GCSE Maths Practice: mutually-exclusive-events

Question 9 of 10

This question tests whether you can recognise when events form a complete sample space.

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\( \begin{array}{l}\textbf{Event X has probability } \frac{7}{12}. \\ \text{Event Y has probability } \frac{5}{12}. \\ \text{No other outcomes are possible.} \\ \text{Find } P(X \text{ or } Y).\end{array} \)

Choose one option:

If all outcomes are included, the probability is 1.

Higher GCSE Probability: Certainty and Complete Sample Spaces

At GCSE Higher level, probability questions are often designed to assess understanding rather than routine calculation. One of the most important ideas students must master is recognising when a probability equals 1 and explaining what this means in context.

A probability of 1 represents a certain event. This occurs when all possible outcomes in the sample space are included. At Higher tier, students are not always told explicitly that events are mutually exclusive or exhaustive. Instead, they must infer this from the information given.

The Key Rule

If two events A and B cannot occur at the same time, then:

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

If the total equals 1, this confirms that one of the events must occur.

Worked Example 1: Complementary Events

A student either passes a driving test or does not pass.

  • The probability of passing is \( \frac{7}{12} \).
  • The probability of not passing is \( \frac{5}{12} \).

These events are complementary: they cannot occur together and they cover every possible outcome. Adding the probabilities gives 1, showing certainty that one of the outcomes will happen.

Worked Example 2: Spinner Interpretation

A fair spinner is divided into 12 equal sections. Seven sections are coloured blue and five are coloured red.

Landing on blue and landing on red cannot occur at the same time, and every spin must land on one of these colours. Therefore, the probability of landing on blue or red is 1.

Common Higher-Tier Mistakes

  • Thinking probability 1 is invalid: A probability of 1 means certainty, not an error.
  • Adding probabilities without checking coverage: Probabilities only sum to 1 if all outcomes are included.
  • Relying on keywords: Higher questions often remove words like “mutually exclusive”.

Why This Is a Higher Question

This question requires students to recognise that the two events form a complete sample space. The challenge is not adding fractions, but interpreting the result correctly and understanding what probability 1 represents.

Frequently Asked Questions

What does probability 1 mean?
It means the event is guaranteed to occur.

What does probability 0 mean?
It represents an impossible event.

Why do examiners include answers like 12/12?
To test understanding of equivalence and certainty.

Study Tip

If your final probability equals 1, always check whether the listed events cover every possible outcome.

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