Expected Value: Dice

Calculate expected value for dice scenarios

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Expected Value of a Single Die Roll

What is "expected value"?

Expected value (E(X)) is the long-run average of a random variable. If you roll a die many times, E(X) is what your average roll will converge to. It's also called the "mean" of the probability distribution.

How do we calculate it?

We multiply each possible outcome by its probability, then add them all up:

This "weights" each outcome by how likely it is.

Question

What is the expected value when rolling a fair die?

Step-by-Step Solution

Write the Expected Value Formula

List All Outcomes and Probabilities

Outcome (x)	Probability P(x)	x × P(x)
1
2
3
4
5
6

Sum All x × P(x) Values

Simplify the Result

Answer

E(X) = 3.5

On average, you expect to roll 3.5 per roll over many trials.

Why 3.5?

The expected value 3.5 is the midpoint of the outcomes 1 through 6:

This shortcut works for any fair die with consecutive integers starting from 1.

Expected Value for Different Dice

Die Type	Outcomes	E(X)
4-sided (d4)	1, 2, 3, 4
6-sided (d6)	1, 2, 3, 4, 5, 6
8-sided (d8)	1, 2, ..., 8
10-sided (d10)	1, 2, ..., 10
12-sided (d12)	1, 2, ..., 12
20-sided (d20)	1, 2, ..., 20