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S5.4StatisticsCore

Tree Diagrams

Tree diagrams map out a sequence of events, with probabilities on each branch. Two simple rules — multiply along branches, add between paths — handle even complex multi-stage problems, including without-replacement situations.

25 min Probability
Edexcel Statistics Y1 — Probability playlist (Zeeshan Zamurred)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Draw a probability tree diagram
  • Multiply along branches for combined events
  • Add across paths for “or” outcomes
  • Handle problems without replacement
1

Multiply along, add between

To find the probability of a particular path through the tree, multiply\textbf{multiply} the probabilities along its branches. To find the probability of an event that can happen via several paths, add\textbf{add} those path probabilities.

along a path: multiply,between paths: add\text{along a path: multiply}, \quad \text{between paths: add}
The two golden rules of tree diagrams.
Example 1Two independent events each with P = 0.5. Find P(both happen).
1Multiply along the path: 0.5×0.50.5 \times 0.5.
Answer0.250.25
2

Branches sum to 1

The probabilities on the branches coming out of any single point must add to 11 (the event and its complement cover all possibilities).

Tip — Each set of branches from a node sums to 1 — a quick way to find a missing branch probability.

3

Without replacement

If an item is not replaced, the probabilities on the second\textbf{second} set of branches change, because the total and the favourable counts have both decreased.

Example 13 red, 2 blue. Draw two without replacement. P(2nd red | 1st red)?
1After one red: 2 red of 4 left.
Answer24=12\tfrac{2}{4} = \tfrac{1}{2}

Formula recap

path probability=product of branches\text{path probability} = \text{product of branches}
Multiply along a path.
event=sum of path probabilities\text{event} = \text{sum of path probabilities}
Add between paths.
branches from a node sum to 1\text{branches from a node sum to } 1
Completeness.

Common mistakes to avoid

Adding along a single path instead of multiplying.
Multiply along a path; add only between separate paths.
Keeping the same probabilities on the second branches without replacement.
Update the second-stage probabilities — the counts have changed.

Key takeaways

  • Multiply along branches; add between paths.
  • Branches out of any node sum to 1.
  • Without replacement, second-stage probabilities change.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.