Dice roller
Roll up to twelve dice, from d2 to d100. Every roll comes back with the odds of that exact total and of beating it, drawn from the complete distribution rather than a simulation.
By Alex Seote, Built and maintains Rule Calculator
Distribution checked against the exact counts · How we check
Every total, and how often it comes up
Computed exactly rather than simulated. The average roll of 3d6 is 10.5 with a standard deviation of 2.96.
Where the odds come from
Adding a die to a roll spreads the probability of every total so far evenly across the faces of the new one. Do that once per die and you have the exact chance of every possible total — a convolution, computed here in full rather than estimated by rolling a million times and counting. For 3d6 the answer is 27 outcomes in 216 for a total of 10, not “about 12%”.
It is also why the bell shape appears without anybody putting it there. One die is flat. Two make a triangle. Three already look like a normal curve, and that is the central limit theorem happening on a tabletop.
Questions people actually ask
- Why is 7 the most common roll on two dice?
- Because it has the most ways of happening. There are 36 ordered outcomes on two six-sided dice and six of them total 7 — 1+6, 2+5, 3+4 and the same three reversed. A 12 has exactly one, 6+6. That is why 7 comes up six times as often as 12, and it is a property of the arithmetic rather than anything about the dice.
- Why does the curve get narrower with more dice?
- Extreme totals need every die to agree and middling totals do not. On 3d6 there is one way to roll a 3 and 27 ways to roll a 10, so a 3 turns up once in 216 rolls and a 10 once in eight. Add more dice and the peak sharpens further: the average stays in the middle while the ends become vanishingly rare. Rolling 5d6 for a total of 8 or lower happens 0.7% of the time.
- Is this actually random?
- Every face comes from crypto.getRandomValues, your browser’s cryptographic generator, drawn through a rejection sampler so no face is favoured. The common shortcut — take a random byte modulo six — gives four faces a 43-in-256 chance and two faces 42, a 2.4% edge that no amount of rolling would make visible to you.
- What are the standard deviation and standard error for?
- The standard deviation says how widely single rolls scatter around the average: for 3d6 it is 2.96, so most rolls land within three of 10.5. The standard error says how far a run of rolls should sit from the average — the standard deviation divided by the square root of how many rolls. It is the number to check before deciding a die is loaded: twenty rolls of 3d6 averaging 11.2 is well inside what fair dice do.
- Can I roll d100 or d2?
- Both, along with d4, d8, d10, d12 and d20, up to twelve at a time. A d2 is a coin flip with a total, and rolling ten of them gives the binomial distribution the bar chart makes obvious: a total of 15 — five of each — comes up 252 times as often as a total of 20.