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Tree Diagrams

FoundationHigherAQAEdexcelOCR

Master tree diagrams for GCSE Maths with structured, exam-style practice. This Foundation and Higher resource covers using probability tree diagrams and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Multiply along the branches, and add between different routes.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A tree diagram lays out the outcomes of two or more events in sequence, with a branch for every possible result and its probability written alongside.

The structure makes combined probabilities straightforward. You multiply along a set of branches to find the probability of that particular path, and you add the results of different paths when more than one satisfies the condition.

The key distinction is whether the events are independent. With replacement, the probabilities on the second set of branches are the same as the first. Without replacement, both the numerator and the denominator change, because an item has been removed and not returned.

Revision notes

Building the diagram

Draw one set of branches for the first event and a full set for each outcome of the second. Write the outcome on each branch and its probability alongside.

The probabilities on any one set of branches must add to \(1\), which is a quick check that you have not missed an outcome or made an arithmetic slip.

Multiplying and adding

Multiply along the branches to find the probability of a complete path, because both events must happen.

When several paths satisfy the condition, work out each and add them. For at least one, it is usually faster to find the probability of none and subtract from \(1\).

With and without replacement

With replacement, the second set of probabilities matches the first, because the item is returned.

Without replacement, both parts of the fraction fall. From \(5\) counters with \(3\) red, the first red has probability \(\frac{3}{5}\) but a second red has probability \(\frac{2}{4}\), since one red and one counter overall have gone.

Key points

  • Each branch shows an outcome and its probability.
  • Probabilities on one set of branches add to \(1\).
  • Multiply along branches for and.
  • Add different completed paths for or.
  • With replacement, the probabilities do not change.
  • Without replacement, both numerator and denominator decrease.

Worked examples

Example 1

A bag holds \(3\) red and \(2\) blue counters. One is taken and replaced, then another is taken. Find the probability that both are red.

Working

\[P(\text{red}) = \tfrac{3}{5}\]the probabilities are unchanged because the counter is replaced
\[\tfrac{3}{5} \times \tfrac{3}{5}\]multiply along the branches for both events
\[= \tfrac{9}{25}\]work out the probability

Example 2

A bag holds \(4\) red and \(6\) blue counters. Two are taken without replacement. Find the probability that both are red.

Working

\[P(\text{first red}) = \tfrac{4}{10}\]four of the ten counters are red
\[P(\text{second red}) = \tfrac{3}{9}\]one red has been removed, so both parts of the fraction fall
\[\tfrac{4}{10} \times \tfrac{3}{9} = \tfrac{2}{15}\]multiply along the branches

Example 3

Using the same bag of \(4\) red and \(6\) blue, find the probability of getting at least one red in two draws without replacement.

Working

\[P(\text{no red}) = \tfrac{6}{10} \times \tfrac{5}{9} = \tfrac{1}{3}\]find the probability of drawing two blues
\[1 - \tfrac{1}{3}\]at least one red is the complement of no reds
\[= \tfrac{2}{3}\]work out the probability

Common mistakes

  • Keeping the same denominator without replacement.

    If a counter is not returned, the total falls by one for the second draw, so 4/10 is followed by 3/9, not 3/10.

  • Adding along a single path.

    Both events must happen for that path, so the probabilities are multiplied, not added.

  • Forgetting a second path.

    One red and one blue can happen in two orders, so both paths must be worked out and added.

  • Listing every path for at least one.

    Finding the probability of none and subtracting from 1 is faster and less error-prone.

Exam tips

  • Label every branch with both the outcome and its probability before calculating.
  • Check each set of branches sums to 1 as a quick error check.
  • Leave answers as fractions unless the question asks otherwise.
  • For at least one, use 1 minus the probability of none.

Key terms

Tree diagram
A diagram showing the outcomes and probabilities of a sequence of events.
Independent events
Events where the outcome of one does not affect the probability of the other.
Without replacement
When an item is not returned, so later probabilities change.
Complement
The probability that an event does not happen, equal to \(1\) minus its probability.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.