GCSE Maths Practice: mutually-exclusive-events

Question 10 of 10

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

‹

\( \begin{array}{l}\textbf{Two events A and B are mutually exclusive.} \\ \text{Event A has probability } \frac{3}{8} \text{ and event B has probability } \frac{5}{8}. \\ \text{No other outcomes are possible.} \\ \text{Find } P(A \text{ or } B).\end{array} \)

Choose one option:

If mutually exclusive events cover all outcomes, the probability is 1.

Higher-Level Reasoning with Mutually Exclusive Events

At GCSE Higher level, probability questions often go beyond simple calculations and instead test whether students understand what probability values represent. One key idea is recognising when a set of events covers the entire sample space. When this happens, the probability of one event or the other occurring is equal to 1.

Two events are mutually exclusive if they cannot occur at the same time. This means there is no overlap between their outcomes. However, mutual exclusivity alone is not enough to conclude that the probability equals 1. Students must also recognise whether the events together include all possible outcomes.

The Core Rule

For mutually exclusive events A and B:

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

If this total equals 1, then the event is certain.

Worked Example 1 (Sample Space Reasoning)

A fair spinner has 8 equal sections. The outcomes are divided into two groups:

  • Landing on a prime number
  • Landing on a non-prime number

These two events are mutually exclusive and together cover every possible outcome on the spinner. Therefore, the probability of landing on a prime or a non-prime number is equal to 1.

Worked Example 2 (Complementary Events)

A student either passes an exam or does not pass the exam.

  • Passing the exam
  • Not passing the exam

These outcomes are mutually exclusive and exhaustive. One of them must happen, so their combined probability is 1.

Common Higher-Tier Mistakes

  • Assuming addition always gives 1: Probabilities only add to 1 if the events cover all outcomes.
  • Ignoring the sample space: Higher questions often test whether students recognise missing outcomes.
  • Thinking probability 1 means "nothing happens": It actually means the event is guaranteed.

Why This Is Higher GCSE Content

This type of question tests understanding rather than routine calculation. Students must interpret the meaning of probability values and connect numerical results to logical conclusions about certainty.

Frequently Asked Questions

Is probability 1 allowed?
Yes. It represents a certain event.

What does probability 0 mean?
An impossible event.

Why do examiners like these questions?
They test conceptual understanding, not just arithmetic.

Study Tip

Always ask yourself whether the events listed account for every possible outcome. If they do, the probability is 1.

Related Quizzes