GCSE Maths Practice: mutually-exclusive-events

Question 4 of 10

This question focuses on using the addition rule for mutually exclusive events.

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\( \begin{array}{l}\textbf{Event G has probability } \frac{3}{10}, \text{ and Event H has probability } \frac{2}{10}. \\ \text{The events are mutually exclusive.} \\ \text{Find the probability of G or H.}\end{array} \)

Choose one option:

Confirm that the events cannot occur together before applying the addition rule.

Recognising Mutually Exclusive Events

In GCSE probability, one of the most important skills is identifying how events are related before carrying out any calculations. Events are described as mutually exclusive when they cannot happen at the same time. If one event occurs, the other event is guaranteed not to occur.

This idea is particularly useful because it allows us to use a simple and reliable rule when calculating probabilities. When events do not overlap, there is no risk of counting the same outcome twice.

The Addition Rule for Mutually Exclusive Events

When two events A and B are mutually exclusive, the probability that either A or B 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 six-sided die is rolled once.

  • The probability of rolling an even number is \( \frac{3}{6} \).
  • The probability of rolling an odd number is \( \frac{3}{6} \).

A number cannot be both even and odd at the same time, so these events are mutually exclusive. Using the rule above, the probability of rolling an even or an odd number is found by adding the two probabilities.

Worked Example 2

A bag contains counters of three different colours.

  • The probability of picking a red counter is \( \frac{2}{8} \).
  • The probability of picking a blue counter is \( \frac{3}{8} \).

When one counter is selected, it cannot be both red and blue at the same time. Therefore, these events are mutually exclusive, and their probabilities can be added to find the probability of selecting a red or blue counter.

Common Mistakes to Avoid

  • Adding probabilities when events overlap: If two events can happen together, the addition rule alone is not correct.
  • Missing key words: Phrases like “only one”, “either”, or “cannot happen together” usually indicate mutually exclusive events.
  • Confusing ‘and’ with ‘or’: Mutually exclusive rules apply to “or” situations, not “and”.

Everyday Examples

Mutually exclusive events occur frequently in daily life. For example, if a student chooses one of two optional subjects, selecting one automatically excludes the other. In sports, a match can end in a win or a loss, but not both at the same time.

Recognising these situations helps students understand probability as a way of modelling real-life choices and outcomes.

Frequently Asked Questions

How can I tell if events are mutually exclusive?
Ask whether both events could occur at the same time. If not, they are mutually exclusive.

Is it always correct to add probabilities?
No. You can only add probabilities directly when events do not overlap.

Why is this important at GCSE level?
This topic appears regularly in foundation and higher exam papers and is essential for later probability topics.

Study Tip

Before calculating, classify the events first. Choosing the correct rule often makes the calculation straightforward.

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