What is a chance node?

A point where the outcome occurs by chance, with probabilities on the branches; drawn as a circle.

The profit (or loss) shown at the end node of a path.

How do you find the EMV at a chance node?

Multiply each pay-off by its probability and add.

What do you do at a decision node when folding back?

Choose the branch with the best value (highest EMV for profits) and write it at the node.

How do you treat a cost paid along a branch?

Subtract it from the pay-offs on that branch.

A measure of the personal value of an outcome, often scaled from 0 to 1.

How do you use utilities to decide?

Replace pay-offs by utilities, find the expected utility at each chance node and choose the highest.

What does risk averse mean?

Preferring a certain outcome to a gamble with the same EMV.

Name a limitation of EMV.

It ignores risk, and for a one-off decision the average outcome may never actually occur.

Decision trees and expected monetary valueEdexcel A-Level Further Maths: Flashcards

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What is a decision node?

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What is a decision node?: A point where the decision maker chooses a branch; drawn as a square.
What is a chance node?: A point where the outcome occurs by chance, with probabilities on the branches; drawn as a circle.
What is a pay-off?: The profit (or loss) shown at the end node of a path.
How do you find the EMV at a chance node?: Multiply each pay-off by its probability and add.
In which direction do you evaluate a tree?: Right to left (folding back).
What do you do at a decision node when folding back?: Choose the branch with the best value (highest EMV for profits) and write it at the node.
What must the probabilities leaving a chance node total?: 1
How do you treat a cost paid along a branch?: Subtract it from the pay-offs on that branch.
What is utility?: A measure of the personal value of an outcome, often scaled from 0 to 1.
How do you use utilities to decide?: Replace pay-offs by utilities, find the expected utility at each chance node and choose the highest.
What does risk averse mean?: Preferring a certain outcome to a gamble with the same EMV.
Name a limitation of EMV.: It ignores risk, and for a one-off decision the average outcome may never actually occur.
EMV of a 0.3 chance of £100 and a 0.7 chance of £20?: $0.3\times100+0.7\times20=£44$

Exam questions on Decision trees and expected monetary value

A florist must decide whether to order a large or a small stock of roses for a weekend. The weather will be fine with probability 0.6 or poor with probability 0.4. A large stock gives a profit of £800 if the weather is fine and £200 if it is poor. A small stock gives a profit of £500 if fine and £350 if poor.
The probability of fine weather is $p$ instead of $0.6$ . Find the value of $p$ for which the two stock sizes have the same EMV.2 marks
A company must choose between two projects. Project X gives a certain profit of £40 000. Project Y gives a profit of £100 000 with probability 0.5, and otherwise a loss of £20 000.
The directors assign a utility of 0 to a loss of £20 000, 0.8 to a profit of £40 000 and 1 to a profit of £100 000. Calculate the expected utility of each project and state which the company should choose.2 marks
A software firm is deciding whether to launch a new app. Without any research, the probability that the app succeeds is 0.5. A success gives a profit of £120 000 and a failure gives a loss of £50 000; not launching gives £0. Alternatively, the firm can first pay £10 000 for a market survey. The survey result is favourable with probability 0.6, and then the probability of success is 0.75; otherwise it is unfavourable, and then the probability of success is 0.125. The profits and losses above do not include the cost of the survey.
Find the EMV of launching the app after an unfavourable survey result, and state the decision the firm should then make.3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).