Dice Roller (RPG Notation), Explained
After reading this, you will know exactly what 3d6+2 or 4d6kh3 means, how to compute the odds of any roll by hand, and how to read the probability distribution the tool draws.
What dice notation says
Dice notation is a compact way to describe a roll. The core form is NdS: roll N dice, each with S sides, and add the faces. So 3d6 rolls three six-sided dice and sums them, giving a total between 3 and 18. A single d20 rolls one twenty-sided die, giving 1 through 20 with equal chance.
You can attach a modifier. 3d6+2 rolls three six-sided dice and adds 2, so the range shifts to 5 through 20. You can also keep only some dice. 4d6kh3 rolls four six-sided dice and keeps the highest three, discarding the lowest. That single rule, common in tabletop character creation, changes the odds in a way most players guess wrong, and the tool shows you the exact numbers.
Here is the hook. Roll 4d6kh3 and the average result is about 12.24, noticeably higher than the 10.5 you get from plain 3d6. The extra die does not just add a point; it reshapes the whole distribution toward high totals.
When to use it, and when not to
Use the roller when you need a fair result and want to see the mechanics: individual dice, the total, and the full distribution behind that total. It suits tabletop play, statistics teaching, and any time you want to check the odds of a rule before you adopt it.
Do not use it as a cryptographic random source. The tool produces results that are fine for games and demonstrations, but they are not meant to generate encryption keys, session tokens, or anything an attacker has reason to predict. For a security decision, use a dedicated cryptographic generator, not a dice widget.
Randomness has no memory. If you roll d20 and get 20 three times in a row, the next roll still has probability 1/20 = 0.05 of being a 20. Dice are not "due" for a low number. Every roll starts fresh.
The math behind a sum of dice
One die is uniform: on a dS, each face has probability 1/S. For d6 that is 1/6 \approx 0.1667 per face, flat across all six values.
Sums are different. The number of ways to roll a total k with N dice of S sides is a counting problem. For two dice the count rises then falls, which is why 7 is the most common result on 2d6. The probability of a total k is the count of favorable outcomes over the total number of outcomes.
Here S is the number of sides, N is the number of dice, and S^N is the total number of equally likely ordered outcomes. For 3d6 that denominator is 6^3 = 216. The numerator is the count of dice combinations that add to k.
The expected value of a single fair die is the average of its faces.
For d6 that is (6+1)/2 = 3.5. Expected values add, so 3d6 averages 3 \times 3.5 = 10.5, and 3d6+2 averages 12.5. A modifier shifts the mean by its own value and changes nothing about the shape.
Why many dice make a bell
A single die is flat. Add dice and the middle fills in, because there are many ways to reach a middle total and only one way to reach each extreme. On 3d6 the total 3 happens exactly one way (1,1,1), so its probability is 1/216 \approx 0.00463. The total 10 happens 27 ways, so its probability is 27/216 = 0.125. That is 27 times more likely than a 3.
Contrast that with a flat d20, where every value from 1 to 20 has probability 0.05. The single die has no peak. The bell shape only appears when you sum several dice, and it grows tighter around the mean as you add more.
Reproducing the demo: 3d6+2
The demo rolls 3d6+2. Suppose the three dice come up 4, 6, and 3. Follow the arithmetic.
- Sum the dice: 4 + 6 + 3 = 13.
- Add the modifier: 13 + 2 = 15. That is your total.
- Locate the odds. The dice summed to 13. From the table below, 13 has probability 21/216 \approx 0.09722 before the modifier.
- The modifier does not change probability, only labels. A dice-sum of 13 (probability 0.09722) is reported as a final total of 15.
So the mean of 3d6+2 is 12.5, the range is 5 to 20, and your specific 15 sits just above center. It is a common result, not a lucky one.
| Dice sum | Ways | Probability | Total after +2 |
|---|---|---|---|
| 10 | 27 | 0.125 | 12 |
| 11 | 27 | 0.125 | 13 |
| 12 | 25 | 0.1157 | 14 |
| 13 | 21 | 0.09722 | 15 |
| 14 | 15 | 0.06944 | 16 |
Reading the results panel
The tool shows three things. First, every individual die, so you can verify the sum and see which dice a keep rule discarded. Second, the total after modifiers. Third, the exact distribution of your whole expression, computed by enumerating outcomes, not by simulation. That distribution is the truth about your roll, independent of the one result you happened to get.
Read the distribution to judge a single roll. If your total lands in the fat middle, it is ordinary. If it lands in a thin tail, it is rare, and you can quote the probability. For 3d6, rolling an 18 has probability 0.00463, so it happens about once in 216 rolls on average. Rolling 10 or 11 happens a quarter of the time combined.
Common mistakes
Confusing keep rules is the frequent one. 4d6kh3 and 3d6 both produce three-dice totals, but their odds differ. The keep-highest version averages about 12.24 versus 10.5, and it almost never yields a very low total, because the worst die is thrown away. If you assumed they were equal, you overestimated how often a character rolls a weak score.
A second mistake is expecting a modifier to change your luck. +2 shifts every possible total up by 2 and leaves the shape untouched. It does not make high rolls more likely relative to the range; it just relabels.
A third is treating percentile dice as special. d% is simply a d100: uniform, every value 1 to 100 at probability 0.01. It is flat like d20, not bell-shaped.
To see how strong a keep-highest rule is, compare the means. Plain 3d6 gives 10.5. Rolling 4d6kh3 gives about 12.24, a gain of roughly 1.74 points from a single extra die you then discard.
Related tools
If you only need a two-way outcome, the Coin Flip tool gives you tallies and streaks across many flips, which is a cleaner way to study a single 50/50 event. If you want to pick from your own list of options rather than a numeric range, the Wheel Spinner draws one entry at a time from whatever you type in.
Frequently asked questions
What does kh3 mean in dice notation?
It means keep the highest 3 dice and discard the rest. In 4d6kh3 you roll four six-sided dice and sum the best three. Use kl to keep the lowest instead, so 2d20kl1 rolls two d20 and keeps the smaller, which is disadvantage in many game systems.
Is 3d6 the same as d18?
No. A hypothetical d18 would be flat, every value 1 to 18 at probability 1/18 \approx 0.0556. 3d6 is bell-shaped: 10 and 11 each have probability 0.125, while 3 and 18 have only 0.00463. Same range, very different odds.
Are the dice truly random?
They are random enough for games and teaching, drawn from your browser's built-in generator. They are not cryptographic randomness, so do not use them where an attacker predicting the outcome would cause harm.
Why did I roll three 20s in a row?
Because it is allowed and not especially rare over many rolls. Three independent 20s in a row have probability (1/20)^3 = 1/8000 = 0.000125. The next roll is still 0.05 for a 20. Past rolls do not change future ones.
Does a +2 modifier improve my chances of a high result?
It raises every total by 2 and keeps the shape identical. On 3d6+2 the mean moves from 10.5 to 12.5 and the range from 3 to 18 becomes 5 to 20, but no total becomes relatively more likely than before.
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