2.5StatisticsStretch

Tree Diagrams

Tree diagrams organise sequential events. The branches on the second layer show conditional probabilities, and multiplying along a path gives the probability of a combined outcome.

24 min Video by Zeeshan Zamurred Conditional Probability

Edexcel A level Maths: 2.5 Tree DiagramWatch the full walkthrough before the notes below.

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What you'll be able to do

Draw a probability tree
Place conditional probabilities on later branches
Multiply along branches
Add path probabilities for combined events

Multiplying along branches

The probability of a sequence of outcomes is the product of the probabilities along that path. Later branches carry $conditional$ probabilities.

P (path) = P (first) \times P (second ∣ first)

Multiply along the branches.

P (1st red) = \frac{3}{5}

P (2nd red ∣ red) = \frac{2}{4}

\frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10}

Answer

\frac{3}{10}

Tip — Without replacement, the second branch probabilities change — they are conditional.

Adding paths

For an event that can occur via several paths (e.g. “exactly one red”), find each path probability and add them.

Formula recap

P (path) = \prod (branch probabilities)

Multiply along a path.

P (event) = \sum (path probabilities)

Add the relevant paths.

Common mistakes to avoid

Keeping the same probabilities on the second branches without replacement.

Update them — later branches are conditional.

Adding along a single path.

Multiply along a path; add across different paths.

Key takeaways

Multiply probabilities along a path.
Later branches are conditional (especially without replacement).
Add path probabilities for a combined event.

Test yourself

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