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

Section 1

Decision trees and their terms

A decision tree models a problem with choices and uncertain events, read from left to right. A decision node (drawn as a square) is a point where the decision maker chooses between branches. A chance node (drawn as a circle) is a point where an outcome happens by chance, with a probability on each branch. The probabilities on the branches leaving a chance node add up to 1. At the end of each path is an end node showing the pay-off, usually the profit for that outcome.

Key termsdecision nodechance nodepay-off

Common mistake

Putting probabilities on the branches of a decision node. Only chance-node branches carry probabilities.

Section 2

Constructing a tree

Build the tree from the left. Start with the first decision as a square. Each option leads either to a pay-off or to a chance node, which splits into the possible outcomes with their probabilities. Where the outcome of one event leads to a further decision, add another square after it. Any cost paid along a branch (for example, the price of a survey) is subtracted from the pay-offs on that branch, or at the node where it is incurred.

Exam tip

List every decision and every uncertain event in time order before drawing anything.

Section 3

Expected monetary value

The expected monetary value (EMV) at a chance node is the sum of each pay-off multiplied by its probability: $\text{EMV}=\sum p\times\text{pay-off}$ . Work right to left, a process called folding back. At each chance node calculate the EMV. At each decision node choose the branch with the best value, which is the highest EMV for a profit, and write that value at the node. Example: a café can hire a marquee for £300. In fine weather (probability 0.7) the profit is £1000 with the marquee and £600 without; in poor weather (0.3) it is £200 with the marquee and £400 without. EMV with marquee: $0.7\times1000+0.3\times200-300=£460$ . EMV without: $0.7\times600+0.3\times400=£540$ . The café should not hire the marquee.

Key termsEMVfolding back

Common mistake

Averaging the pay-offs without weighting them by their probabilities, or forgetting to subtract a cost.

Section 4

Multi-stage decisions

In a multi-stage tree, later decisions are made first when folding back. For a decision about commissioning a survey, find the best decision after each possible survey result, then use those values to find the EMV at the survey's chance node, then subtract the survey cost, and compare with the best option without the survey. The difference between the two values is how much the survey adds. A negative EMV on a branch is a loss, so not proceeding (value 0) may be better.

Key termssurvey node

Exam tip

Write the value of each node next to it as you fold back, and cross out the rejected branches.

Section 5

Utility

EMV treats £1 as equally valuable whatever the amount, and ignores risk. Utility is a measure of the personal value of an outcome, usually scaled from 0 (worst) to 1 (best). The decision maker gives a utility to each pay-off, then the expected utility of a chance node is $\sum p\times\text{utility}$ and the option with the highest expected utility is chosen. A risk-averse decision maker prefers a certain sum to a gamble with the same EMV, so the certain sum has a higher utility than the gamble's expected utility. A risk-seeking decision maker does the opposite.

Key termsutilityexpected utilityrisk averse

Section 6

Comparing and evaluating courses of action

EMV is suitable for repeated decisions, because the long-run average matters. For a one-off decision it may mislead, because one outcome occurs, not the average. Probabilities may be estimates, and different utility values can change the decision. When asked to evaluate, state the decision from each method, say whether they agree, and justify a final recommendation with a reason such as risk or the sensitivity of the values.

Key termssensitivity

Common mistake

Stopping at the numbers. Evaluate and compare questions need a conclusion with a reason.

That's the notes covered.

Carry on to the next subtopic.

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).