Probability Tree Diagrams –

Probability tree diagrams are a powerful tool for solving multi-stage probability problems in GCSE Maths. They appear on both Foundation and Higher papers across AQA, Edexcel and OCR, and the Higher tier frequently includes "without replacement" questions. A tree diagram organises all possible outcomes, making it easy to apply the AND and

§Key definitions

AND rule (multiplication):

To find the probability of a sequence of outcomes, multiply along the branches.

OR rule (addition):

To find the probability of one outcome or another (when they are mutually exclusive), add the probabilities of the relevant end results.

Question:

A bag contains 4 red and 6 blue counters. A counter is taken out, its colour noted, and it is replaced. A second counter is then taken. Find the probability that (a) both counters are red, (b) the counters are different colours.

(a)

P(both red) = 4/25.

(b)

P(different colours) = P(Red, Blue) + P(Blue, Red) = 6/25 + 6/25 = 12/25.

§Formulas to memorise

P(A and B) = P(A) × P(B)

P(A or B) = P(A) + P(B)

With replacement — the probabilities stay the same at each stage because the item is put back. The events are independent.

Without replacement — the probabilities change at the second stage because the item is not put back. The total number of outcomes decreases by one. This is tested heavily at Higher level.

AND rule (multiplication):: To find the probability of a sequence of outcomes, multiply along the branches.

OR rule (addition):: To find the probability of one outcome or another (when they are mutually exclusive), add the probabilities of the relevant end results.

Multiply along branches — to find the probability of each path.

Add the path probabilities — for paths that satisfy the event you are asked about.

Worked example

Working:

⚠ Common mistakes

✗Forgetting to adjust probabilities for "without replacement" — the denominator and sometimes the numerator must change for the second stage.
✗Adding instead of multiplying along a branch — multiply along, add between paths.
✗Missing a path — for "at least one", there are usually multiple paths to add. Using the complement (1 minus the opposite) is often quicker and avoids this error.
✗Not simplifying fractions — always reduce to lowest terms unless told otherwise.
✗Probabilities at a branch point not adding to 1 — this is a quick self-check. If they do not add to 1, you have made an error.

✦ Exam tips

→Draw the diagram neatly — use a ruler for branches and clearly label outcomes and probabilities. A messy diagram leads to reading errors.
→Use the complement for "at least one" — it is much faster to calculate 1 − P(none) than to add multiple paths.
→Three-stage trees — some Higher questions have three stages. The method is the same: multiply along and add between. Just be careful with the arithmetic.
→Link to Venn diagrams — some problems can also be solved using Venn diagrams. See Venn diagrams.
→Show all working — write out each path probability and clearly state which paths you are adding. This earns method marks.

MMXXVI specification · AQA · Edexcel · OCRgcsemathsai.co.uk/formulas/probability-tree-diagrams

← All formula sheets Read the full topic guide →