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Dice Mechanics

Source GameMastery Guide pg. 34
The Pathfinder Roleplaying Game uses dice to resolve events during the course of a game, such as whether the fighter hits the vampire or the vampire makes its save against the wizard’s spell. However, the type and number of dice used determines the statistical probability for each numerical outcome, and fiddling with these probabilities can introduce interesting effects.

A single die has an equal chance to produce any of its results; if you roll a d20, there is a 1-in-20 chance for a 1, or a 2, or a 20, and so on. That means those dreaded fumbles and beloved crits come up just as often as an unremarkable 7, 11, or 16. This type of roll result is called a discrete uniform distribution.

Two dice added together do not create an equal distribution of results; if you roll 3d6, there is only a 1-in- 216 chance for an 18 (by rolling three 6s), but a 27-in-216 chance to get an 11 (from multiple combinations of 3 3 4, 2 4 5, 2 3 6, and so on). That means the extreme values at the low and high end are much rarer than the middle values. This type of roll result is called a normal distribution, commonly known as a bell curve because graphing the results gives a line with a hump in the middle that tapers off toward the ends. The more dice you have in a roll, the more probable the middle results become (in the bell curve, the “bell” becomes taller and more narrow, and the rest of the curve is shorter and flatter).

Note that even though a d% is normally generated by two d10s, the result is still a discrete uniform distribution rather than a bell curve because the numbers on the two dice aren’t added together. It’s also worth noting that, when estimating average values such as damage, the average of a d6 is 3.5 rather than 3, as the lowest value possible on most dice is 1, not 0.

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Dice Mechanics