STATISTICSProbability TheoryStatistics Calculator
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Dice Probability Calculator

Free dice probability calculator. Exact P(sum), P(sum≥k), P(sum≤k), P(a≤sum≤b). 2d6, 3d6, 4d6 drop l

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Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

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STATISTICSProbability Theory

Exact Sum Probabilities for Any Dice Combination

P(exact sum), P(sum≥k), P(sum≤k), P(a≤sum≤b). 2d6, 3d6, 4d6 drop lowest, d20. PMF bar chart, CDF, odds for/against.

Dice Average →Coin Flip Probability →

Real-World Scenarios — Click to Load

Dice Configuration

Calculation Mode

Target

dice_prob.sh
CALCULATED
$ dice_prob 2d6
Primary Probability
16.6667%
= 1667/10000
Complement
83.33%
Odds For
6:30
Favorable / Total
6 / 36
Share:
Dice Probability
2d6
16.6667%
or 1667/10000
6 / 36 outcomes
numbervibe.com/calculators/statistics/dice-probability-calculator

Full PMF Bar Chart

Cumulative Distribution (CDF)

Calculation Breakdown

PARAMETERS
Dice notation
2d6
Total outcomes
36
f^n
COMPUTATION
Favorable outcomes
6
Primary probability
16.6667%
ext{favorable} / ext{total}
SUMMARY
Odds for
6:30
Odds against
30:6

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • P(2d6 = 7) = 6/36 = 1/6 ≈ 16.67% — 7 is the most likely sum for two dice
  • • More dice = more bell-shaped distribution (Central Limit Theorem)
  • • P(at least one 6 on 2d6) = 11/36 ≈ 30.6% — NOT 1/3
  • • The number of outcomes for n dice with f faces = f^n (grows exponentially)
  • • Dice probabilities are symmetric: P(2d6 = 5) = P(2d6 = 9)
  • • 4d6 drop lowest: D&D ability scores. Average ~12.24 per stat

Did You Know?

🎰Craps is the fastest-moving casino game — all based on 2d6 probability. The house edge on "pass" is only 1.41%Source: Casino math
📊The famous Dice Problem solved by Galileo: why does 3d6 sum to 10 more often than 9? (27 ways vs 25 ways)Source: Galileo, 1620
⚔️In D&D, 4d6 drop lowest averages ~12.24 per stat, higher than the 3d6 average of 10.5Source: D&D rules
🧮The generating function for a d6 is (x + x² + x³ + x⁴ + x⁵ + x⁶)/6 — elegant algebra gives exact probabilitiesSource: Probability theory
🎲With 2d6, you're more likely to roll doubles (1/6) than any specific non-double sum (max 5/36)Source: Combinatorics
📐The variance of 1d6 = 35/12 ≈ 2.917. For nd6, variance = n × 35/12Source: Variance formula
🏛️The oldest known dice (astragali) were animal knucklebones used ~5000 BC in MesopotamiaSource: Archaeology

Expert Tips

Dice vs Simulator

This calculator gives exact probabilities. A dice roller simulates rolls — use this for planning and analysis.

4d6 Drop Lowest

D&D ability scores use 4d6 drop lowest — we enumerate all 1296 outcomes to get exact probabilities.

Craps Quick Ref

Win on come-out: 7 or 11. Lose: 2, 3, 12. P(win) = 8/36 ≈ 22.2%, P(lose) = 4/36 ≈ 11.1%.

Modifier

Add +N or -N to shift the entire distribution. E.g. 2d6+3 has range 5–15, mean 10.

How It Works

1. Convolution

The probability distribution of a sum of dice is computed by convolving the distributions of each die. Start with P(0)=1, then fold in each die.

2. Favorable Outcomes

Count outcomes that match your target. For 2d6=7, there are 6 ways: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1). P = 6/36 = 1/6.

3. Symmetry

For 2d6, P(sum=k) = P(sum=14-k). The distribution is symmetric around 7.

4. Central Limit Theorem

As you add more dice, the sum distribution approaches a normal (bell) curve. 3d6 is already quite bell-shaped.

Why Use This Calculator vs. Other Tools?

FeatureThis CalculatorDice RollerAnyDice
Exact probability (not simulation)✅❌✅
PMF + CDF charts✅⚠️ Frequency only✅
4d6 drop lowest✅✅✅
Copy & share results✅❌❌
AI-powered interpretation✅❌❌
Educational content✅⚠️ Limited❌
Odds for/against✅❌⚠️ Manual

Frequently Asked Questions

What is P(2d6 = 7)?

6/36 = 1/6 ≈ 16.67%. There are 6 ways to roll 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

Why is 7 the most likely sum for 2d6?

7 has the most combinations (6). Sums 2 and 12 have only 1 combination each.

What is P(at least one 6 on 2d6)?

11/36 ≈ 30.6%. Count: (6,1),(6,2),...,(6,6) and (1,6),...,(5,6) — 11 outcomes. NOT 1/3.

How does 4d6 drop lowest work?

Roll 4d6, discard the lowest, sum the top 3. Used for D&D ability scores. Average ~12.24 per stat.

What are the odds of rolling 7 or 11 in Craps?

P(7)=6/36, P(11)=2/36. P(7 or 11) = 8/36 ≈ 22.2%.

Why do dice probabilities form a bell curve?

Central Limit Theorem: sums of many independent random variables approach a normal distribution.

How many outcomes does 5d6 have?

6^5 = 7,776 outcomes. The number grows exponentially with more dice.

What is the difference from Dice Average Calculator?

This calculator gives exact P(sum) for specific targets. Dice Average gives E(X), variance, and full distribution.

Dice Probability by the Numbers

1/6
P(2d6 = 7)
6/36
Ways to Roll 7
7,776
Total 5d6 Outcomes
5000 BC
Oldest Known Dice

Disclaimer: This calculator provides exact dice probabilities for educational and gaming reference. Different from a dice roller (which simulates rolls) and dice average (which computes expected values). For critical applications, verify results against established tools like AnyDice.

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