Edexcel A-level Further Maths revision notes

Decision analysis

Section 7

Year 2

2 specification points

Notes and three levels of exam-style practice for each registered specification point in this section.

Checked against Edexcel 9FM0 section 7

Checked against Edexcel 9FM0 section 7. Review basis: the qualification registry sourced from the Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) specification; registry verification recorded 17 July 2026.

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7.1

Year 2

Use, construct and interpret decision trees.

Notes

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Explanation

A decision tree lays out a sequence of decisions and uncertain events as a branching diagram read from left to right. A decision node, drawn as a square, is a point where the decision maker chooses which branch to take.
A chance node, drawn as a circle, is a point where the outcome is decided by chance, and the probabilities on the branches leaving a chance node must sum to $1$ . The end nodes at the right show the profit or loss for each complete sequence of choices and outcomes.
The tree is then evaluated backwards, from right to left. At a chance node the expected monetary value is calculated, by multiplying each branch's value by its probability and adding.
At a decision node the most favourable value is chosen, and the branches not chosen are marked with two short strokes to show that they are rejected. Working back to the first node gives the value of the whole problem, and the surviving branches give the plan of action.
Costs incurred along a branch, such as a fee for market research, are subtracted from the values that pass through it.

Worked example

A firm may launch product $A$ or product $B$ . Product $A$ succeeds with probability $0.6$ , giving a profit of $£80\,000$ , and otherwise loses $£20\,000$ . Product $B$ succeeds with probability $0.3$ , giving $£150\,000$ , and otherwise loses $£15\,000$ . Which should it launch?

1.Draw a decision node with two branches, $A$ and $B$ , each leading to a chance node.
2.At the chance node for $A$ : $0.6\times80\,000+0.4\times(-20\,000)=48\,000-8\,000$ .
3.So the EMV of $A$ is $£40\,000$ .
4.At the chance node for $B$ : $0.3\times150\,000+0.7\times(-15\,000)=45\,000-10\,500$ .
5.So the EMV of $B$ is $£34\,500$ , and at the decision node the larger value is chosen.

Answer: Launch product $A$ , with an expected monetary value of $£40\,000$ .

Common mistakes

Don't fall into the trap of choosing the branch with the largest possible pay-off rather than the largest EMV.
Don't fall into the trap of taking an expectation at a decision node, which would assume the decision maker chooses at random.
Don't fall into the trap of forgetting to subtract a cost that is incurred on only some of the branches.

Exam tip

Write the value of every node next to it on the diagram and strike through the rejected branches; the mark scheme awards marks for the annotated tree, not only for the final decision.

Tier 1 · Easy

ORIGINAL

State the shapes used for a decision node and for a chance node in a decision tree, and state what calculation is carried out at each when the tree is evaluated.

(3)

(Total for Question 1 is 3 marks)

Tier 2 · Standard

ORIGINAL

A market trader may hire a small stall costing

£200

or a large stall costing

£500

. Demand is high with probability

0.4

and low with probability

0.6

. With the small stall the takings are

£900

if demand is high and

£600

if it is low; with the large stall they are

£1600

and

£700

. Draw the structure of the decision tree in words and determine which stall the trader should hire.

(6)

(Total for Question 1 is 6 marks)

Tier 3 · Hard

ORIGINAL

A firm may commission market research at a cost of

£5000

before deciding whether to launch a product. Without research, the launch succeeds with probability

0.5

, giving

£60\,000

, and otherwise loses

£20\,000

; the alternative is not to launch, which gives nothing. Research reports favourably with probability

0.6

, after which the launch succeeds with probability

0.7

; it reports unfavourably with probability

0.4

, after which the launch succeeds with probability

0.2

. Determine whether the research is worth commissioning.

(8)

(Total for Question 1 is 8 marks)

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7.2

Year 2

Use of expected monetary values (EMVs) and utility to compare alternative courses of action.

Notes

Evidence from your answers: none yet

Your confidence:

Explanation

The expected monetary value of a course of action is the sum of each possible pay-off multiplied by its probability, and choosing the largest EMV is the standard criterion for comparing alternatives. It is the right criterion when the decision is one of many similar ones, because the average is then actually realised.
It is a poor criterion for a single large decision, because it treats a certain $£5000$ as identical to a gamble with a $50\%$ chance of $£10\,000$ and a $50\%$ chance of nothing, whereas most people and most firms would not be indifferent.
Utility replaces money by a measure of its usefulness to the decision maker, through a utility function $u$ , and the criterion becomes the largest expected utility.
A concave utility function, such as $u(x)=\sqrt{x}$ , models a decision maker for whom each extra pound is worth slightly less than the one before; such a decision maker is risk averse and will sometimes reject a gamble with the larger EMV.
The certainty equivalent of a gamble is the sure amount whose utility equals the gamble's expected utility, and the difference between the EMV and the certainty equivalent is the risk premium.

Worked example

A decision maker with utility $u(x)=\sqrt{x}$ must choose between a certain $£5625$ and a gamble giving $£14\,400$ or $£400$ with equal probability. Compare the two using EMV and using utility.

1.EMV of the gamble $=0.5\times14\,400+0.5\times400=7200+200=£7400$ .
2.EMV of the certain amount is $£5625$ , so EMV prefers the gamble.
3.Expected utility of the gamble $=0.5\sqrt{14\,400}+0.5\sqrt{400}=0.5(120)+0.5(20)=70$ .
4.Utility of the certain amount $=\sqrt{5625}=75$ .
5. $75>70$ , so expected utility prefers the certain amount.

Answer: EMV prefers the gamble at $£7400$ ; utility prefers the certain $£5625$ , since $75>70$ . The certainty equivalent of the gamble is $70^2=£4900$ .

Common mistakes

Don't fall into the trap of taking the utility of the expected value instead of the expected value of the utility.
Don't fall into the trap of comparing a utility with a sum of money, which are different quantities.
Don't fall into the trap of assuming a larger EMV always means the better decision.

Exam tip

Set the calculation out as a table of pay-off, probability, utility and product; the marks are for $\sum p_iu(x_i)$ , and taking $u$ of the mean is the error the question is testing for.

Tier 1 · Easy

ORIGINAL

A project gives a profit of

£40\,000

with probability

0.35

, a profit of

£10\,000

with probability

0.45

and a loss of

£20\,000

with probability

0.2

. Find its expected monetary value.

(3)

(Total for Question 1 is 3 marks)

Tier 2 · Standard

ORIGINAL

A decision maker with utility

u(x)=\sqrt{x}

must choose between a certain

£3600

and a gamble giving

£8100

£1600

with equal probability. Determine the choice made under the EMV criterion and under the expected utility criterion, and comment.

(6)

(Total for Question 1 is 6 marks)

Tier 3 · Hard

ORIGINAL

A decision maker with utility

u(x)=\sqrt{x}

chooses between a certain

£5625

and a gamble giving

£14\,400

£400

with equal probability. Show that the two criteria disagree, find the certainty equivalent and the risk premium, and interpret the risk premium.

(8)

(Total for Question 1 is 8 marks)

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