Probability Tree Diagrams: How to Draw Them, Fill Them In and Answer Every Follow-Up Question

Probability tree diagrams turn up in almost every GCSE Maths series, on both Foundation and Higher papers. They’re also one of the most reliable sources of marks. Once you know the two rules (multiply along, add between), the questions become routine. This guide goes through the method step by step, using four worked examples of the kinds of questions exam boards ask.

What is a probability tree diagram?

A tree diagram shows two or more events that happen in order, like spinning a spinner twice, rolling a dice and then flipping a coin, or picking two counters from a bag.

Each event gets its own set of branches. You write an outcome at the end of each branch and its probability on the branch itself. If you follow a route from left to right, you get one complete outcome.

Say Sam spins a spinner with 5 equal sections, 2 marked W (win) and 3 marked L (lose), and plays twice. There are four possible outcomes:

Route	Outcome	Calculation	Probability
Win → Win	WW	2/5 × 2/5	4/25
Win → Lose	WL	2/5 × 3/5	6/25
Lose → Win	LW	3/5 × 2/5	6/25
Lose → Lose	LL	3/5 × 3/5	9/25

Notice that the four probabilities add up to 25/25, which is 1. That’s a useful check.

How to draw a tree diagram step by step

1. First event: draw one branch for each possible outcome (here, Win and Lose).
2. Second event: from the end of every first-stage branch, draw another full set of branches.
3. Label the outcomes at the end of each branch.
4. Write the probabilities on the branches. Every Win branch gets 2/5 and every Lose branch gets 3/5.

Remember: each pair of branches coming from the same point must add up to 1. If they don’t, one of your probabilities is wrong.

In exams, the diagram is often drawn for you, either fully or partly. Almost every question asks you to fill in the probabilities and then answer follow-up questions using them.

Rule 1: Multiply along the branches

Ria rolls a fair six-sided dice and then flips a fair coin. She wins a prize if she rolls a 6 and then gets tails.

• First stage: P(6) = 1/6, P(not 6) = 5/6
• Second stage: P(heads) = 1/2, P(tails) = 1/2, on both sets of branches

To find P(6, T), follow that route and multiply:

1/6 × 1/2 = 1/12

To multiply fractions, multiply the numerators together and the denominators together.

Tip: a common mistake is adding probabilities along a route. Adding makes the answer bigger, but needing two things to happen should make it less likely. If your “both happen” answer is bigger than either single probability, you’ve added when you should have multiplied.

Rule 2: Add between routes

A bag contains 3 green and 5 red counters. A counter is taken out, replaced, and a second counter is taken. The counter goes back, so the probabilities are the same for both picks: P(green) = 3/8 and P(red) = 5/8.

Here are three typical follow-up questions:

Both counters red: there’s only one route.

P(R, R) = 5/8 × 5/8 = 25/64

Exactly one counter red: there are two routes, green-then-red and red-then-green.

P(G, R) = 3/8 × 5/8 = 15/64 P(R, G) = 5/8 × 3/8 = 15/64 Total = 15/64 + 15/64 = 30/64

Both counters the same colour: again there are two routes, both green or both red.

P(G, G) = 3/8 × 3/8 = 9/64 P(R, R) = 25/64 Total = 9/64 + 25/64 = 34/64

You can simplify 34/64 to 17/32, but you don’t need to unless the question asks. If you need a reminder on adding fractions quickly without a calculator, our non-calculator tactics guide covers it.

Did you know? The most common reason students lose marks on these questions is spotting only one route. Before you calculate anything, trace every route with your finger and tick the ones that fit the question.

Tree diagrams with decimals

Sometimes the probabilities are decimals instead of fractions. Luke sits a Maths test and an English test. P(passes Maths) = 0.8 and P(passes English) = 0.7.

You aren’t given the fail probabilities, so subtract from 1:

Test	Pass	Fail
Maths	0.8	1 − 0.8 = 0.2
English	0.7	1 − 0.7 = 0.3

The method stays the same:

• Passes both: 0.8 × 0.7 = 0.56
• Fails both: 0.2 × 0.3 = 0.06
• Passes at least one: “at least one” means one or more, so there are three routes:
  • Pass, Pass: 0.56
  • Pass, Fail: 0.8 × 0.3 = 0.24
  • Fail, Pass: 0.2 × 0.7 = 0.14
  • Total = 0.56 + 0.24 + 0.14 = 0.94

Tip: there’s a faster way to answer “at least one” questions. The only route that doesn’t count is failing both, so P(at least one pass) = 1 − 0.06 = 0.94. You get the same answer from one subtraction, with fewer chances to make a slip. Adding it both ways is also a good check.

When branches stop early

This version comes up often in AQA papers. Lauren rolls a dice at the school fair. If she rolls a 5, she loses straight away. If she rolls anything else, she rolls again and wins if she gets an even number.

• First roll: P(5) = 1/6, P(not 5) = 5/6
• Second roll (only after “not 5”): P(even) = 3/6, P(odd) = 3/6

The “5” branch has no second set of branches, because the game is already over. Don’t draw them just because the other diagrams had them. The diagram looks unfinished, but it’s correct.

P(Lauren wins) = P(not 5, even) = 5/6 × 3/6 = 15/36

P(Lauren loses) has two routes:

• Not 5, then odd: 5/6 × 3/6 = 15/36
• Rolls a 5 first: 1/6 (nothing to multiply, because the route ends there)

To add them, use a common denominator: 1/6 = 6/36.

15/36 + 6/36 = 21/36

Check: 15/36 + 21/36 = 36/36 = 1. Winning and losing cover every possibility, so the check works.

Quick reference: tree diagram rules

Situation	What to do
Along one route	Multiply
More than one route fits	Add the route totals
Missing branch probability	1 − the other branch
With replacement	Second-stage probabilities match the first
”At least one”	1 − P(none)
Event ends early	No further branches on that route
Final check	All route totals add up to 1

Probability is one part of the Statistics content. If you’re revising the whole strand, our guides to mean, median, mode and range and box and whisker plots cover the other topics that come up most often.

Practise tree diagrams with our GCSE apps

The best way to get fluent with tree diagrams is to do lots of them, with different contexts, fractions and decimals, and every type of follow-up question.

• GCSE Statistics covers the full GCSE Statistics and Probability content, including surveys, sampling, charts, averages, spread and probability theory. Tree diagrams, independent events and “at least one” questions sit alongside everything else in the Statistics strand.
• GCSE Maths is the complete revision bundle, with Number, Algebra, Geometry and Statistics in one app and 2900+ questions across the four subject apps. It’s ideal for mixed practice, where a probability question might also need confident fraction arithmetic.

Aim for a handful of tree diagram questions a few times a week. Before long, “multiply along, add between” will come automatically, and those marks will be yours on exam day.