What is the probability of rolling a 7 with two six-sided dice?

Six out of thirty-six, which is 1 in 6 or about 16.67%. Two d6 have 36 equally likely face pairs, and six of them total seven: 1+6, 2+5, 3+4, 4+3, 5+2 and 6+1. That is why 7 is the most common result in games like craps and Settlers of Catan. Type N=2, S=6 and pick «sum = 7» and the tool returns 6/36 with the matching bar highlighted.

How do I calculate the expected value of a dice roll?

For a single fair die with S faces the average is (S+1)/2, so a d6 averages 3.5 and a d20 averages 10.5. For N dice the expected sum is just N times that, which is N·(S+1)/2: three d6 average 10.5, two d6 average 7. The tool shows this in the summary card, and because the distribution is symmetric the expected value also sits at the centre of the bar chart.

What does «at least X» mean and how is it computed?

«Sum ≥ X» is the cumulative probability of every total from X up to the maximum N·S. With 2d6, the chance of rolling at least 7 is the sum of the ways to make 7, 8, 9, 10, 11 and 12 (6+5+4+3+2+1 = 21) over 36, so 21/36 ≈ 58.3%. Pick the ≥ comparator and the tool adds up all qualifying bars and shows both the percentage and the exact fraction.

How do dice odds change as I add more dice?

One die is flat: every face is equally likely. Add dice and the sum distribution bunches toward the middle and the extremes get rare, because there are many ways to make a middling total and only one way to make the minimum or maximum. By the central limit theorem the shape approaches a bell curve. Roll 10d6 in the tool and you will see a near-normal distribution centred on 35.

Why use exact counting instead of rolling thousands of times?

Simulation gives an estimate whose error shrinks slowly, roughly with the square root of the number of rolls, and the answer wobbles every run. Exact counting enumerates all Sᴺ outcomes through a convolution, so 2d6 sum=7 is reported as precisely 6/36 every time, never 16.6% one run and 16.9% the next. For odds you can quote in a rules discussion, exact beats approximate.

How large can N and S be here?

Up to 20 dice and up to 100 faces, so 20d100 with sums from 20 to 2000 is fine. The internal way-counts can reach dozens of digits, far past what a normal JavaScript number holds without losing precision, so the tool counts in BigInt and only converts to a decimal probability at the very end. The cap is a deliberate UI choice, not a math limit; the bar chart stays readable below it.

Balance a D&D encounter before the session

You want to know how hard a save-or-suck spell really is. Set the attack as the right NdS, ask «sum ≥ DC», and you get the exact hit chance instead of a gut feel. Comparing 3d6 against 1d20 for an ability check shows why the bell-shaped 3d6 makes average outcomes far more common and crits far rarer — useful when you are deciding which resolution system fits the tone of your table.

Tune a board-game dice curve

Designing a game where players roll for movement or combat? The 2d6 curve peaks at 7 and tapers symmetrically, which is why so many classics use it. Punch in your candidate dice, read the distribution shape, and check whether the modal result lands where your reward structure expects it. Switching from 2d6 to 1d12 flattens the curve and changes the whole feel of the game.

Check a probability homework answer

A stats assignment asks for P(sum of three dice ≥ 15). Rather than enumerate 216 cases by hand and risk a slip, set N=3, S=6, pick the ≥ comparator and read both the fraction and the percentage. The exact way-count next to it lets you show your working: numerator over Sᴺ.

Settle a tabletop rules argument with a real number

Two players disagree on whether rolling at least a 10 on 2d6 is likely. Instead of arguing, share the tool URL with N=2, S=6 and «≥ 10» already filled in. The link reopens to exactly that query — 6/36, about 16.7% — so the answer is reproducible and not a matter of opinion.

Dice Probability Calculator — Exact Odds for Any NdS Sum

Exact odds for any NdS roll: full sum distribution, expected value, and P(sum = / ≥ / ≤ X), counted not simulated

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Roll N dice with S sides each and read the exact probability of every possible sum — computed by combinatorial counting, not simulation, so the numbers never drift.

Number of dice (N)

Sides per die (S)

Notation: 2d6

Presets

Summary

Expected sum E[X]

Std deviation σ

2.41523

Variance

5.83333

Sum range

2 – 12

Most likely sum

7 (16.67%)

Total outcomes (Sᴺ)

Cumulative probability query

Pick a comparator and a target sum. Example: with 2d6 and «= 7» the answer is 6/36 = 16.67%.

Probability

16.67%

as a fraction: 6 / 36

Distribution

Bar height is the probability of each sum. Hover or read the table below for exact values.

Probability table

Sum	Ways	Percent
2	1	2.78%
3	2	5.56%
4	3	8.33%
5	4	11.11%
6	5	13.89%
7	6	16.67%
8	5	13.89%
9	4	11.11%
10	3	8.33%
11	2	5.56%
12	1	2.78%

Every count and probability is computed in your browser with BigInt math. Nothing is uploaded or logged.

What this tool does

A dice probability calculator that gives the exact odds for rolling N dice with S sides each, for any combination from a single d6 up to 20 dice with 100 faces. It does not simulate rolls. Instead it counts every possible outcome with a dynamic-programming convolution, so the probability of each sum is a true fraction (ways to make that sum divided by the total Sᴺ outcomes), not a noisy estimate that changes every time you reload. The page draws the full distribution as a bar chart, reports the expected value, variance and standard deviation, and answers the three questions players actually ask: what is the chance the total is exactly X, at least X, or at most X. Counts use BigInt, so even the hundred-digit way-counts inside a 20d20 roll stay precise with no floating-point drift. Useful for D&D and tabletop GMs balancing encounters, board-game designers tuning a 2d6 curve, and statistics students checking homework. Everything runs client-side, with a copy button for the whole table and a shareable URL that reproduces your exact dice and query.

Tool details

Input: Files + Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy + Preview; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 11 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Dice Probability Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Balance a D&D encounter before the session
You want to know how hard a save-or-suck spell really is. Set the attack as the right NdS, ask «sum ≥ DC», and you get the exact hit chance instead of a gut feel. Comparing 3d6 against 1d20 for an ability check shows why the bell-shaped 3d6 makes average outcomes far more common and crits far rarer — useful when you are deciding which resolution system fits the tone of your table.
Tune a board-game dice curve
Designing a game where players roll for movement or combat? The 2d6 curve peaks at 7 and tapers symmetrically, which is why so many classics use it. Punch in your candidate dice, read the distribution shape, and check whether the modal result lands where your reward structure expects it. Switching from 2d6 to 1d12 flattens the curve and changes the whole feel of the game.
Check a probability homework answer
A stats assignment asks for P(sum of three dice ≥ 15). Rather than enumerate 216 cases by hand and risk a slip, set N=3, S=6, pick the ≥ comparator and read both the fraction and the percentage. The exact way-count next to it lets you show your working: numerator over Sᴺ.
Settle a tabletop rules argument with a real number
Two players disagree on whether rolling at least a 10 on 2d6 is likely. Instead of arguing, share the tool URL with N=2, S=6 and «≥ 10» already filled in. The link reopens to exactly that query — 6/36, about 16.7% — so the answer is reproducible and not a matter of opinion.

Common pitfalls

Treating «exactly X» and «at least X» as the same thing. P(sum = 7) on 2d6 is 6/36, but P(sum ≥ 7) is 21/36 — almost four times larger. Always pick the comparator that matches the rule you are checking; a saving throw of «DC 7 or higher» is an «at least» question, not an «exactly».
Assuming every sum is equally likely. Only a single die is flat. With two or more dice the middle sums are far more common than the extremes, so guessing that 2d6 has a 1-in-11 chance of any given total is wrong — 7 is six times as likely as 2.
Confusing the number of dice with the number of faces. «3d6» means three six-sided dice (sums 3 to 18), while «6d3» means six three-sided dice (sums 6 to 18). They share an upper bound but have completely different shapes and means; enter N and S in the right boxes.

Privacy

Every count, probability and statistic is computed in your browser with plain JavaScript and BigInt arithmetic. No roll, query or result is ever sent to a server, and nothing is logged. The one caveat: the shareable URL encodes your dice count, face count and the cumulative query in the query string, so a link pasted into chat records those values in the recipient server's access log. The inputs here are not sensitive, so for this tool that is harmless — but it is the same mechanism that makes «share my exact odds» work.

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Dice Probability Calculator — Exact Odds for Any NdS Sum

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Dice Probability Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Balance a D&D encounter before the session

Tune a board-game dice curve

Check a probability homework answer

Settle a tabletop rules argument with a real number

Common pitfalls

Privacy

FAQ

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