GCSE Maths Practice: tree-diagrams

Question 2 of 9

This is a two-stage probability question. Think carefully about how the contents of the bag change after the first draw.

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\( \begin{array}{l}\textbf{A bag contains 2 red, 3 blue, and 5 green balls.} \\ \text{Two balls are drawn one after the other without replacement.} \\ \text{Find the probability that the first ball is red and the second ball is blue.}\end{array} \)

Diagram

Choose one option:

Use a tree diagram and multiply the probabilities along the correct path.

Sequential Probability Using Tree Diagrams

This question involves drawing two balls one after another from a bag, where the first ball is not replaced before the second is drawn. Problems like this are known as sequential probability questions and are a key topic in GCSE Maths.

When objects are drawn without replacement, the total number of possible outcomes changes after the first draw. This means the probability of the second event depends on what happened in the first event. These events are called dependent events.

Understanding the First Draw

At the start, the bag contains red, blue, and green balls. The probability of drawing a specific colour is found by dividing the number of balls of that colour by the total number of balls in the bag.

Once the first ball is removed, the total number of balls decreases by one. This directly affects the probabilities for the second draw.

Worked Example (Different Numbers)

Suppose a bag contains 4 red balls and 6 blue balls. One ball is drawn and not replaced, then another ball is drawn.

  • Probability of red first = 4/10
  • After removing one red ball, there are 3 red balls left out of 9
  • Probability of red second = 3/9

The probability of drawing two red balls in a row is found by multiplying:

4/10 × 3/9

Using Tree Diagrams Effectively

Tree diagrams are extremely helpful for organising sequential probability questions. Each branch shows a possible outcome, and the probability for that outcome is written on the branch. To find the probability of a specific sequence, you multiply the probabilities along the path.

For example, a path might show:

  • First draw: Red
  • Second draw: Blue

The probability of this sequence is the product of the two branch probabilities.

Common Mistakes

  • Using the original total for the second draw
  • Adding probabilities instead of multiplying them
  • Forgetting that the first ball is not replaced

Real-Life Connections

Sequential probability is used in many real-world situations, such as drawing cards from a deck, selecting items from a box in quality control, or modelling repeated events in games and experiments.

Quick Self-Check

  • Does the total number change after the first event?
  • Are the events dependent?
  • Have you multiplied along the correct path?

Study Tip: If the question says "without replacement", always expect the probabilities to change after the first step.

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