GCSE Maths Practice: mutually-exclusive-events

Question 10 of 10

This question checks your understanding that probabilities of mutually exclusive events can add up to 1.

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\( \begin{array}{l}\textbf{Event X has probability } \frac{4}{7}, \text{ and Event Y has probability } \frac{3}{7}. \\ \text{The events are mutually exclusive.} \\ \text{Find the probability of X or Y.}\end{array} \)

Choose one option:

If the probabilities add to 1, the outcome is certain.

When Probabilities Add Up to 1

In probability, events are described as mutually exclusive if they cannot happen at the same time. This means that if one event occurs, the other event definitely does not. At GCSE level, recognising mutually exclusive events is important because it tells you which probability rule to use.

Sometimes, adding probabilities of mutually exclusive events results in a total probability of 1. This does not mean the calculation is wrong. Instead, it means that one of the listed events is guaranteed to happen.

The Addition Rule

For two mutually exclusive events A and B, the probability that either event happens is:

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

If this sum equals 1, it shows that all possible outcomes are covered.

Worked Example 1

A fair coin is flipped once.

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

These events are mutually exclusive, and together they cover all possible outcomes. Adding the probabilities gives a total of 1, showing that one of these outcomes must occur.

Worked Example 2

A spinner has four equal sections labelled red, blue, green, and yellow.

  • The probability of landing on red is \( \frac{1}{4} \).
  • The probability of landing on blue, green, or yellow combined is \( \frac{3}{4} \).

The spinner must land on one of these colours, so the total probability adds up to 1.

Common Mistakes

  • Thinking probability cannot equal 1: A probability of 1 simply means the event is certain.
  • Adding probabilities without checking exclusivity: Only add probabilities directly when events cannot overlap.
  • Believing a total of 1 means “nothing happens”: It actually means something must happen.

Real-Life Meaning

Probabilities that add up to 1 occur often in everyday situations. For example, when choosing between staying at home or going out, one of the two must happen. In exams, a student either passes or does not pass — these outcomes cover all possibilities.

Understanding this helps students interpret probability values correctly rather than seeing them as abstract fractions.

Frequently Asked Questions

Is a probability of 1 allowed?
Yes. It represents a certain event.

What does a probability of 0 mean?
It represents an impossible event.

Why is this tested at GCSE level?
It checks both calculation skills and understanding of what probability values mean.

Study Tip

If your answer equals 1, check whether the events cover all possible outcomes. If they do, your answer is reasonable.

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