Lena and Marco have $45,000 to spend: Station Road opened in September 2026. For 2027, Lena and Marco have set aside $45,000 and have two ideas for it.
Option 1: a delivery van
- The Mill Lane bakehouse can supply bread to hotels and cafés
- Costs $30,000
Option 2: a café corner
- In Aisha's High Street shop, with six tables and a coffee machine
- Costs $45,000
Neither result is certain, so they draw a decision tree and put numbers on it.
You already know the parts of a tree: a square where Lena and Marco choose, a circle where chance decides, and a probability on each branch leaving a circle. Now the numbers go in.
At the van's circle, Marco estimates the hotel orders over the next three years. Strong orders (probability 0.6) would bring in $70,000. Weak orders (probability 0.4) would bring in $20,000.
The expected value at the van's circle
Multiply the first branch
Probability times outcome: 0.6 × $70,000 = $42,000.
Multiply the second branch
0.4 × $20,000 = $8,000.
Add the answers
$42,000 + $8,000 = $50,000. This is the van's expected value.
Check the probabilities
0.6 + 0.4 = 1. The branches leaving one circle must always add up to 1, because one of them is certain to happen.
The rule, in one line: Expected value = (probability × outcome) + (probability × outcome) + … for every branch leaving the circle.
Two branches or three, the rule is the same: multiply along each branch, then add.
The café corner has a branch that loses money. If it is busy (probability 0.7) it brings in $90,000. If it is quiet (probability 0.3), the extra staff cost more than the coffee brings in, and it loses $10,000. A table may print that loss as (10,000): brackets mean minus.
Keep the minus sign all the way through: 0.7 × $90,000 = $63,000, and 0.3 × −$10,000 = −$3,000. So the café's expected value is $63,000 − $3,000 = $60,000.
Nobody ever receives the expected value: The van will bring in $70,000 or $20,000. It will never bring in $50,000.
The expected value is an average: what the van would earn per go if Lena could make this decision many times over. It is a way to compare options, not a forecast of what will happen.
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The café's expected value, $60,000, beats the van's $50,000. So the café wins? Not yet.
The café costs $45,000 and the van only $30,000, and that money is spent whatever happens. Take the cost away and you get what each option is expected to add (the net expected value).
| Option 1: delivery van | Option 2: café corner | |
|---|---|---|
| Expected value | (0.6 × $70,000) + (0.4 × $20,000) = $50,000 | (0.7 × $90,000) + (0.3 × −$10,000) = $60,000 |
| Cost | $30,000 | $45,000 |
| Net expected value | $50,000 − $30,000 = $20,000 | $60,000 − $45,000 = $15,000 |
| Decision | Chosen: the higher net expected value | Rejected: crossed out on the tree |
Choose on the net figure: Net expected value = expected value − cost of the option.
On the numbers, Lena and Marco should buy the van: it is expected to add $20,000, against $15,000 for the café. The café looked better only because its cost had not yet been taken off.
Three things to check before you choose
- What the question calls the figure. 'Expected return' and 'net predicted outcome' mean after the cost. 'Expected outcome' may mean before it: look at any figure the table already gives for the other option and match it.
- A negative net figure. If an option's net expected value is below zero, it is expected to lose money, so doing nothing (net $0) would beat it.
- Units. If the table is in $ m, keep the answer in $ m, and put the $ sign in your working or answer.
On the tree, the loser is not rubbed out. It is crossed with two short lines (//) across its branch, next to the square, so anyone reading the tree can see which option was turned down and why.
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A question that says 'construct a fully labelled decision tree' wants the drawing, the working and the choice, all on the page. Here is Lena's tree built in the order you would draw it, from left to right.
Drawing Lena's tree
The square
Draw a small square on the left: the decision Lena and Marco make. Label it 'Spend $45,000 in 2027'.
One branch per option
Two straight lines out of the square. Write the option and its cost on each: 'Van, cost $30,000' and 'Café corner, cost $45,000'.
A circle on each branch
At the end of each option's branch, a circle: the point where chance takes over.
The outcome branches
Out of each circle, one line per outcome, with its probability on it: strong 0.6 and weak 0.4; busy 0.7 and quiet 0.3.
The end figures
At the end of every branch, the outcome: $70,000, $20,000, $90,000, −$10,000.
The results
Above each circle, its expected value and net expected value: $50,000 and $20,000 for the van; $60,000 and $15,000 for the café.
The choice and the key
Cross the café's branch with //. Add a key: square = decision, circle = chance, // = rejected option.
| Branch | Probability | Outcome | At the circle |
|---|---|---|---|
| Van, cost $30,000 | EV $50,000; net $20,000 (chosen) | ||
| Strong orders | 0.6 | $70,000 | |
| Weak orders | 0.4 | $20,000 | |
| // Café corner, cost $45,000 | EV $60,000; net $15,000 (rejected) | ||
| Busy | 0.7 | $90,000 | |
| Quiet | 0.3 | −$10,000 |
Write the working under the tree: Van: (0.6 × $70,000) + (0.4 × $20,000) − $30,000 = $42,000 + $8,000 − $30,000 = $20,000.
Café: (0.7 × $90,000) + (0.3 × −$10,000) − $45,000 = $63,000 − $3,000 − $45,000 = $15,000.
Then one sentence: the best option is the van, because its net expected value, $20,000, is the higher.
What a fully labelled tree needs
- A square for the decision and a circle for every uncertain outcome
- Every probability and every figure in place, and the cost of each option
- The working for each option's expected value, with the $ sign
- The rejected option crossed out with //, and a key explaining the symbols
- The best option named in a sentence, not only marked on the tree
The tree says van, by $5,000. Before Lena signs anything, she asks the question every tree needs: how far can these numbers be trusted, and what do they leave out?
The numbers are estimates
- Marco guessed the 0.7 chance of a busy café; nobody has run a café in the shop before
- The outcomes are forecasts three years ahead, and costs can overrun
- A small change in one estimate can flip the choice
The average hides the risk
- Only one branch will happen, so the expected value never actually arrives
- The café's worst case is −$10,000 − $45,000 = −$55,000; the van's is −$10,000
- A bakery that has just spent $120,000 on Station Road may not survive the worst branch
Some things cannot be counted
- Space and staff: the café takes room from Aisha's counter and needs more of her time
- Fit with the plan, reputation, the hotels' reliability as customers
- Timing: money in 2029 is treated as if it arrived today
| Chance the café is busy | Café: expected value | Café: net expected value | Tree's choice |
|---|---|---|---|
| 0.6 | $50,000 | $5,000 | Van ($20,000) |
| 0.7 (Marco's estimate) | $60,000 | $15,000 | Van ($20,000) |
| 0.75 | $65,000 | $20,000 | A tie |
| 0.8 | $70,000 | $25,000 | Café |
One estimate decides it: Move the café's chance of being busy from 0.7 to 0.8 and the tree changes its mind. The whole answer rests on one guess by Marco.
So a tree's result is only as good as the probabilities and outcomes put into it. When two net figures are close, the numbers alone should not settle the decision.
Here, the numbers and the judgement point the same way. The van has the higher net expected value, the smaller worst case and leaves Aisha's shop as it is. The café is the bigger bet: better if all goes well, much worse if it does not.
Use the tree, then judge it: Calculate, choose the highest net expected value, then ask three questions: how reliable are the estimates, can the business survive the worst branch, and what does the tree leave out?
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How this comes up: This always comes in a case with a table of options, costs, probabilities and outcomes.
How the parts usually run
A short calculation
The expected value, expected return or net predicted outcome of one option, showing all your working, for two marks.
The full job
Construct a fully labelled decision tree for two or three options and identify the best one, for four or six marks.
A later part
Often asks you to use your tree to recommend an option, weighing its limits.
The pattern for a construct question
- Draw it. Square, one branch per option with its cost, a circle on each, the outcome branches with probabilities, every end figure.
- Work it out. For each option: (probability × outcome) + (probability × outcome) − cost, each line written out, with the $ sign.
- Mark it. Each net expected value above its circle, // across the rejected branch, a key for the symbols.
- Say it. One sentence naming the best option and its net expected value.
Three traps: Forgetting the cost. The option with the bigger expected value is not the best until its cost is taken off.
Dropping the minus. A loss of (4,000) at probability 0.5 is −$2,000, not +$2,000.
A bare number. '1,000' on its own is not an answer. Show the working and write '$1,000'.
Using the information in the case, construct a fully labelled decision tree for both options and identify the best option for Lena's Bakery (show all your working).
Model answer plan
See the answer broken into clear points, with guidance on how each point earns marks.