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ScotlandApplications of MathematicsSyllabus dot point

How do you use expected values to choose between options when outcomes are uncertain?

Calculating expected values to compare decisions under risk, using decision tables and decision trees, and justifying a choice while recognising the limits of an expected-value approach.

A focused answer to the SQA Higher Applications of Mathematics decision-making content, covering expected value as a decision criterion, decision tables and trees, comparing options under risk, and the limits of using expected value alone.

Generated by Claude Opus 4.89 min answer

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  1. What this dot point is asking
  2. Decisions under risk
  3. Comparing options by expected value
  4. Decision tables and trees
  5. The limits of expected value
  6. Try this

What this dot point is asking

The SQA wants you to use expected values to compare decisions when outcomes are uncertain, set out the options in a decision table or tree, recommend the choice with the best expected value, and recognise that expected value alone does not capture risk. This applies the probability work to real decision making, the second strand of the planning area.

Decisions under risk

A decision under risk offers several options (actions you can choose), each leading to outcomes that depend on chance events with known probabilities. For example a shop chooses how much stock to order, and the profit depends on uncertain demand. The aim is to choose the option that performs best on average.

Comparing options by expected value

To decide, compute the expected value of each option separately, then compare. For profit you choose the option with the highest expected value; for cost or loss you choose the lowest.

Decision tables and trees

A decision table lists the options as rows and the chance outcomes as columns, with the payoff in each cell, so the expected value of each row can be computed. A decision tree draws the same information as branches: a decision point splits into options, and each option branches into chance outcomes with their probabilities and payoffs.

The limits of expected value

Expected value is a long-run average and ignores risk, the spread of possible outcomes. Two options with the same expected value can differ hugely in risk: one almost certain, the other all-or-nothing. A cautious decision maker (or one who cannot afford a bad outcome) may rationally prefer a safer option with a lower expected value.

The SQA expects you to recommend on expected value but also to comment on risk when the outcomes vary widely, especially where a single play, not a long run, is involved.

Try this

Q1. An option gives £100\pounds 100 with probability 0.40.4 and £300\pounds 300 with probability 0.60.6. Find its expected value. [2 marks]

  • Cue. E=100×0.4+300×0.6=40+180=£220E = 100 \times 0.4 + 300 \times 0.6 = 40 + 180 = \pounds 220.

Q2. Option A has expected profit £500\pounds 500, option B £480\pounds 480. On expected value alone, which is chosen? [1 mark]

  • Cue. Option A, because it has the higher expected profit.

Q3. Two options both have expected value £200\pounds 200, but one is certain and the other ranges from £0\pounds 0 to £400\pounds 400. Why might a cautious person prefer the certain one? [2 marks]

  • Cue. Expected value ignores risk; the certain option avoids the chance of getting nothing, which a risk-averse person values.

Exam-style practice questions

Practice questions written in the style of SQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

SQA Higher Apps style5 marksA trader can stock few, medium or many units. Profit depends on demand. Stocking medium gives £200\pounds 200 if demand is low (probability 0.30.3) and £500\pounds 500 if demand is high (probability 0.70.7). Stocking many gives £50\pounds 50 if low and £800\pounds 800 if high. Find the expected profit of each and recommend a choice.
Show worked answer →

Medium: E=200×0.3+500×0.7=60+350=£410E = 200 \times 0.3 + 500 \times 0.7 = 60 + 350 = \pounds 410 (2 marks).

Many: E=50×0.3+800×0.7=15+560=£575E = 50 \times 0.3 + 800 \times 0.7 = 15 + 560 = \pounds 575 (2 marks).

Stocking many has the higher expected profit, £575\pounds 575 against £410\pounds 410, so on an expected-value basis the trader should stock many (1 mark). Markers reward both expected-value calculations and a recommendation based on the larger expected profit.

SQA Higher Apps style4 marksTwo investments have the same expected return of £1000\pounds 1000, but one has possible outcomes of £950\pounds 950 or £1050\pounds 1050 and the other of £0\pounds 0 or £2000\pounds 2000. Explain why a decision maker might not treat them as equally good despite the equal expected value.
Show worked answer →

Expected value measures only the long-run average, and here both average £1000\pounds 1000 (1 mark).

The second investment is far riskier: its outcomes range from £0\pounds 0 to £2000\pounds 2000, so there is a real chance of getting nothing, while the first is almost certain to return close to £1000\pounds 1000 (2 marks).

A cautious decision maker prefers the low-risk option even at the same expected value, so expected value alone does not capture attitude to risk (1 mark). Markers reward noting the equal expected value, the difference in spread or risk, and that expected value ignores risk preference.

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