D20 Dice Roller Calculator - Icosahedron Roll Simulator

A d20 dice roller is an interactive simulator that rolls one or many fair 20-sided dice and reports each face, the batch total, crit and fumble highlights, expected value, and a 1 to 20 empirical frequency table.

• • D&D attack and save rolls: Roll a single d20 for attacks, ability checks, and saving throws when you cannot or do not want to use a physical die.
• • Pathfinder and OSR game nights: Roll advantage or disadvantage (two d20 dice and take the higher or lower) or combine a d20 with other dice for mixed pools.
• • Probability class demos: Run a 1000 or 10000 roll batch and watch the empirical face frequencies converge to the theoretical 5 percent uniform distribution.
• • Dice balance testing: Use the seed input to reproduce a specific batch of rolls and compare physical d20 dice against an unbiased random sequence.

The tool runs entirely in the browser with no install, so it works on a phone at the table, a laptop during prep, or a Chromebook. Each batch uses a seeded xorshift32 generator so the same seed reproduces the same dice sequence.

The d20 dice roller calculator also tracks running totals and the empirical probability of each face, so a single click becomes a small experiment that students, game masters, and dice collectors can compare against the 5 percent reference.

For the percentile-style sibling of this d20 dice roller, see the D100 Dice Roller, which simulates the standard two-d10 reading style and supports 1 to 10 d100 dice per batch.

Each d20 die has twenty faces numbered 1 through 20, so a single die is a discrete uniform distribution on 1..20 with probability 1/20 = 5 percent per face. For n dice the total follows the n-fold convolution of the uniform distribution.

• k: Face value of a single d20, integer from 1 to 20 inclusive.
• n: Number of d20 dice rolled in the batch, integer from 1 to 10.
• P(X = k): Probability that a single d20 shows face k, equal to 1/20 or 5 percent for every face.

The expected value of a single fair d20 is (1 + 20) divided by 2, which works out to 10.5 exactly. The variance of a single die is (N squared minus 1) divided by 12, which for N = 20 gives 33.25, so the standard deviation is the square root of 33.25, about 5.766. The sum of n dice has variance n times 33.25 and standard deviation sqrt(n times 33.25).

The calculator repeats the same draw process inside a seeded xorshift32 generator so the seed lets you reproduce the exact same batch later, which makes it possible to share a specific Monte Carlo run or replay a contested attack roll.

According to Wolfram MathWorld - Dice, the 20-sided die is a regular icosahedron with twenty equilateral-triangle faces numbered 1 through 20, and the expected value of a single fair d20 is 10.5.

When you want to extend these single-die results to general probability expressions, the Probability Calculator handles unions, intersections, and complements using the same 1/20 building block.

Four short definitions anchor the rest of the page. Keep them next to the calculator so a beginner can refer back without leaving the page.

If you already understand those four ideas, you can read the rest of the page without a probability textbook. If not, treat them as a glossary and come back as needed.

When your game moves past a single d20 and you need to mix in d4, d6, d8, d10, or d12 dice, the Custom Dice Roller Calculator handles the full polyhedral pool with one shared expected-value readout.

Use the calculator below to roll d20 dice in three different ways depending on what you need at the table.

1. 1 Pick the number of dice: Type the number of d20 dice you want to roll. Leave it at 1 for a single attack roll, or set it to 2 for advantage or disadvantage.
2. 2 Set the batch size for the frequency table: Use 1 for a one-off roll, or bump it to 100, 1000, or 10000 to build the empirical face frequency table.
3. 3 Pick a seed if you want to reproduce the run: Leave the seed at 42 for a casual batch, or change it to any integer between 0 and 999999 so the same sequence can be replayed later.
4. 4 Click Calculate to roll the batches: The Results panel refreshes with the latest face, the latest total, the crit and fumble counts, and the empirical mean.
5. 5 Read the face frequency table: Faces 1 through 20 are listed with simulated count, simulated probability, and the theoretical 5 percent reference.
6. 6 Adjust and rerun: Change the die count to 2 for an advantage test, switch the seed, or bump the batch size to 5000 to confirm the face probabilities stay near 5 percent.

For a classroom demo, run 100 batches with dieCount 1 and seed 42 to show the rough per-face spread, then rerun with 10000 batches to demonstrate the empirical probabilities converging to 5 percent.

For a hands-on Monte Carlo demo of the law of large numbers that this d20 dice roller also supports, the 2 Dice Roller Calculator shows the symmetric triangular distribution of two six-sided dice and tracks the empirical probability of a sum of 7.

A digital d20 brings four advantages that a physical die cannot match at the table.

• • Crit and fumble at a glance: The page tracks natural 20s and natural 1s as separate counters, so the game master can spot the moment a crit lands without reading every face value.
• • Reproducible sequences: The seed lets you share the exact same batch with a player who missed the session, or save a batch for a probability lab write-up.
• • Fast batch sizing: Roll 1000 or 10000 dice in milliseconds, which lets you demonstrate uniform convergence far faster than hand rolling.
• • Built-in frequency table: The face frequency table compares simulated count and probability for every face against the theoretical 5 percent, so you can test dice balance.
• • Browser-only, no install: Runs on any device with a modern browser, including phones at the game table, tablets during prep, and Chromebooks in a classroom.

These benefits make this calculator useful both as a quick decision tool for a game and as a teaching aid for introductory probability.

If you want a simpler uniform distribution example to compare the d20 against, the Coin Flip Probability Calculator tracks empirical versus theoretical probabilities for the basic 1/2 outcome.

A few factors drive the gap between simulated and theoretical face counts, and the calculator handles them transparently.

• • The page models fair d20 dice only. For non-Platonic polyhedra such as d24, d30, or d100, switch to the matching specialty calculator.
• • The xorshift32 generator is fast and reproducible but is not cryptographically secure. Use a different tool when true randomness matters.
• • Empirical probabilities in small batches are noisy by design. Treat any single batch below 1000 rolls as a rough illustration rather than a final probability estimate.

Keeping those caveats in mind is what turns this calculator from a fun toy into a useful study tool.

According to Wikipedia - Discrete uniform distribution, the mean of a uniform distribution on the integers 1 through N is (N + 1) divided by 2 and the variance is (N squared minus 1) divided by 12.

According to Omni Calculator - d20 dice roller, a d20 dice roller simulates the standard 20-sided die used in D&D and other tabletop role-playing games and supports single, multi-die, and batch modes.

When you need to combine the d20 result with an independent event such as a critical hit and a sneak attack, the And Probability Calculator computes the joint probability without forcing you to enumerate the full outcome grid.