Lesson 14 — Expected Value: Reading Risk and Decisions

Summary

Learners meet expected value as the weighted average of outcomes, use it to judge games, insurance, and real decisions, and see that “fair” is a matter of both the arithmetic and of who bears the risk. The lesson closes with a spaced, interleaved review of the unit’s four strands and assembly of the portfolio.

Objectives

• Use probability rules to compute probabilities of compound events and interpret expected value in everyday decisions. (D05.S4.10.02)

Connection

A lottery ticket costs more than it is, on average, worth — the expected value is less than the price, which is exactly how a lottery can fund anything. An insurance premium costs more than the average payout, because the insurer must pay for the service and the risk; yet insurance is still wise for you, because a rare-but-ruinous loss is worth paying to avoid. Expected value turns “should I?” into a number you can read: multiply each outcome by its chance and add them up. But the number is not the whole story — who can afford the loss changes what “fair” means. The arithmetic is for everyone; so is the right to decide with it.

Materials

• Expected-value worksheet
• Math journal

Preparation

• Copy or draw the expected-value worksheet.
• Retrieval (interleaved, brief): from Lesson 1, exponent rules; Lesson 4, complex numbers; Lesson 5/6, function features; Lesson 9/10, trig; Lesson 12/13, probability rules. Today: the decision side of chance, and the unit’s close.
• Prepare the game and insurance worked examples.

Facilitator note

This lesson is written to the learner (“you”). The idea to land: expected value is the probability-weighted average of outcomes, E = Σ p(x)·x, and it lets us compare risky options — but fairness also depends on who bears the risk, not only on the average. Teach the computation with worked examples (S-011), then let learners evaluate a real decision.

The intellectual lens: expected value is a weighted average — each outcome counts in proportion to its likelihood. The critical-thinking lens: a decision can be arithmetically “unfair” and still personally wise (insurance) — the number informs judgment, it does not replace it. The ethics lens: a game rigged against players, or a loan whose average cost falls hardest on the poor, is a moral question the arithmetic can reveal but not settle. The egalitarian lens: who can absorb a loss changes what “fair risk” means; a 1% chance of ruin is trivial to the wealthy and devastating to the poor — so equal odds are not always equal fairness. The global lens: the mathematical study of expected value grew from questions about dividing stakes in interrupted games of chance, worked out by Pascal and Fermat in the 17th century (S-301) — one chapter in a worldwide history of deciding under uncertainty. Close by assembling the unit portfolio and doing a spaced, interleaved recall of all four strands (S-039).

Procedure

1. Recall, interleaved (5 min). One line each: what is the growth factor for 5% growth? What is i²? What is an asymptote? What is the sine of 30°? What is P(A|B)?
2. Meet expected value (15 min). The expected value of a choice is the weighted average of its outcomes: E = Σ p(x)·x — multiply each outcome by its probability and add. Worked example: a game pays 10 with probability 1/4 and 0 with probability 3/4; its expected value is (1/4)(10) + (3/4)(0) = 2.5. A game is fair if its expected value equals what you pay to play.
3. Read a lottery and an insurance (15 min). Worked examples:
   • Lottery: a ticket costs 5 and has a 1-in-1,000 chance of 1,000 (and otherwise nothing). Expected value = (1/1000)(1000) + (999/1000)(0) = 1 — less than the 5 price, so the ticket is, on average, a losing bet.
   • Insurance: a 1% chance of a 10,000 loss. Expected loss = 100. A premium above 100 pays for the insurer’s cost — yet buying it can still be wise if a 10,000 loss would ruin you. The arithmetic is one thing; your ability to bear the loss is another.
4. Decide and defend (10 min). Choose one real decision — a bet, a repair now vs. later, a crop choice, a journey — compute its expected value, and write two sentences: what the number says, and what it leaves out (who bears the risk).
5. Assemble the portfolio (5 min). Gather this unit’s artifacts — the power-rules card, the complex-number map, the function charts, the surveying work, the probability tables — into a portfolio. On the cover, write one sentence you now believe about what mathematics is for.
6. Close (5 min). Say the difference between “expected value” and “what will happen,” and name one decision the number alone cannot make for you.

Differentiation

• Support: Compute a two-outcome expected value with a worked layout first; use whole-number payoffs.
• Extension: Design a game whose expected value is exactly fair, then modify one payoff to make it unfair and explain the change.

Assessment

• Formative (peer + self): Can the learner compute an expected value, judge a game or premium against it, and explain what the number leaves out?
• Summative (portfolio, unit): The assembled unit portfolio, reviewed for completeness across all four strands and for the learner’s own closing statement.

Home connection

Ask an adult about a real risk they weigh — a purchase, a trip, a repair. Compute its expected value together, and talk about what the number does and does not tell you about the decision.

Resources

• On expected value and the origin of probability theory (Pascal and Fermat): Encyclopaedia Britannica, “Probability and statistics,” https://www.britannica.com/science/probability (S-301).
• On retrieval practice and spacing: Dunlosky et al. (2013), https://doi.org/10.1177/1529100612453266 (S-039).
• On worked examples and guided practice: Kirschner, Sweller & Clark (2006), https://doi.org/10.1207/s15326985ep4102_1 (S-011).