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 Video by Zeeshan Zamurred Probability

Edexcel Statistics Y1 — Probability playlist (Zeeshan Zamurred)Watch the full walkthrough 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

Multiply along, add between

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

along a path: multiply, between paths: add

The two golden rules of tree diagrams.

1Multiply along the path:

0.5 \times 0.5

Answer

0.25

Branches sum to 1

The probabilities on the branches coming out of any single point must add to $1$ (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.

Without replacement

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

1After one red: 2 red of 4 left.

Answer

\frac{2}{4} = \frac{1}{2}

Formula recap

path probability = product of branches

Multiply along a path.

event = sum of path probabilities

Add between paths.

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

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