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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