GCSE Maths Practice: tree-diagrams

Question 5 of 11

A bag contains 3 red, 4 green and 5 blue balls. Two balls are drawn one after another without replacement. Use a tree diagram to compare two outcomes.

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\( \begin{array}{l}\textbf{A bag contains 3 red, 4 green and 5 blue balls.}\\ \text{Two balls are drawn one after another without replacement.}\\ \text{Which is more likely: drawing a green ball then a red ball, or drawing a green ball then a blue ball?}\\ \text{(You may use a tree diagram.)} \end{array} \)

Choose one option:

Both options start with Green, so the first probability is the same. Multiply along each path and compare the results using equivalent fractions.

Higher Tree Diagrams: Comparing Two Paths

This question is GCSE Higher because you are not just finding one probability. You must calculate two different path probabilities from the same tree diagram and then compare them. This requires careful handling of dependent events and a clear comparison of fractions.

Why the Events Are Dependent

The balls are drawn without replacement, so after the first draw the total number of balls decreases by 1. The number of each colour remaining may also change depending on what was drawn. Because the second probability depends on the first draw, these are dependent events.

Tree Diagram Structure

A good tree diagram for this situation starts with the first draw. Because the question focuses on outcomes that begin with Green, it is sensible to split the first stage into:

  • Green
  • Not Green

Then, from the Green branch, you expand the second draw into the colours you want to compare (for example, Red and Blue). This keeps the diagram clear and highlights the relevant paths.

Multiplying Along a Path

For a path such as “Green then Red”, you multiply:

P(Green then Red) = P(Green) × P(Red | Green)

The “| Green” part reminds you that the second probability is calculated after one green ball has been removed.

Worked Example (Different Numbers)

A jar contains 2 red, 5 green and 3 blue counters. Two counters are drawn without replacement. Which is more likely: Green then Red or Green then Blue?

  • P(G then R) = 5/10 × 2/9
  • P(G then B) = 5/10 × 3/9

Notice both paths start with the same first step (5/10). The comparison comes down to the second fraction.

Common Mistakes

  • Not changing the denominator: after the first draw, the total becomes one less.
  • Changing the wrong numerator: only the colour drawn first decreases.
  • Comparing fractions incorrectly: use a common denominator or convert to equivalent fractions.
  • Over-expanding the tree: for Higher questions you want enough detail to compare paths, but you still should focus on the relevant branches.

Exam Tip

If two paths share the same first probability (for example, both start with Green), you can often compare the second-step probabilities directly. But you must still show the multiplication and a clear fraction comparison for full marks.

Study tip: In comparison questions, calculate both paths, simplify if helpful, then compare using equivalent fractions.

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