Expected Value Formula

The expected value of a random variable is the long-run average outcome of a random process, calculated as the weighted sum of each possible outcome times.

The Formula

E(X)=∑x⋅P(x)E(X) = \sum x \cdot P(x)

When to use: If you played a game forever, expected value is your average result per play. Positive EV = profitable long-term. Negative EV = you'll lose over time. It's the mathematical way to evaluate risky decisions.

Quick Example

Lottery: $1 ticket, 1/1000 chance of $500. EV=11000(500)+9991000(−1)=0.50−0.999=−0.499EV = \frac{1}{1000}(500) + \frac{999}{1000}(-1) = 0.50 - 0.999 = -0.499 You lose ~50¢ per ticket on average.

What This Formula Means

The expected value of a random variable is the long-run average outcome of a random process, calculated as the weighted sum of each possible outcome times its probability. It represents what you would earn or lose on average per trial if the process were repeated infinitely many times.

If you played a game forever, expected value is your average result per play. Positive EV = profitable long-term. Negative EV = you'll lose over time. It's the mathematical way to evaluate risky decisions.

Formal View

For a discrete random variable XX with outcomes x1,x2,…,xkx_1, x_2, \ldots, x_k and probabilities p1,p2,…,pkp_1, p_2, \ldots, p_k, the expected value is E[X]=∑i=1kxi piE[X] = \sum_{i=1}^{k} x_i \, p_i. For a continuous random variable with density f(x)f(x), E[X]=∫−∞∞x f(x) dxE[X] = \int_{-\infty}^{\infty} x \, f(x) \, dx.

Worked Examples

Example 1

medium
A coin is flipped until heads appears. The number of flips NN has P(N=k)=(1/2)kP(N=k)=(1/2)^k for k=1,2,…k=1,2,\dots. Find E(N)E(N).

Answer

22

First step

1
This is geometric with p=0.5p=0.5.

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

hard
You draw cards one at a time without replacement from a standard 52-card deck until you draw an ace. What is the expected number of draws?

Example 3

medium
A game costs \$2 to play. You roll a fair die: if you roll a 6, you win \$10; otherwise, you win nothing. Find the expected value per game.

Common Mistakes

  • Confusing EV with most likely outcome - The safer move is to ask "Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?" and then state the data source, denominator, or variable before interpreting the result.
  • Ignoring EV for emotional decisions - The safer move is to ask "Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?" and then state the data source, denominator, or variable before interpreting the result.
  • Not accounting for all outcomes - The safer move is to ask "Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?" and then state the data source, denominator, or variable before interpreting the result.
  • Choosing expected value from a keyword alone - Keywords like chance, probability, outcome are only clues; the data structure must match the concept.

Why This Formula Matters

Expected Value helps students reason about uncertainty without guessing. It connects outcomes, sample spaces, and event rules so students can decide whether to add, multiply, condition, simulate, or compare long-run behavior.

Frequently Asked Questions

What is the Expected Value formula?

The expected value of a random variable is the long-run average outcome of a random process, calculated as the weighted sum of each possible outcome times its probability. It represents what you would earn or lose on average per trial if the process were repeated infinitely many times.

How do you use the Expected Value formula?

If you played a game forever, expected value is your average result per play. Positive EV = profitable long-term. Negative EV = you'll lose over time. It's the mathematical way to evaluate risky decisions.

Why is the Expected Value formula important in Statistics?

Expected Value helps students reason about uncertainty without guessing. It connects outcomes, sample spaces, and event rules so students can decide whether to add, multiply, condition, simulate, or compare long-run behavior.

What do students get wrong about Expected Value?

Students often know a procedure related to expected value but skip the recognition step: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition? That leads to a calculation or graph that looks reasonable but answers a different question.

What should I learn before the Expected Value formula?

Before studying the Expected Value formula, you should understand: probability basic, weighted average.

← Back to Expected Value See All Examples Practice Problems