AP Stats Expected Value Calculator

An expected value is the long-run average of a discrete random variable. If you repeat a game or experiment many times, E(X) is the average result you should expect per trial — not the most likely single result.

What Is Expected Value?

Expected value is the long-run average value of a random variable, what you would get on average if you repeated the same random process many times. You find it by multiplying each possible value by its probability and adding the results. A game with an expected value of -0.50 dollars means you lose about 50 cents per play on average, even though no single play pays exactly -0.50 dollars.

How to Use This Calculator

1. Type every possible value of X in the first box, separated by commas. 2. Type the matching probability for each value in the second box, in the same order. 3. Click Calculate. Check that the probabilities sum to 1; if they do not, the calculator will show an error instead of a misleading answer.

How to Calculate Expected Value

Formula: E(X) = Σ x · p, variance Var(X) = Σ (x − E(X))² · p, and standard deviation SD(X) = √Var(X).

Worked example: A game pays 0 with probability 0.5, 10 with probability 0.3, and 20 with probability 0.2. Then E(X) = (0 × 0.5) + (10 × 0.3) + (20 × 0.2) = 0 + 3 + 4 = 7. Variance = (0−7)²×0.5 + (10−7)²×0.3 + (20−7)²×0.2 = 24.5 + 2.7 + 33.8 = 61, so SD ≈ 7.81. The long-run average payout is 7 per game, even though 7 is never paid on a single play.

Tips and Common Mistakes

• Do not forget a value with probability 0 — it contributes 0 to E(X) but still belongs in the distribution.
• Probabilities must be decimals (0.5), not percentages (50), in this calculator.
• Pair each x with the correct p; swapping two probabilities changes E(X).
• E(X) does not have to be a possible value — 7 above is an average, not an outcome.
• Variance uses squared distances from E(X), so always find E(X) first.

Frequently Asked Questions

What is expected value in AP Statistics?

It is the mean of a discrete random variable: the average value over many repetitions, calculated as Σ x·p.

Must probabilities sum to 1?

Yes. A complete probability distribution must sum to 1. If yours sums to 0.99 or 1.01, a value or probability is missing or mistyped.

Can expected value be negative?

Yes, if losses outweigh gains — for example, a game where you usually lose money has a negative expected value.

What is the difference between E(X) and a sample mean?

E(X) comes from a known probability distribution. A sample mean comes from observed data. They estimate the same idea, but one is theoretical and one is observed.

How is variance different from standard deviation?

Variance is in squared units. Standard deviation is its square root, back in the original units, so it is easier to interpret.

When do AP Statistics questions ask for expected value?

In probability, random variables, and binomial/geometric settings — for example, expected profit, expected number of successes, or expected winnings.

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