Dice Roller

Preset:

Pick a normal roll like 1d6 or 2d6, an RPG starter, or an odds-check example. You can still edit the expression after loading it.

Roll expression:

Examples: 1d20+5, 4d6kh3, 8d10dl2-1, or d100.

Roll mode:

Choose Single, Advantage, or Disadvantage to match the table rule.

Simulations: {{ simulationsValueText }}

Enter 1 for one live roll; use up to 5000 for probability checks.

runs

Success check: {{ successTargetValueText }}

Pick an operator and enter a target number, such as >= 15.

Exploding dice: {{ explodeValueText }}

Select Off, On max face, or Custom trigger such as 6 on d6.

Reroll threshold: {{ rerollValueText }}

Use 0 to disable; values like 1 or 2 model low-face reroll rules.

reroll ≤

RNG source:

Use Crypto for live rolling, Seeded deterministic for reproducible tests.

Seed:

Enter a campaign/session label when RNG source is Seeded deterministic.

Session log size: {{ historyLimitValueText }}

Keep between 10 and 200 recent entries in the session history.

entries

Field	Value	Copy
{{ row.label }}	{{ row.value }}

Increase simulations above 1 to render the probability distribution.

Total	Count	Share	Cumulative ≤	Cumulative ≥	Target	Copy
{{ row.total }}	{{ row.count }}	{{ row.share }}	{{ row.cumulativeAtMost }}	{{ row.cumulativeAtLeast }}	{{ row.target }}

Increase simulations above 1 to build the probability table.

#	Kept	Roll chain	Die total	Copy
{{ die.index }}	{{ die.kept ? 'Yes' : 'No' }}	{{ die.chain }}	{{ die.total }}

When	Expression	Total	Simulations	Success rate	Copy
{{ entry.when }}	{{ entry.expression }}	{{ entry.total }}	{{ entry.simulations }}	{{ entry.successRate }}

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Tabletop dice notation compresses a chance rule into a short expression. Instead of writing "roll four six-sided dice, keep the highest three, then add two," players can write 4d6kh3+2. The shorthand is compact enough for character sheets, encounter notes, solo games, teaching examples, and quick probability checks, but the small suffixes matter because they change which dice count toward the final total.

The most basic form is NdS: N is how many dice are rolled and S is how many faces each die has. 1d20 rolls one twenty-sided die, 2d6 rolls two six-sided dice and adds them, and d100 is common shorthand for one percentile die. A signed modifier such as +5 or -1 moves the final total after the dice have been counted.

Single outcome: The result of one roll, useful when a game needs an answer right now.
Distribution: The pattern of totals produced by many repeated rolls under the same rule.
Target number: A threshold such as >= 15 or <= 65 used to turn a total into pass or fail.
Selection rule: A keep/drop suffix such as kh3 or dl1 that decides which completed dice are counted.

Repeated rolls answer a different question from a live roll. A single 1d20+5 total tells a player whether the current check succeeded. Hundreds or thousands of repeated 1d20+5 rolls show the likely spread, the average total, and the share that meet a target number. Those statistics are estimates unless the rule is simple enough for an exact shortcut.

Random dice are useful for games, classroom demonstrations, and informal planning. They are not a certified draw method for wagering, regulated prize selection, security tokens, or any decision that needs an audited random source and a verifiable record.

How to Use This Tool:

Start with the roll rule, then choose whether you need one result, a repeatable run, or a probability estimate.

Choose a Preset for common rolls such as 1d6, 1d20+5, 4d6kh3, 2d6+3, or 1d100. Leave it on Custom when typing your own notation.
Enter one Roll expression. Supported examples include 1d20+5, 4d6kh3, 8d10dl2-1, and d100. Shorthand such as d100 is normalized to 1d100 after a valid roll.
Set Roll mode to Single roll, Advantage, or Disadvantage. Advantage and disadvantage roll the full expression twice and keep the higher or lower final total.
Open Advanced when the run needs more than a basic result. Set Simulations from 1 to 5000, add a Success check, enable Exploding dice, set a Reroll threshold, choose an RNG source, or change the Session log size.
Use Crypto for ordinary live rolling, Math.random only when browser entropy is not required, and Seeded deterministic when the same seed and settings should replay the same sequence.
Press Roll. Read Latest roll for the immediate total, Outcome Ledger for totals and statistics, Probability Lens for the distribution chart, Probability Table for cumulative odds rows, and Roll Trace for die-by-die chains.
If an error says the expression is unsupported or a drop count leaves no kept dice, reduce the notation to one NdS expression or lower the drop count. Then roll again and confirm the normalized expression in the result summary.

Interpreting Results:

Last total is the latest selected roll after modifiers, keep/drop rules, rerolls, exploding dice, and advantage or disadvantage have already been applied. Mean total, Standard deviation, P10 total, Median total, P90 total, and the histogram describe the simulated sample, so read them together with Simulations.

A high Success rate does not guarantee the next roll. It means that the simulated final totals passed the selected operator and target that often. Check Roll Trace when a surprising total appears, because the trace shows which dice were kept, which were dropped, and whether a reroll or explosion changed a die chain.

Dice result fields and confidence checks
Result field	How to read it	What to verify
`Last total`	The final answer for the most recent selected roll.	Open `Roll Trace` if the total seems too high, too low, or affected by a special rule.
`Mean total`	The average final total across the completed simulations.	Compare with `Analytical mean` only when that field appears for a simple sum.
`Standard deviation`	The spread of final totals around the sample mean.	Larger values mean swingier rolls even when the average looks favorable.
`Success rate`	The share of final totals that passed the selected target rule.	Confirm the operator is the one intended, especially `>` versus `>=` and `<` versus `<=`.
`Probability Table`	Counts and cumulative percentages for each simulated total.	Use the cumulative columns when a threshold matters more than one exact total.

Seeded deterministic results are repeatable demonstrations, not proof that a draw was fair. For important public draws, use a process that records the random source, participants, time, and verification method outside the browser session.

Technical Details:

A fair die with s faces is a discrete uniform random variable over the integers from 1 through s. Summing several independent dice produces a distribution whose center and spread depend on the number of dice, the number of faces, and any fixed modifier. Selection rules such as keep-high and drop-low change the distribution because completed die totals are sorted before the final sum is chosen.

Reroll and explosion rules alter each die before selection. A low-face reroll replaces the first face once when it is at or below the chosen threshold. An exploding die adds another face when the trigger appears, and repeated triggers continue the chain up to a guard limit. Advantage and disadvantage sit above those die-level rules: the whole expression is rolled twice, then the higher or lower final total is selected.

Rule Core

Dice notation stages and boundaries
Stage	Mechanism	Boundary
Expression	Accepts one `NdS` term, optional `kh`, `kl`, `dh`, or `dl` suffix, and one signed integer modifier.	Chained dice such as `2d6+1d4` and parenthesized arithmetic are outside the accepted notation.
Bounds	Dice count is clamped from 1 to 2000, faces from 2 to 100000, simulations from 1 to 5000, and session log size from 10 to 200.	A drop rule must leave at least one kept die.
Reroll	A die is rolled once more when its first face is less than or equal to the threshold.	The replacement face is the value that can later explode.
Explosion	The die adds another face while the trigger repeats.	The trigger can be the maximum face or a custom face, and each die chain stops after 40 extra faces.
Selection	`kh` keeps highest totals, `kl` keeps lowest totals, `dh` drops highest totals, and `dl` drops lowest totals.	Selection uses completed die totals after reroll and explosion chains.
Target check	The final total is compared with `>`, `>=`, `=`, `<`, or `<=`.	Pass/fail is based on the final total, not on individual die faces.

Formula Core For Simple Sums

Exact mean and standard deviation are available only for a plain fair-dice sum with a fixed modifier. Keep/drop rules, rerolls, exploding dice, advantage, and disadvantage require sampled statistics because they change the distribution after the original face rolls.

E [T] = n \frac{s + 1}{2} + m, σ [T] = \sqrt{n \frac{s^{2} - 1}{12}}

n is dice count, s is faces per die, m is the modifier, and T is the final total.

For 2d6+3, the expected total is 2 x (6 + 1) / 2 + 3 = 10. The variance is 2 x (36 - 1) / 12, so the standard deviation is about 2.42. A rule such as 4d6kh3 has a higher center than 3d6, but the exact shortcut above no longer applies because the lowest completed die is removed.

The simulation histogram counts final totals. Percentiles are linearly interpolated from sorted totals, cumulative percentages are derived from the same simulated totals, and the target success rate is the number of passing final totals divided by the simulation count.

The random integer draw rejects values that would create uneven modulo buckets. Crypto mode uses browser cryptographic randomness when available, seeded deterministic mode uses a repeatable number stream from the seed text, and the Math.random option follows the browser's standard non-cryptographic generator.

Worked Examples:

A d20 check with 1d20+5, Success check set to >= 15, and 1000 simulations should usually show Mean total near 15.50. Totals from 15 through 25 pass, so Success rate should hover around 55% as the simulation count grows.

For 2d6 with target >= 7, the most common middle totals carry more weight than the edges. Probability Table should put cumulative-at-least odds for total 7 near 58.33% in a large run, while an exact total of 12 remains much rarer.

Ability-score rolling with 4d6kh3 rolls four dice and keeps the highest three completed die totals. Roll Trace should mark one die as not kept, and Mean total usually settles in the low twelves rather than near the 3d6 average of 10.5.

If 4d6dl4 returns a drop-count error, the rule tried to drop every die. Change it to 4d6dl1 or 4d6kh3, roll again, and use Roll Trace to confirm that exactly one die was excluded.

FAQ:

What dice expressions are accepted?

Use one NdS expression with optional kh, kl, dh, or dl, plus one optional signed modifier. Examples include 1d20+5, 4d6kh3, 8d10dl2-1, and d100.

Why do analytical fields disappear?

Analytical mean and Analytical stdev appear only for simple summed dice with a fixed modifier. Keep/drop rules, rerolls, explosions, advantage, and disadvantage use sampled statistics instead.

Does advantage reroll only one die?

No. Advantage and Disadvantage roll the full expression twice, including modifiers and special die rules, then select the higher or lower final total.

Why did my exploding die get very large?

An exploding die keeps adding faces while the trigger repeats. Each die chain has a 40-extra-face guard, but rare repeated triggers can still create much larger totals than a normal die.

Does the seed make a roll fair?

No. The seed makes the deterministic sequence repeatable for the same seed and settings. It is useful for demonstrations and test notes, not for certifying a public draw.

Are roll results sent to a dice server?

No dice service is used for parsing, rolling, simulation, the trace, or the session log. The expression and results stay in the browser unless you copy or download them.

Glossary:

Dice notation: A compact expression such as NdS that states how many dice to roll and how many faces each die has.
Modifier: A fixed number added to or subtracted from the kept dice subtotal.
Keep/drop rule: A suffix that keeps or removes the highest or lowest completed die totals before the modifier is applied.
Exploding dice: A rule that adds another face when a selected trigger face appears.
Distribution: The frequency pattern of final totals across repeated rolls under the same settings.
Target number: The threshold used with an operator to decide whether a final total passes.
Seed: Text used to replay a deterministic random sequence when the same roll settings are used.

References:

Crypto: getRandomValues() method, MDN Web Docs, last modified 19 August 2025.
Dice Reference, Roll20 Help Center, updated 20 February 2024.
Dice, AnyDice documentation.
explode DIE, AnyDice documentation.