GCSE Maths Practice: mutually-exclusive-events

Question 2 of 10

This question focuses on adding probabilities for events that cannot occur at the same time.

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

Choose one option:

Check that the events cannot happen at the same time, then apply the addition rule.

Mutually Exclusive Events Explained

In GCSE Maths, probability questions often require you to decide how events are related before choosing the correct method. One of the most important relationships is when events are mutually exclusive. This means that the events cannot happen at the same time. If one event occurs, the other definitely does not.

Understanding this idea helps prevent a very common mistake: using the wrong probability rule. When events are mutually exclusive, there is no overlap between them, so outcomes are never counted twice.

The Key Rule (with LaTeX)

For mutually exclusive events A and B, the probability rule is:

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

This rule works because there are no shared outcomes between A and B.

Worked Example 1

A fair six-sided die is rolled.

  • The probability of rolling a 1 is \( \frac{1}{6} \).
  • The probability of rolling a 4 is \( \frac{1}{6} \).

Since a die cannot show both numbers on one roll, these events are mutually exclusive. Using the rule:

\[ P(1 \text{ or } 4) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} \]

This result could then be simplified if required.

Worked Example 2

A spinner is divided into 10 equal sections.

  • The probability of landing on green is \( \frac{3}{10} \).
  • The probability of landing on yellow is \( \frac{2}{10} \).

Because the spinner can only land on one colour at a time, these outcomes are mutually exclusive. The probability of landing on green or yellow is found by adding the probabilities.

Common Mistakes

  • Adding when events overlap: If events can happen together, you must subtract the overlap instead.
  • Ignoring the wording: Phrases like “cannot happen together” or “only one outcome” are clues.
  • Confusing ‘and’ with ‘or’: “Or” usually means add, but only when events are mutually exclusive.

Why This Matters in Real Life

Mutually exclusive events appear in everyday decision-making. For example, when choosing a single transport option for a journey, you might take a bus or a train, but not both at the same time. In games, sports outcomes, and surveys, recognising mutually exclusive choices ensures probabilities are calculated correctly.

Frequently Asked Questions

How can I quickly check if events are mutually exclusive?
Ask yourself whether both events could happen at once. If not, they are mutually exclusive.

Do probabilities always add up neatly?
Yes, but remember the total probability of all possible outcomes cannot exceed 1.

Is this tested at foundation level?
Yes. This is a core GCSE probability skill and often appears in foundation exams.

Study Tip

Before calculating anything, decide which probability rule applies. Identifying whether events are mutually exclusive is often more important than the calculation itself.

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