Dice & Coin Expected Value Calculator

How it works

E[one die] = (s + 1) ÷ 2 · Var[one die] = (s² − 1) ÷ 12 · E[total] = n · E[one] · Var[total] = n · Var[one]

Each face of a fair s-sided die is equally likely, so a single die averages (s + 1)/2 — 3.5 for a standard d6. Because expectation adds, the expected total of n independent dice is just n times that. Variance also adds for independent rolls, so the total’s standard deviation grows with the square root of the number of dice, which is why sums of many dice cluster tightly around their mean (the bell-shaped curve of the central limit theorem). A coin is the s = 2 case, averaging 1.5 with faces labelled 1 and 2.

Worked example

Two standard six-sided dice: each averages (6 + 1)/2 = 3.5, so the expected total is 2 × 3.5 = 7. Each die has variance (36 − 1)/12 = 2.917, so the total variance is 5.833 and the standard deviation is √5.833 = 2.42. Totals range from 2 to 12.

Sources: Expected value — Wikipedia

Frequently asked questions