GCSE Maths Practice: mutually-exclusive-events

Question 9 of 10

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

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\( \begin{array}{l}\textbf{Event X has probability } \frac{2}{7}, \text{ and Event Y has probability } \frac{3}{7}. \\ \text{The events are mutually exclusive.} \\ \text{Find the probability of X or Y.}\end{array} \)

Choose one option:

Confirm that the events cannot overlap before adding their probabilities.

Mutually Exclusive Events: A Clear Explanation

In probability, events are described as mutually exclusive when they cannot happen at the same time. If one event occurs, the other event definitely does not. This idea is a key foundation topic in GCSE Maths because it determines which rule you should use when combining probabilities.

When events are mutually exclusive, they have no overlap. Each possible outcome belongs to only one event, meaning no outcome can be counted twice. This makes probability calculations simpler and helps avoid common errors.

The Addition Rule for Mutually Exclusive Events

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

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

This formula works because there are no shared outcomes between the events.

Worked Example 1

A fair spinner is divided into 7 equal sections numbered 1 to 7.

  • The probability of landing on number 2 is \( \frac{1}{7} \).
  • The probability of landing on number 6 is \( \frac{1}{7} \).

The spinner can only land on one number at a time, so these outcomes are mutually exclusive. The probability of landing on 2 or 6 is found by adding the two probabilities.

Worked Example 2

A bag contains counters of three different colours.

  • The probability of selecting a red counter is \( \frac{3}{10} \).
  • The probability of selecting a green counter is \( \frac{4}{10} \).

Only one counter is chosen, so it cannot be both colours at once. These events are mutually exclusive, so their probabilities can be added.

Common Mistakes Students Make

  • Adding probabilities when events overlap: If two events can happen together, simple addition will give an incorrect result.
  • Not checking the wording: Phrases such as “either”, “only one”, or “cannot occur together” are strong clues.
  • Mixing up ‘or’ and ‘and’: The addition rule applies to “or” situations, not “and”.

Real-Life Situations

Mutually exclusive events appear frequently in everyday life. When choosing a single dessert, you might choose cake or ice cream, but not both at the same time. In sports, a match might end in a win or a loss, but not both.

Recognising these situations helps students understand why probabilities can be added and how probability models real-world decisions.

Frequently Asked Questions

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

Can probabilities ever add up to more than 1?
No. The total probability of all possible outcomes cannot exceed 1.

Why is this topic important at GCSE level?
It forms the basis for more advanced probability topics such as non-mutually exclusive events and Venn diagrams.

Study Tip

Always decide whether events overlap before calculating. Choosing the correct rule first makes probability questions much easier.

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