GCSE Maths Practice: mutually-exclusive-events

Question 7 of 10

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

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\( \begin{array}{l}\textbf{Event X has probability } \frac{2}{3}. \\ \text{Event Y has probability } \frac{1}{6}. \\ \text{Only one event can occur at a time.} \\ \text{Find } P(X \text{ or } Y).\end{array} \)

Choose one option:

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

Higher GCSE Probability: Combining Fractions with Interpretation

At GCSE Higher level, probability questions are designed to test reasoning as well as calculation. Students must decide whether events can be combined by addition, whether they overlap, and what the final probability tells them about the situation being modelled.

Two events can be added together if they cannot occur at the same time. In Higher-tier questions, this is often implied rather than stated explicitly. Recognising this condition is a key part of the challenge.

The Probability Rule

If two events A and B do not overlap, then:

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

However, the result must always be interpreted.

Worked Example 1: Different Denominators

A bag contains counters of different colours.

  • The probability of selecting a red counter is \( \frac{2}{3} \).
  • The probability of selecting a green counter is \( \frac{1}{6} \).

Because a counter cannot be both red and green at the same time, the probabilities can be added. Before adding, it is helpful to recognise that different denominators may require careful handling.

Worked Example 2: Interpreting the Result

A student estimates the probability of revising maths in the evening as \( \frac{2}{3} \) and revising English as \( \frac{1}{6} \).

Only one subject is revised each evening. Adding the probabilities gives the chance of revising either maths or English, but there is still a chance of revising another subject or not revising at all.

Common Higher-Tier Mistakes

  • Assuming the total must be 1: Probabilities only sum to 1 if all outcomes are included.
  • Adding overlapping events: If events can occur together, overlap must be subtracted.
  • Ignoring interpretation: Higher questions often assess what the answer means, not just how to calculate it.

Why This Is a Higher Question

This question requires students to recognise non-overlapping events, combine fractions with different denominators, and interpret a probability that is less than 1. The challenge lies in reasoning about the sample space rather than performing routine addition.

Frequently Asked Questions

Does a probability of \( \frac{5}{6} \) mean the event is almost certain?
It means the event is very likely, but not guaranteed.

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

Why include \( \frac{1}{2} \) as an option?
To test careful fraction handling and reasoning.

Study Tip

At Higher level, always interpret your final probability by asking whether all possible outcomes have been included.

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