GCSE Maths Practice: mutually-exclusive-events

Question 5 of 10

This question tests your ability to apply the addition rule for mutually exclusive events.

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\( \begin{array}{l}\textbf{Event J has probability } \frac{1}{6}, \text{ and Event I has probability } \frac{2}{6}. \\ \text{The events are mutually exclusive.} \\ \text{Find the probability of J or I.}\end{array} \)

Choose one option:

Check that the events cannot happen together before adding their probabilities.

Mutually Exclusive Events in Probability

In probability, understanding the relationship between events is just as important as performing calculations. Two events are described as mutually exclusive when they cannot happen at the same time. If one event occurs, the other is guaranteed not to occur.

This concept is frequently tested in GCSE Maths because it helps students choose the correct rule when combining probabilities. When events are mutually exclusive, their outcomes do not overlap, making calculations straightforward.

The Addition Rule

When 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 rule works because there are no shared outcomes between the two events. Each probability represents a separate set of outcomes.

Worked Example 1

A fair spinner is divided into 6 equal sections.

  • The probability of landing on section 2 is \( \frac{1}{6} \).
  • The probability of landing on section 5 is \( \frac{1}{6} \).

The spinner can only land on one section at a time, so these outcomes are mutually exclusive. To find the probability of landing on section 2 or section 5, the probabilities are added.

Worked Example 2

A bag contains counters numbered from 1 to 4.

  • The probability of picking a counter numbered 1 is \( \frac{1}{4} \).
  • The probability of picking a counter numbered 3 is \( \frac{1}{4} \).

When one counter is chosen, it cannot be both numbers at once. Therefore, these events are mutually exclusive, and their probabilities can be combined using addition.

Common Mistakes

  • Adding probabilities when events overlap: If two events can happen at the same time, this rule does not apply.
  • Ignoring key words: Phrases such as “only one outcome” or “cannot occur together” signal mutually exclusive events.
  • Confusing ‘and’ with ‘or’: The addition rule applies to “or” situations, not “and”.

Real-Life Contexts

Mutually exclusive events are common in everyday life. When choosing a seat in a cinema, you may sit in seat A or seat B, but not both at once. In games, a player may win or lose, but cannot do both simultaneously.

Recognising these situations helps students connect probability theory with real-world decision-making.

Frequently Asked Questions

How can 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?
It forms the foundation for more advanced probability topics, including non-mutually exclusive events.

Study Tip

Always decide which probability rule applies before doing any calculations. Correct classification makes probability questions much easier.

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