Dice Probability Calculator

Adding dice does not give a flat distribution

One six sided die gives every result equal probability. Two dice do not: seven comes up six times as often as two, because there are six ways to make it and one way to make twelve. Three dice concentrate further, and the distribution approaches a bell curve as more are added, which is the central limit theorem showing up in the most tangible possible form. This is why game designers choose a dice pool rather than a single die when they want results clustered around a typical value rather than spread evenly.

Keep highest is not the same as adding a bonus

Rolling two dice and keeping the higher shifts the average of a six sided die from 3.5 to about 4.47, which sounds like a bonus of one. It is not equivalent: the mechanic raises low results substantially while barely changing high ones, so it compresses the distribution rather than sliding it. A flat bonus moves every outcome equally. The difference matters whenever there is a target number, because the two mechanics change the chance of hitting different targets by different amounts.

Exact enumeration rather than simulation

These probabilities are computed by convolving the distributions exactly, so the results are precise rather than estimated. A simulation of a million rolls still leaves visible error in the tails, which is exactly where the interesting questions are: the chance of a critical result or of a total failure. Exact computation gives those to full precision at no extra cost.

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