The Math of Roleplaying Games
Understanding standard deviation and bell curves in analog gaming mechanics.
The Expected Value
The foundation of table-top odds is the "expected value" (EV) of a die roll. For a fair die, the EV is simply the average of all faces:
A standard 6-sided die (d6) has an EV of 3.5. A 20-sided die (d20) has an EV of 10.5. When you roll multiple dice, such as 3d6, the expected value is naturally the sum of their individual EVs: 3 * 3.5 = 10.5.
Combinatorics and The Bell Curve
When you roll a single d20, every outcome from 1 to 20 has the exact same probability of 5%. This is a "flat distribution."
However, when you roll multiple dice, the distribution shifts dramatically.
Consider rolling 2d6. There are $6^2 = 36$ total possible outcomes. There is only ONE way to roll a sum of 2 (1+1) yielding a $1/36 = 2.77\%$ chance. There are SIX ways to roll a 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) yielding a $6/36 = 16.66\%$ chance.
The more dice you add, the more extreme the "Bell Curve" (normal distribution) becomes. Central numbers become exponentially more likely, and extremes become vanishingly rare.