Expected Value: Dice

Calculate expected value for dice scenarios

Expected Value of the Sum of Two Dice

When you roll two fair dice and add the results, what is the expected sum?

Expected value is linear: the expected value of a sum equals the sum of expected values.

This works even if X and Y are not independent!

Let = result of die 1, = result of die 2.

We want .

We know and from the single die calculation.

Answer

E(X + Y) = 7

On average, the sum of two dice is 7.

We can verify by computing :

Sum (s)	Ways	P(S=s)	s × P(s)
2	1
3	2
4	3
5	4
6	5
7	6
8	5
9	4
10	3
11	2
12	1

Linearity of expectation is one of the most powerful tools in probability: