P3ProbabilityFoundation & Higher

Tree Diagrams

Tree diagrams organise the probabilities of two or more events happening one after another. The two golden rules: multiply along the branches, and add between different paths.

40 min Video by Maths Genie AQA GCSE Maths

Probability Tree DiagramsWatch the walkthrough, then read the notes below.

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What you'll learn

Draw and label a tree diagram
Multiply probabilities along branches
Add probabilities of different paths
Handle “with” and “without” replacement

Multiply along, add between

Each set of branches must add to 1. To find the probability of a particular path, $multiply$ the probabilities along it. To combine several paths (e.g. “exactly one red”), $add$ the path probabilities.

P (path) = P_{1} \times P_{2}

Multiply along the branches.

1Multiply along the H–H path.

0.5 \times 0.5 = 0.25

Answer

0.25

Without replacement

If an item is not replaced, the second set of probabilities changes (the total goes down by one). This is the most common place to lose marks — update the fractions carefully.

Tip — Branches from the same point always add to 1 — a quick check.

Remember these

P (path) = \prod branches

Multiply along a path.

P (event) = \sum paths

Add the relevant paths.

Watch out for these

Adding along a single path.

Multiply along a path; add across different paths.

Keeping the same fractions without replacement.

Reduce the totals on the second set of branches.

Key takeaways

Branches from a point add to 1.
Multiply along a path; add between paths.
Without replacement, the second probabilities change.

Test yourself

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