GCSE Maths Practice: mutually-exclusive-events

Question 1 of 10

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

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\( \begin{array}{l}\textbf{Event A has probability } \frac{4}{5}. \\ \text{Event B has probability } \frac{1}{10}. \\ \text{Only one event can occur at a time.} \\ \text{Find } P(A \text{ or } B).\end{array} \)

Choose one option:

Use a common denominator before adding probabilities.

Higher GCSE Probability: Combining Events Carefully

At GCSE Higher level, probability questions often require more than simply adding fractions. Students must interpret the structure of the problem, recognise how events relate to one another, and apply probability rules correctly. A key skill is identifying when events are mutually exclusive even if this is not stated directly.

Two events are mutually exclusive if they cannot occur at the same time. In this case, their outcomes do not overlap. When events are mutually exclusive, the probability that one or the other occurs can be found by adding their probabilities.

The Key Rule

For two mutually exclusive events A and B:

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

However, at Higher tier, you are often expected to justify why this rule applies.

Worked Example 1 (Different Denominators)

A bag contains red and blue counters.

  • The probability of picking a red counter is \( \frac{3}{5} \).
  • The probability of picking a blue counter is \( \frac{1}{10} \).

Because a single counter cannot be both red and blue, the events are mutually exclusive. Before adding, convert to a common denominator:

\[ \frac{3}{5} = \frac{6}{10} \]

Then add the probabilities to find the probability of selecting a red or blue counter.

Worked Example 2 (Interpreting the Result)

A student estimates the probability of being late for school as \( \frac{1}{10} \), and the probability of arriving early as \( \frac{4}{5} \).

These events cannot occur at the same time. Adding the probabilities gives a result less than 1, showing that there is still a chance of arriving exactly on time. This interpretation is often what examiners are testing at Higher level.

Common Higher-Tier Errors

  • Adding without checking overlap: Only mutually exclusive events can be added directly.
  • Forgetting to convert fractions: Different denominators must be handled carefully.
  • Assuming the result must be 1: Only exhaustive events sum to 1.

Why This Is a Higher Question

This question tests more than calculation. Students must recognise mutual exclusivity without being told, handle fractions with different denominators, and interpret what the final probability means.

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 remove keywords?
To test understanding rather than memory.

Study Tip

At Higher level, always ask: do these events overlap, and do they cover the whole sample space?

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