GCSE Maths Practice: mutually-exclusive-events

Question 4 of 10

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

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\( \begin{array}{l}\textbf{Event M has probability } \frac{7}{12}. \\ \text{Event N has probability } \frac{2}{12}. \\ \text{Only one event can occur at a time.} \\ \text{Find } P(M \text{ or } N).\end{array} \)

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 often require students to interpret what a probability represents rather than simply performing calculations. One key skill is recognising when events can be combined directly and understanding what the final value tells you about certainty.

Two events can be added together when they do not overlap, meaning they cannot occur at the same time. In these situations, the probability that one event or the other occurs is found by adding their probabilities. However, Higher-tier questions often test whether students can decide if the combined events represent some outcomes or all outcomes in the sample space.

The Probability Rule

When events A and B are mutually exclusive:

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

If the result is less than 1, there are still outcomes not covered by the events.

Worked Example 1: Incomplete Sample Space

A fair spinner has 12 equal sections.

  • The probability of landing on a multiple of 3 is \( \frac{4}{12} \).
  • The probability of landing on a prime number is \( \frac{3}{12} \).

These outcomes do not overlap, but they do not include every possible result on the spinner. Adding the probabilities gives a value less than 1, showing that some outcomes are not included.

Worked Example 2: Interpreting the Result

A student estimates the probability of catching the bus on time as \( \frac{7}{12} \) and the probability of cycling to school as \( \frac{2}{12} \).

Only one method of transport is used each day, so the events do not overlap. Adding the probabilities gives the chance that the student either catches the bus or cycles, but it does not guarantee arrival, because other options still exist.

Common Higher-Tier Mistakes

  • Assuming the total must be 1: Probabilities only sum to 1 if all outcomes are included.
  • Adding overlapping events: Overlap must be subtracted when events are not mutually exclusive.
  • Ignoring the meaning of the answer: Higher questions often assess interpretation, not just calculation.

Why This Is a Higher Question

This question requires students to recognise that the events do not overlap but also do not cover the entire sample space. The key challenge is understanding what the final probability represents.

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 do examiners include an answer of 1?
To test whether students understand certainty versus likelihood.

Study Tip

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

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