GCSE Maths Practice: tree-diagrams

Question 1 of 9

This question involves two draws without replacement. Think about how the contents of the bag change after the first draw.

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

Diagram

Choose one option:

Use a tree diagram to organise the outcomes and multiply along the branches.

Understanding Probability with Sequential Events

When probability questions involve more than one step, such as drawing two objects one after another, it is important to think carefully about how the situation changes after each step. These types of problems are known as sequential probability questions and are very common in GCSE Maths.

In questions where objects are drawn without replacement, the key idea is that once an object is taken out, it does not go back in. This means the total number of possible outcomes changes, and so do the probabilities for the next step.

Why the Total Changes

At the start, all objects in the bag are available. After the first draw, one object has been removed, leaving fewer objects behind. Because probability is calculated as:

Probability = number of favourable outcomes ÷ total possible outcomes

both the numerator and the denominator may change between draws.

Worked Example (Different Situation)

Suppose a bag contains 4 red balls and 6 yellow balls. One ball is taken out and not replaced, then a second ball is taken.

  • Probability of drawing a red ball first = 4/10
  • After removing one red ball, there are now 3 red balls left out of 9 total
  • Probability of drawing another red ball = 3/9

The probability of drawing two red balls in a row would be:

4/10 × 3/9

Using Tree Diagrams

Tree diagrams are one of the best tools for organising sequential probability questions. Each branch represents a possible outcome, and probabilities are written along the branches. To find the probability of a specific path, you multiply along the branches.

For example, one branch might show:

  • First draw: Blue
  • Second draw: Green

The probability of this path is found by multiplying the probability on the first branch by the probability on the second branch.

Common Mistakes to Avoid

  • Forgetting to reduce the total number after the first draw
  • Using the same probability for both draws
  • Adding probabilities instead of multiplying them

Real-Life Applications

Sequential probability appears in many real-life situations, such as selecting items from a box, quality control in factories, card games, and even genetics. Any situation where one event affects the next can be modelled using this approach.

Quick Check Questions

  • Does the total number of objects change after the first step?
  • Are the events dependent or independent?
  • Should the probabilities be multiplied or added?

Study Tip: If the problem says "without replacement", always expect the probabilities to change between steps.

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