GCSE Maths Practice: mutually-exclusive-events

Question 6 of 10

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

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\( \begin{array}{l}\textbf{Event S has probability } \frac{3}{7}. \\ \text{Event T has probability } \frac{2}{7}. \\ \text{Only one event can occur at a time.} \\ \text{Find } P(S \text{ or } T).\end{array} \)

Choose one option:

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

Higher GCSE Probability: Combining Non-Exhaustive Events

At GCSE Higher level, probability questions often require students to interpret what their answers mean rather than simply perform calculations. A common task is deciding whether events can be combined by addition and whether the result represents certainty or only a partial set of outcomes.

Two events can be added together if they do not overlap, meaning they cannot occur at the same time. At Higher tier, this fact is often implied rather than stated directly. Students must recognise this from the context of the problem.

The Probability Rule

When two events A and B do not overlap, the probability that either event occurs is:

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

However, Higher-level questions also require you to decide whether the combined events cover all possible outcomes.

Worked Example 1: Partial Coverage

A fair spinner is divided into 7 equal sections.

  • The probability of landing on a multiple of 2 is \( \frac{3}{7} \).
  • The probability of landing on a multiple of 5 is \( \frac{1}{7} \).

These events do not overlap, but they do not include every possible number on the spinner. Adding the probabilities gives the chance of landing on either type of number, but not certainty.

Worked Example 2: Real-Life Interpretation

A student estimates the probability of travelling to school by bus as \( \frac{3}{7} \) and by car as \( \frac{2}{7} \).

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 total 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 test meaning, not just arithmetic.

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 the result as a non-certain outcome. The challenge lies in understanding the sample space, not simply adding fractions.

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 answers close to the correct one?
To test understanding of probability limits and reasoning 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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