GCSE Maths Practice: tree-diagrams

Question 6 of 11

A bag contains 4 red, 2 green and 1 blue ball. Two balls are drawn one after another without replacement.

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\( \begin{array}{l}\textbf{A bag contains 4 red, 2 green and 1 blue ball.}\\ \text{Two balls are drawn one after another without replacement.}\\ \text{Find the probability that the first ball is green and the second ball is red.}\\ \text{(You may use a tree diagram.)} \end{array} \)

Diagram

Choose one option:

At Higher tier, make sure your tree shows all possible colours at each stage before selecting the correct path.

Higher Tree Diagrams with Multiple Outcomes

This question is GCSE Higher because it involves a full tree diagram with three possible outcomes on the second draw. Unlike simpler Foundation questions, you must correctly account for all colours remaining after the first draw, even though only one final path is required.

Understanding the First Stage

The bag contains 7 balls in total: 4 red, 2 green and 1 blue. The first stage of the tree must therefore show three branches: Red, Green and Blue. Each branch must use the correct probability based on the original total of 7.

Why the Second Stage Is Harder

Once a ball has been drawn, it is not replaced. This means both the total number of balls and the number of each colour can change. For example, if a green ball is drawn first, there is now one fewer green ball, but the numbers of red and blue balls stay the same.

Constructing the Tree Diagram

To build the tree correctly:

  • Start with three branches: Red (4/7), Green (2/7), Blue (1/7).
  • From the Green branch, show all possible second outcomes using the new total of 6.
  • Label the second-stage probabilities carefully, making sure only the colour drawn first is reduced.

This structure is what makes the question suitable for Higher tier.

Multiplying Along a Path

Once the tree is drawn, the probability of a specific sequence is found by multiplying along that path. For “Green then Red”, you multiply the probability on the Green branch by the probability on the Red branch coming from it.

Worked Example (Different Numbers)

A box contains 5 red, 3 green and 2 blue counters. Two counters are taken without replacement. Find the probability of drawing green then red.

  • P(Green) = 3/10
  • After green is removed, red = 5 out of 9
  • P(Green then Red) = 3/10 × 5/9

Common Mistakes at Higher Tier

  • Only drawing two branches instead of all possible colours
  • Reducing the wrong colour after the first draw
  • Keeping the denominator the same on the second draw
  • Trying to simplify before multiplying, leading to arithmetic errors

Exam Tip

For Higher tree-diagram questions, always show all possible outcomes at each stage, even if you only need one final path. This demonstrates full understanding and helps avoid logical errors.

Study tip: When there are three or more outcomes, slow down and label every branch carefully before doing any multiplication.

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