GCSE Maths Practice: mutually-exclusive-events

Question 8 of 10

This question tests your understanding of adding probabilities for mutually exclusive events.

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\( \begin{array}{l}\textbf{Event P has probability } \frac{1}{8}, \text{ and Event Q has probability } \frac{3}{8}. \\ \text{The events are mutually exclusive.} \\ \text{Find the probability of P or Q.}\end{array} \)

Choose one option:

Confirm that the events cannot overlap before adding their probabilities.

Understanding Mutually Exclusive Events

In probability, events are described as mutually exclusive when they cannot occur at the same time. This means that if one event happens, the other definitely does not. Identifying this relationship correctly is essential at GCSE level because it tells you exactly which probability rule to apply.

When events are mutually exclusive, they have no overlap. Each outcome belongs to one event only, which means there is no risk of counting the same outcome twice. This makes probability calculations simpler and more reliable.

The Addition Rule

If two events A and B are mutually exclusive, the probability that either event occurs is calculated using:

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

This formula works because the events do not share any outcomes.

Worked Example 1

A fair six-sided die is rolled once.

  • The probability of rolling a 2 is \( \frac{1}{6} \).
  • The probability of rolling a 6 is \( \frac{1}{6} \).

A single roll cannot result in both numbers at once, so these events are mutually exclusive. The probability of rolling a 2 or a 6 is found by adding the two probabilities.

Worked Example 2

A bag contains counters of different colours.

  • The probability of choosing a red counter is \( \frac{2}{10} \).
  • The probability of choosing a blue counter is \( \frac{3}{10} \).

Only one counter is chosen, so it cannot be both colours at the same time. These events are mutually exclusive, allowing their probabilities to be added.

Common Mistakes to Avoid

  • Adding probabilities when events overlap: If two events can happen together, simple addition is not correct.
  • Not reading the question carefully: Words such as “either”, “only one”, or “cannot occur together” usually indicate mutual exclusivity.
  • Confusing ‘or’ with ‘and’: The addition rule applies to “or”, not “and”.

Real-Life Examples

Mutually exclusive events appear frequently in everyday life. When choosing a single mode of transport, you may walk or take a bus, but not both at the same time. In competitions, a participant may either win first prize or not win it, but cannot do both.

Seeing these examples helps students understand why adding probabilities makes sense in these situations.

Frequently Asked Questions

How do I check if events are mutually exclusive?
Ask whether both events could happen at the same time. If not, they are mutually exclusive.

Can probabilities ever total more than 1?
No. Probabilities represent chances, so the total cannot exceed 1.

Why is this topic important?
It forms the foundation for more advanced probability topics, including non-mutually exclusive events and Venn diagrams.

Study Tip

Always identify the relationship between events before calculating. Choosing the correct rule first makes probability questions much easier.

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