Algebra 2
Statistics and probability
Expected value
20 practice questions
0 video lessons
Theory + worked examples
Theory
The expected value is the long-run average outcome, weighting each value by its probability:
\[E(X)=\sum x\cdot P(x).\]
It need not be a possible outcome — a die's expected value is \(3.5\).
Each die value weighted by \(\dfrac16\) gives \(E=3.5\).
Computing an expected value.
The definition:
\[E(X)=\sum x\cdot P(x)\]
Subtract any cost to get the expected net value.
How to find expected value
- List each outcome and its probability.
- Multiply each outcome by its probability.
- Add the products.
- Subtract any cost for a net value.
Example 1 — Fair die
Find the expected value of a fair die roll.
Solution
Average the outcomes weighted equally.
| \(E(X)\) | \(=\) | \(\dfrac{1+2+3+4+5+6}{6}\) |
| \(=\) | \(3.5\) |
Example 2 — A simple game
A game pays \(\$5\) with probability \(0.2\) and \(\$0\) otherwise. Find \(E\).
Solution
Weight each payout by its probability.
| \(E\) | \(=\) | \(5(0.2)+0(0.8)\) |
| \(=\) | \(\$1\) |
Example 3 — Net value with a cost
The game in Example 2 costs \(\$2\) to play. Find the expected net.
Solution
Subtract the cost from the expected payout.
| \(E_{\text{net}}\) | \(=\) | \(1-2=-\$1\) |
Example 4 — Fair game
When is a game “fair”?
Solution
When the expected net value is \(\$0\) — no advantage to either side.
Common pitfalls
Weight by probability — don't just average the outcomes unless equally likely.
Include a cost for the net value.
The expected value may not be an achievable outcome.
Frequently asked questions
What is expected value?
The long-run average of a random variable, \(\sum x\,P(x)\).
How do you compute it?
Multiply each outcome by its probability and add.
Can the expected value be impossible?
Yes — a die's expected value is \(3.5\), not a face.
When is a game fair?
When the expected net value is zero.
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