Dice Probability Calculator
Calculate the probability of rolling specific outcomes with any number and type of dice. Enter values for instant results with step-by-step formulas.
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer
Dice Probability Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: P(sum=k) = count of combinations that sum to k / total outcomes (sides^dice)
Worked example — 16.67% (1 in 6)
Formula
P(sum=k) = count of combinations that sum to k / total outcomes (sides^dice)
Total outcomes = sides^numDice. The number of ways to achieve a specific sum follows a bell curve centered on the average roll.
Worked Examples
Example 1: 2d6 rolling 7
Problem:2 six-sided dice, target sum 7
Solution:6 ways out of 36 = 16.67%
Result:16.67% (1 in 6)
Frequently Asked Questions
What is the most common sum with 2d6?
7 is the most common (6 ways out of 36 = 16.67%). The probability curve is triangular for 2 dice and approaches normal for more dice.
What are probabilities for common tabletop RPG dice?
Common RPG dice probabilities for rolling a specific number: a d4 (4-sided) has a 25% chance per face; a d6 has 16.67% per face; a d8 has 12.5% per face; a d10 has 10% per face; a d12 has 8.33% per face; and a d20 has 5% per face. For advantage in D&D 5th Edition (roll 2d20 and take the higher), the probability of rolling at least 15 jumps from 30% with one die to approximately 51% with advantage.
How does the central limit theorem apply to dice rolling?
The Central Limit Theorem states that the sum of many independent random variables tends toward a normal (bell curve) distribution regardless of the shape of individual distributions. One die produces a flat uniform distribution where every outcome is equally likely. Two dice create a triangular distribution peaked at the middle sum. By six or more dice, the distribution becomes visibly bell-shaped. This explains why rolling multiple dice and adding them (common in RPGs for damage) produces results clustered near the average, making extreme highs and lows increasingly rare.
What is the gambler's fallacy in dice probability?
The gambler's fallacy is the mistaken belief that past independent events influence future probabilities. When rolling dice, each roll is completely independent — rolling a 6 on one throw does not affect the 1-in-6 probability of the next roll. Many gamblers incorrectly think that after a long streak without rolling a 6, a 6 becomes 'due.' In reality, dice have no memory. This fallacy also applies to slot machines, roulette wheels, and coin flips. Understanding independent probability is essential for making rational decisions in games of chance and avoiding costly betting mistakes based on false patterns.
What is the difference between odds and probability?
Probability is expressed as a number between 0 and 1 (or a percentage), representing the likelihood of an event. Odds compare favorable outcomes to unfavorable ones — odds of 3:1 means 3 wins for every 1 loss, which is a probability of 3/(3+1) = 75%. Casinos often express odds differently from true probability to build in their house edge.
What is the probability of rolling a specific number on a standard die?
A fair six-sided die has 1/6 ≈ 16.67% probability for each face. Rolling at least one specific number in two rolls = 1 − (5/6)² ≈ 30.6%. Rolling two specific numbers on two dice = 1/36 ≈ 2.78%. These calculations multiply individual probabilities for independent events.
What is a fair game in probability theory?
A fair game is one where the expected value for all players is zero — no participant has a mathematical advantage. In practice, most casino games are unfair (negative EV for players) due to the house edge. Flipping a coin for even money is a fair game; flipping for $0.90 per win and $1 per loss is unfair.
What is the birthday problem in probability?
The birthday problem asks: how many people are needed for a 50% chance two share a birthday? The answer is just 23 people — surprising because there are 365 days. The probability no two people share a birthday with n people = (365/365)(364/365)(363/365)...(365−n+1)/365. With 23 people this equals ≈50.7%, meaning a shared birthday is more likely than not.
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
Related Calculators
🧮Dice Odds Calculator
Calculate dice odds with inputs, formulas, and instant results.
🧮Card Hand Probability Calculator
Calculate card hand probability with inputs, formulas, and instant results.
🧮Gacha Probability Calculator
Calculate gacha probability with inputs, formulas, and instant results.
🧮Match Win Probability Calculator
Calculate match win probability with inputs, formulas, and instant results.
🧮Card Probability Calculator
Calculate the probability of drawing specific cards from a standard 52-card deck.
🧮Coin Toss Probability Calculator
Calculate the probability of getting a specific sequence of heads and tails in multiple flips.
🧮Gaming Pc Fps Calculator
Estimate frames per second based on GPU, CPU, resolution, and game optimization level.
🧮Lottery Odds Calculator
Calculate lottery odds with inputs, formulas, and instant results.