GCSE Maths Practice: mutually-exclusive-events

Question 2 of 10

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

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\( \begin{array}{l}\textbf{Event F has probability } \frac{2}{3}. \\ \text{Event G has probability } \frac{1}{3}. \\ \text{No other outcomes are possible.} \\ \text{Find } P(F \text{ or } G).\end{array} \)

Choose one option:

If all outcomes are included, the probability is 1.

Higher GCSE Probability: Recognising Certainty

At GCSE Higher level, probability questions often focus on interpretation rather than simple calculation. One important idea students must understand is what it means when probabilities add up to 1. A probability of 1 does not indicate an error — it indicates certainty.

Two events are mutually exclusive if they cannot happen at the same time. However, at Higher level, students are often not told this directly. Instead, they must infer it from the context of the problem. More importantly, they must decide whether the events together cover the entire sample space.

The Core Rule

When two events A and B are mutually exclusive:

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

If this sum equals 1, then one of the events must occur.

Worked Example 1: Complementary Events

A student either passes a test or fails it.

  • The probability of passing is \( \frac{2}{3} \).
  • The probability of failing is \( \frac{1}{3} \).

These two outcomes are mutually exclusive and exhaustive. Together, they cover all possible outcomes. Adding the probabilities gives a total of 1, showing that one of the two outcomes is guaranteed.

Worked Example 2: Sample Space Reasoning

A spinner is divided into three equal sections labelled A, B, and C.

  • The probability of landing on A or B combined is \( \frac{2}{3} \).
  • The probability of landing on C is \( \frac{1}{3} \).

Every spin must result in one of these outcomes, so the probability of landing on A, B, or C is 1.

Common Higher-Tier Mistakes

  • Thinking probability 1 is invalid: A probability of 1 simply means certainty.
  • Adding probabilities without checking coverage: Events must cover all outcomes to sum to 1.
  • Relying on keywords: At Higher level, words like “mutually exclusive” may not be given.

Why This Is a Higher-Level Question

This question requires students to recognise that the events are not only mutually exclusive, but also exhaustive. The key challenge is interpreting the result, not performing the addition.

Frequently Asked Questions

What does probability 1 represent?
A guaranteed event.

What does probability 0 represent?
An impossible event.

Why do examiners test this?
To assess understanding of probability as a model of certainty and uncertainty.

Study Tip

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

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