolls are one key.
• Set the target number to whatever you need to beat, and the tool tells you your real chance before you roll.
• Watch the highlighted bar. After each roll the chart marks where you landed, which builds an intuition for what is ordinary and what was lucky.
• Use the history to check a disputed roll, or to see how a run of luck actually looked.
• Try 2d20kh1 and 2d20kl1 back to back if you want to feel the difference advantage makes.

Related tools

The Coin Flip covers two-way decisions and keeps a running fairness check. The Name Wheel picks one name from a list, and the Team Splitter divides a group into balanced teams. If you need names for the characters you have just rolled, the Fantasy Name Generator is built for it.

Common questions

Are these dice actually random?

They use your browser’s cryptographic random number generator, not the ordinary one, and the raw value is rejection-sampled so that no face is even fractionally more likely than another. That last step matters more than it sounds: the obvious way to turn a random 32-bit number into a number from 1 to 6 is to take the remainder, and that quietly favours the low faces. We discard the values that would cause the skew and draw again.

In practice this is fairer than a physical die. Real dice have rounded corners, pips drilled out of one side, and small density differences from moulding, all of which bias them slightly.

What does 4d6dl1 mean?

Roll four six-sided dice and drop the lowest one, then add the other three. It is the standard way to roll a character’s ability scores in most editions of D&D, because it produces better numbers than 3d6 without allowing anything higher than 18. You can also write it 4d6kh3, which means the same thing from the other direction: keep the highest three.

How much does advantage actually help?

More than most people assume, and not evenly. Rolling two d20 and keeping the higher raises the average from 10.5 to 13.825, but the average is the wrong way to think about it. What matters is the chance of hitting the number you need, and there advantage helps most in the middle: on a target of 11 it adds about 25 percentage points, while on a target of 20 it adds under five. If you need a 2, you were going to succeed anyway.

Why does the chart have a hump in the middle?

Because there are more ways to make a middling total than an extreme one. On two six-sided dice there is exactly one way to roll 2 and six ways to roll 7, so 7 comes up six times as often. Add more dice and the effect gets stronger. A single die is flat — every face equally likely — but the moment you add dice together you get a hump, and that is why 3d6 behaves nothing like a single eighteen-sided die.

Are the probabilities exact or simulated?

Exact. The curve is computed from the mathematics of the roll, not by rolling a million times and counting. For plain dice that means convolving the distributions together; for keep-highest and drop-lowest rolls it means a dynamic program over the dice in rank order. Both give the true answer rather than an estimate, which is why the average for 4d6 drop lowest shows as 12.24 rather than something that wobbles each time you load the page.

Can I roll dice with more than 100 sides?

Yes, up to 1,000 sides and 100 dice in a group. Very large rolls still roll, but above a few thousand possible totals the tool stops drawing the probability curve, because computing it exactly would take long enough to freeze the page. You still get the total, the individual dice and the range.

Is anything I roll sent anywhere?

No. Every roll happens in your browser and nothing is transmitted or stored on a server. Your roll history lives in the page and disappears when you close the tab.