GCSE Maths Practice: mutually-exclusive-events

Question 8 of 10

This question tests your ability to combine probabilities and interpret a non-certain result.

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\( \begin{array}{l}\textbf{Event T has probability } \frac{5}{9}. \\ \text{Event U has probability } \frac{3}{9}. \\ \text{Only one event can occur at a time.} \\ \text{Find } P(T \text{ or } U).\end{array} \)

Choose one option:

If the total probability is less than 1, some outcomes are not included.

Higher GCSE Probability: Combining Events Without Overlap

At GCSE Higher level, probability questions are designed to test whether students understand how probability models real situations. This involves more than simply adding fractions. Students must decide whether events overlap, whether they can be combined directly, and what the final probability represents.

Two events can be combined by addition only if they cannot occur at the same time. In Higher-tier questions, this information is often implied rather than stated directly, so students must recognise it from the context. Even when events can be added, it is essential to decide whether the combined events include all possible outcomes.

The Probability Rule

When two events A and B do not overlap:

\[ P(A \text{ or } B) = P(A) + P(B) \]

If the result is less than 1, this shows that some outcomes are not included.

Worked Example 1: Partial Coverage

A fair spinner has 9 equal sections.

  • The probability of landing on an even number is \( \frac{4}{9} \).
  • The probability of landing on a multiple of 3 is \( \frac{2}{9} \).

These outcomes do not overlap, but they do not include every number on the spinner. Adding the probabilities gives the chance of landing on an even number or a multiple of 3, but not certainty.

Worked Example 2: Real-Life Interpretation

A student estimates the probability of revising maths in the evening as \( \frac{5}{9} \) and the probability of revising science as \( \frac{3}{9} \).

Only one subject is revised each evening, so the events do not overlap. Adding the probabilities gives the chance of revising either maths or science, but there is still a chance of revising a different subject or not revising at all.

Common Higher-Tier Mistakes

  • Assuming the result must be 1: Probabilities only add to 1 when all outcomes are included.
  • Adding overlapping events: If events can occur together, overlap must be subtracted.
  • Ignoring interpretation: Higher questions often assess understanding of what the answer means.

Why This Is a Higher Question

This question requires students to recognise that the events do not overlap, apply the correct probability rule, and interpret a result that is less than 1. The challenge lies in reasoning about the sample space, not simply performing the calculation.

Frequently Asked Questions

Does a probability less than 1 mean the event is unlikely?
No. It simply means the event is not guaranteed.

When does probability equal 1?
When all possible outcomes are included.

Why include similar fractions as options?
To test careful reasoning and accuracy.

Study Tip

At Higher level, always ask whether your combined events cover all possible outcomes or only some of them.

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