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Algebra 2 Statistics and probability

Expected value

20 practice questions 0 video lessons Theory + worked examples

Expected Value

California Algebra 2 • Standard S-MD.2 • Statistics & Probability

Expected Value is a topic in Statistics & Probability in the California Common Core State Standards. It is aligned to Standard S-MD.2, which requires students to calculate the expected value of a random variable and interpret it as a long-run mean.

Expected value is the long-run average of a random variable — each outcome weighted by its probability, \(E(X)=\sum x\cdot P(x)\).

California Algebra 2 › Statistics & Probability › Expected Value  —  Standard S-MD.2

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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 \(3.5\).
Expected value of a die The expected value weights each outcome by its probability. 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 fair die: each value, probability 1/6 expected value = 3.5
Each die value weighted by \(\dfrac16\) gives \(E=3.5\).
Expected value Expected value Expected value E(X) = Σ x · P(x) weight each value by its probability the long-run average outcome
Computing an expected value.

The definition:

\[E(X)=\sum x\cdot P(x)\]
expected value is the sum of each value times its probability
Subtract any cost to get the expected net value.

How to find expected value

  1. List each outcome and its probability.
  2. Multiply each outcome by its probability.
  3. Add the products.
  4. 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.

\(E(X)\)\(=\)\(\dfrac{1+2+3+4+5+6}{6}\)
\(=\)\(3.5\)
the expected value is 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.

\(E\)\(=\)\(5(0.2)+0(0.8)\)
\(=\)\(\$1\)
the expected value is 1 dollar
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.

\(E_{\text{net}}\)\(=\)\(1-2=-\$1\)
the expected net is negative 1 dollar
Example 4 — Fair game
When is a game “fair”?
Solution

When the expected net value is \(\$0\) — no advantage to either side.

a game is fair when the expected net value is zero

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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