Probability Updated November 2026

Expected value

Expected value is the long-run average outcome: multiply each possible value by its probability and add the products, E(X) = sum(x * P(x)). It is what you would average per play if you repeated the situation many times.

Syntax

E(X) = sum( x * P(x) )

Part	What it is
`x`	each possible outcome value
`P(x)`	its probability; all probabilities add to 1

Examples

One fair die
(1 + 2 + 3 + 4 + 5 + 6) * 1/6 = 21 / 6

Result: 3.5

Raffle: 100 tickets at $2 each, one $100 prize
E(profit) = 0.01 * 100 - 2

Result: -$1 per ticket

Excel: =SUMPRODUCT(values, probabilities)

Bet: win $10 with probability 0.3, lose $5 with probability 0.7
0.3 * 10 + 0.7 * (-5) = 3 - 3.5

Result: -$0.50

Expected number of successes in n trials: n * p
20 emails with a 10% reply rate

Result: 2 replies

Common mistakes

Using probabilities that do not sum to 1; check the list of outcomes is complete.
Forgetting the cost of playing; the expected value of a $2 ticket is the expected prize minus $2.
Expecting the expected value to happen on a single play; 3.5 never shows on a die, it is the average over many rolls.

Practice Expected value

These drills check your answer on live data, free, no account needed.

Expected value of a raffle ticketA charity sells 500 raffle tickets at $2 each. One ticket wins $2,500 and 5 other tickets win $100 each. What is the expected net gain (prize minus ticket price) for one ticket? A loss is negative. Give dollars and cents.

Questions

What is the difference between expected value and mean?

They are the same idea. The mean describes data you have; the expected value is the mean of a probability distribution, before the outcome is known.

How is expected value used in decisions?

Compare the expected value of each option and pick the highest, adjusted for risk. A bet with a negative expected value loses money on average, however it turns out once.

Weighted averageHow to calculate a weighted average: multiply each value by its weight and divide by the total weight. Grade and price examples plus Excel SUMPRODUCT.
Binomial distributionThe binomial formula P(X = k) = nCk p^k (1 - p)^(n - k), its mean np and standard deviation sqrt(np(1 - p)), exact worked examples and Excel BINOM.DIST.
Probability rulesThe basic probability rules with dice and coin examples: the addition rule for A or B, the multiplication rule for independent events and the complement rule.
VarianceHow to calculate variance step by step, why the sample version divides by n - 1, how it relates to standard deviation, and Excel VAR.S and VAR.P.