GCSE Maths Practice: mutually-exclusive-events

Question 3 of 10

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

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\( \begin{array}{l}\textbf{Event H has probability } \frac{4}{9}. \\ \text{Event I has probability } \frac{5}{9}. \\ \text{No other outcomes are possible.} \\ \text{Find } P(H \text{ or } I).\end{array} \)

Choose one option:

If all outcomes are included, the probability is 1.

Higher GCSE Probability: Complete Sample Spaces

At GCSE Higher level, probability questions often assess whether students understand what probability values represent, not just how to calculate them. A key concept is recognising when a set of events forms a complete sample space. When all possible outcomes are included, the total probability is equal to 1.

Two events are mutually exclusive if they cannot occur at the same time. However, Higher-level questions often do not state this directly. Instead, students must infer it from the situation and decide whether the events also cover every possible outcome.

The Core Rule

If events A and B are mutually exclusive, then:

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

If the result equals 1, this shows that one of the events is guaranteed to occur.

Worked Example 1: Exhaustive Events

A student estimates the probability that it will rain tomorrow as \( \frac{5}{9} \).

The probability that it will not rain is therefore \( \frac{4}{9} \).

These two outcomes are mutually exclusive and exhaustive. Together, they cover all possibilities, so the probability that it rains or does not rain is equal to 1.

Worked Example 2: Spinner Outcomes

A fair spinner is divided into 9 equal sections. Four sections are coloured red, and the remaining sections are coloured blue.

Landing on red and landing on blue cannot happen at the same time, and every spin must result in one of these outcomes. Therefore, the combined probability of landing on red or blue is 1.

Common Higher-Tier Errors

  • Thinking probability 1 is incorrect: A probability of 1 means certainty, not a mistake.
  • Adding probabilities without checking coverage: Events must include all outcomes to sum to 1.
  • Relying on keywords: Higher questions may not explicitly say “mutually exclusive”.

Why This Is a Higher Question

This question requires interpretation of probability values and understanding of sample spaces. The challenge lies in recognising that the events are both mutually exclusive and exhaustive, not in performing the addition.

Frequently Asked Questions

What does probability 1 represent?
A certain event that must happen.

Can probabilities ever be greater than 1?
No. The total probability of all possible outcomes is always 1.

Why do examiners include answers like 9/9?
To check whether students understand equivalence and certainty.

Study Tip

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

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