GCSE Maths Practice: mutually-exclusive-events

Question 5 of 10

This question tests your ability to combine probabilities and interpret the result.

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\( \begin{array}{l}\textbf{Event P has probability } \frac{1}{4}. \\ \text{Event Q has probability } \frac{1}{2}. \\ \text{Only one event can occur at a time.} \\ \text{Find } P(P \text{ or } Q).\end{array} \)

Choose one option:

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

Higher GCSE Probability: Interpreting Combined Events

At GCSE Higher level, probability questions are designed to test understanding rather than routine calculation. Students must decide whether events overlap, whether they can be combined directly, and what the final probability represents in terms of certainty.

Two events can be combined by addition only if they do not overlap. This means they cannot occur at the same time. At Higher level, this information is often implied rather than stated, so students must infer it from the context.

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: Incomplete Outcomes

A fair spinner is divided into 4 equal sections.

  • The probability of landing on a shaded section is \( \frac{1}{4} \).
  • The probability of landing on a striped section is \( \frac{1}{2} \).

The spinner can land on other types of sections as well. Adding the probabilities gives the chance of landing on either a shaded or striped section, but not certainty.

Worked Example 2: Real-Life Interpretation

A student estimates the probability of travelling to school by bus as \( \frac{1}{4} \) and by car as \( \frac{1}{2} \).

Only one mode of transport is used each day, so the events do not overlap. Adding the probabilities gives the chance of travelling by bus or car, but there is still a chance of walking or cycling.

Common Higher-Tier Mistakes

  • Assuming the result must be 1: Only exhaustive events sum to 1.
  • Adding overlapping events: Overlap must be subtracted if events are not mutually exclusive.
  • Ignoring the meaning of the answer: Higher questions often test interpretation.

Why This Is a Higher Question

This question requires students to recognise that the events do not overlap, apply the correct rule, and interpret a probability that is less than 1. The challenge lies in understanding what the result means, not in performing the addition.

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 an impossible answer like \( \frac{5}{4} \)?
To test whether students understand that probabilities cannot exceed 1.

Study Tip

At Higher level, always check whether all outcomes have been included before deciding if a probability represents certainty.

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