Dice Roller: Probability, Expected Values, and Why Advantage Matters More Than You Think
Dice are probability machines. The distribution of outcomes from different dice combinations has interesting mathematical properties that affect game design and play strategy.
Single Die Distributions
A single fair die produces a uniform distribution — every face equally likely. A d6 (standard six-sided die) has expected value of 3.5 (average roll). A d20 has expected value of 10.5.
Common dice and their ranges:
- d4: 1–4 (expected: 2.5)
- d6: 1–6 (expected: 3.5)
- d8: 1–8 (expected: 4.5)
- d10: 1–10 (expected: 5.5)
- d12: 1–12 (expected: 6.5)
- d20: 1–20 (expected: 10.5)
- d100: 1–100 (expected: 50.5)
Multiple Dice: Why 2d6 ≠ 1d12
Rolling 2d6 (two six-sided dice, sum the results) produces a bell curve distribution — outcomes near 7 are much more likely than outcomes near 2 or 12. Rolling a 7 is 6× more likely than rolling a 12.
Rolling 1d12 produces a flat distribution — every result equally likely.
Same range (2–12 vs 1–12), completely different probability distributions. Game designers use this intentionally: 2d6 damage is more predictable; 1d12 is "swingy."
Advantage in D&D
Rolling with advantage means rolling two d20s and taking the higher result. This shifts the distribution significantly:
- Probability of rolling 20 normally: 5%
- Probability of rolling 20 with advantage: 1 - (19/20)^2 ≈ 9.75%
Expected value of a d20: 10.5
Expected value of d20 with advantage: 13.83
Advantage effectively adds about +3.3 to your average roll — but it's worth more when you need high numbers (nat 20 for critical hits) and less valuable when any success works.
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Frequently Asked Questions
Why is rolling 2d6 better than 1d12 if they have the same range?
2d6 creates a bell curve where results near 7 are 6× more likely than extreme values like 2 or 12, making outcomes more predictable. 1d12 gives every number equal probability, so you're much more likely to roll very high or very low numbers. Game designers prefer 2d6 for damage because it feels more consistent, while 1d12 is "swingy" and creates more dramatic swings in outcomes.
How much does advantage actually improve my chances in D&D?
Advantage increases your expected value from 10.5 to 13.83 on a d20 — roughly a +3.3 bonus on average. More dramatically, your chance of rolling a 20 jumps from 5% to 9.75%, nearly doubling your critical hit probability. The benefit is strongest when you need high numbers and weakest when you just need to beat a low target.
What's the expected value of a d20, and why does it matter?
A standard d20 has an expected value of 10.5, meaning that's your average roll over many attempts. This matters for game balance because it's the midpoint of the die's range (1–20), helping you predict long-term performance and compare bonuses fairly. A +3 bonus is worth 3/20 of your total range, so knowing the baseline helps you assess whether advantage or a flat bonus is better.
Is a +4 bonus better than advantage on a d20?
No — advantage is worth approximately +3.3 on average, but advantage is actually more valuable because it scales with what you need. Advantage dramatically increases your chances of rolling 15+ or 20, while a flat +4 bonus helps equally across all target numbers. For critical hits and high-DC checks, advantage outperforms a +4 bonus despite the lower average value.
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