GCSE Maths Practice: mutually-exclusive-events

Question 3 of 10

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

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\( \begin{array}{l}\textbf{Event C has probability } \frac{1}{5}, \text{ and Event D has probability } \frac{2}{5}. \\ \text{The events are mutually exclusive.} \\ \text{Find the probability of C or D.}\end{array} \)

Choose one option:

Confirm that the events cannot overlap before adding their probabilities.

What Are Mutually Exclusive Events?

In probability, two events are described as mutually exclusive if they cannot happen at the same time. This means that when one event occurs, the other event is guaranteed not to occur. Recognising mutually exclusive events is a key skill in GCSE Maths because it determines which probability rule should be used.

For example, when selecting a single card from a deck, drawing a heart and drawing a spade are mutually exclusive events. A card cannot belong to both suits at once. Because there is no overlap between these outcomes, calculating the probability of one event or the other becomes much simpler.

The Probability Rule

When two events are mutually exclusive, the probability of either event happening is found by adding their probabilities:

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

This rule works because there are no outcomes shared between the events. Nothing is counted twice.

Worked Example 1

A fair coin is flipped once.

  • The probability of getting heads is \( \frac{1}{2} \).
  • The probability of getting tails is \( \frac{1}{2} \).

Getting heads and getting tails cannot happen at the same time on a single flip, so these events are mutually exclusive. Using the rule:

\[ P(\text{heads or tails}) = \frac{1}{2} + \frac{1}{2} = 1 \]

This makes sense because one of the two outcomes must occur.

Worked Example 2

A spinner has 5 equal sections labelled A, B, C, D, and E.

  • The probability of landing on A is \( \frac{1}{5} \).
  • The probability of landing on D is \( \frac{1}{5} \).

Since the spinner can only land on one letter at a time, these events are mutually exclusive. The probability of landing on A or D is found by adding the probabilities.

Common Errors Students Make

  • Adding probabilities when events overlap: If two events can happen together, this rule does not apply.
  • Not reading the question carefully: Words such as “only one”, “cannot”, or “either” often indicate mutual exclusivity.
  • Mixing up ‘or’ and ‘and’: Mutually exclusive rules apply to “or” situations, not “and”.

Real-Life Situations

Mutually exclusive events occur frequently in everyday life. When choosing one mode of transport for a journey, you may walk or take a bus, but not both at the same time. In school choices, a student might choose one optional subject from a list, meaning selecting one excludes the others.

Understanding these situations helps students model real-world decisions using probability and apply the correct calculations.

Frequently Asked Questions

How can I quickly tell if events are mutually exclusive?
Ask whether both events could happen at the same time. If the answer is no, they are mutually exclusive.

Can I always add probabilities?
Only when events are mutually exclusive. Otherwise, a different rule is needed.

Why is this important for GCSE exams?
This topic is a foundation concept and often appears in multiple-choice and short-answer questions.

Study Tip

Before calculating, classify the events first. Choosing the correct rule is often more important than the arithmetic itself.

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