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Contents · 0% complete
  1. Learning objectives
  2. Core idea
  3. Key formulae
  4. Worked examples
  5. Explore the graph
  6. Common mistakes
  7. Exam tip
  8. Guided practice
  9. Practice questions
  10. Challenge
  11. Summary
  12. Quick quiz
  13. Same topic, different board
  14. Past paper questions
Learn9FM0Option · Decision Mathematics 2Decision analysisDecision trees and utility
Decision analysis · Subtopic 1 of 1

Decision Analysis

Decision trees show choices (square nodes), chance events (circle nodes) and pay-offs. Roll back from the right: at chance nodes take expected values (EMV); at decision nodes take the best option. Utility replaces money with a measure of satisfaction, capturing attitudes to risk.

Edexcel 9FM0 Option≈ 40 minRevision 1 · updated 2026-10-27
Section 1 of 14

Learning objectives

  • Draw decision trees with decision, chance and end nodes.
  • Find EMVs and roll back to the optimal decision.
  • Use utility functions to compare decisions under risk.
Pause and think

Would you take a guaranteed £500, or a 60% chance of £1000 and a 40% chance of losing £200?

Section 2 of 14

Core idea

Start hereTake the money or gamble?

A guaranteed £500 or a gamble that usually pays £1000 but sometimes loses £200? Decisions like this mix choices you control with chances you don't. A decision tree lays them out so you can compare them fairly.

The big idea

Draw decision (square) and chance (circle) nodes. Roll back from the end: at chance nodes take the expected value; at decision nodes choose the best branch. Utility functions refine this for attitudes to risk.

1. Expected monetary value

EMV=∑pixi\text{EMV}=\sum p_ix_i
where
xix_i
net pay-off of outcome ii.
Worked exampleRolling back

Certain £500 or gamble (0.60.6: £1000; 0.40.4: −-£200)?

Goal
Best option by EMV.
Strategy
EMV at the chance node.
  1. 0.6(1000)+0.4(−200)0.6(1000)+0.4(-200)
    Weight each outcome.
  2. =520=520
    EMV of the gamble.
  3. 520>500520>500
    Decision node: pick the larger.
AnswerGamble, EMV £520.
✓ CheckIndifference when 1200p−200=5001200p-200=500, p=712≈0.58<0.6p=\frac7{12}\approx0.58<0.6 ✓.

2. Utility

Risk aversion with a concave utility
  1. u(x)=x+200u(x)=\sqrt{x+200}
    Concave: each extra pound adds less satisfaction.
  2. u(500)=700≈26.5u(500)=\sqrt{700}\approx26.5
    Certain option.
  3. 0.61200+0.40≈20.80.6\sqrt{1200}+0.4\sqrt{0}\approx20.8
    Expected utility of the gamble.
Utility favours the certain £500, even though its EMV is lower.
Decision tree

Roll back from right to left.

Common misconceptionAveraging decisions Never average at a decision node; you choose, so take the best.
Key takeawayValue of information A survey is worth paying for if EMV with it, minus its cost, beats EMV without it.
Pause and think

Which utility function would describe a risk-seeker?

Summary

  • Roll back.
  • Chance: EMV; decision: best.
  • Utility for risk attitudes.
Key words
EMV
Probability-weighted average pay-off.
Utility
Numerical measure of satisfaction.
HistorySt Petersburg paradox. Daniel Bernoulli (1738) introduced utility to explain why nobody pays a fortune to play a game with infinite expected winnings.
Fun factGame shows. Contestants on "Deal or No Deal" regularly accept bank offers below the EMV: real-life risk aversion.
Section 3 of 14

Key formulae

EMVEMV=∑pi xi\text{EMV}=\sum p_i\,x_i
Expected utilityE(U)=∑pi u(xi)E(U)=\sum p_i\,u(x_i)
TrickRoll back. Start at the right-hand end. Replace each chance node by its expected value and each decision node by its best branch; cross out rejected branches with //.
Section 4 of 14

Worked examples

Example 1 · EMV0 / 2 steps
Certain £500, or gamble: 0.60.6 → £1000, 0.40.4 → −-£200.
    Example 2 · Utility0 / 3 steps
    With u(x)=x+200u(x)=\sqrt{x+200}, compare the options.
      Section 5 of 14

      Explore the graph

      EMV of the gamble against p
      Loading the interactive graph…
      Try this
      1. xx is the probability of winning £1000.
      2. The gamble beats £500 on EMV when p>712p>\frac7{12}.

      Indifference at p=712p=\frac7{12}.

      In the real worldOil drilling. Oil firms decide whether to pay for seismic surveys before drilling using decision trees with test and no-test branches.
      Section 6 of 14

      Common mistakes

      IncorrectAveraging at decision nodes
      CorrectChoose the best branch
      You control decisions; only chance is averaged.
      IncorrectForgetting costs
      CorrectSubtract costs along the branch
      Pay-offs are net.
      IncorrectProbabilities not summing to 1
      CorrectCheck each chance node
      Missing outcomes.

      EMV says gamble

      My utility function: absolutely not.

      Section 7 of 14

      Exam tip

      Exam tipNode shapes. Square = decision, circle = chance, triangle or end = pay-off. Examiners expect the right shapes.
      Section 8 of 14

      Guided practice

      SA-DM-DA-G012 marksL1 FoundationNon-calculatorEMV
      A gamble pays £1000 with probability 0.60.6 and loses £200 with probability 0.40.4. Find its EMV in pounds.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 9 of 14

      Practice questions

      SA-DM-DA-0012 marksL3 ExamNon-calculatorEMV
      For what probability pp of winning £1000 (else losing £200) is the gamble's EMV equal to a certain £500?

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      Salam Academy Original Practice · Proposed mark scheme
      SA-DM-DA-0022 marksL2 StandardCalculatorUtility
      With u(x)=x+200u(x)=\sqrt{x+200}, find the expected utility of the gamble (3 s.f.).

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      Salam Academy Original Practice · Proposed mark scheme

      More questions on every skill in this subtopic are in the question bank under Practise.

      Section 10 of 14

      Challenge

      SA-DM-DA-0032 marksL3 ExamNon-calculatorEMV
      A firm can pay £100 for a survey before deciding. Without it the best EMV is £520; with it the EMV (before paying) is £650. Find the net EMV of surveying.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 11 of 14

      Summary

      • Roll back from the right.
      • Chance: EMV; decision: best.
      • Utility models risk attitude.
      Section 12 of 14

      Quick quiz

      Q1 · SA-DM-DA-Q011 markL1 FoundationNon-calculatorEMV
      When rolling back a decision tree, at a decision node you
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Section 13 of 14

      Same topic, different board

      Specification
      A level Further Mathematics 9FM0
      Where
      Decision 2 · Decision analysis
      Expected
      Decision trees with decision, chance and pay-off nodes; expected monetary value (EMV); roll-back; utility functions and expected utility.
      Calculator
      Permitted; show enough working to earn the method marks.
      Typical style
      Edexcel often opens with a routine part and builds to a "show that" or "prove" part.
      Open in
      9FM0 · Option
      Section 14 of 14

      Past paper questions

      No past paper questions are tagged to this subtopic yet. We add papers every week.

      Browse past papers

      Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel 9FM0, First teaching 2017 (official specification).