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The Addition Rule

FoundationHigherAQAEdexcel

Master The Addition Rule for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. For mutually exclusive events, the probability of A or B is P(A) + P(B).

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

The addition rule finds the probability that one event or another occurs. Which form to use depends on whether the events can both happen.

For mutually exclusive events, \(P(A \text{ or } B) = P(A) + P(B)\). The probabilities simply add, because there is no overlap.

For events that can both occur, \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\). The overlap must be subtracted because it has been counted in both terms. Rolling an even number or a number above 4 illustrates this: 6 satisfies both conditions, so adding without subtracting would count it twice.

Revision notes

Mutually exclusive events

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

The events cannot both happen, so there is no overlap and the probabilities add directly. Check this condition holds before using the simple form.

Events that can overlap

\(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\).

The overlap appears in both \(P(A)\) and \(P(B)\), so it has been counted twice. Subtracting it once corrects this.

Recognising which to use

Ask whether both events could happen on the same trial.

Rolling a 2 or a 5 on one dice: impossible to do both, so add. Rolling an even number or a number above 4: the value 6 does both, so subtract the overlap.

Key points

  • The addition rule handles 'or' questions.
  • For mutually exclusive events, add the probabilities.
  • Otherwise subtract the overlap.
  • \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\).
  • The overlap is counted twice without subtraction.
  • Check whether both events can happen first.

Worked examples

Example 1

A dice is rolled. Work out the probability of an even number or a number greater than 4. [3 marks]

Working

\(P(\text{even}) = \frac{3}{6}\) and \(P(>4) = \frac{2}{6}\)find each probability
The overlap is 6, so \(P(\text{both}) = \frac{1}{6}\)identify and quantify the overlap
\(\frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\)apply the addition rule

Example 2

Two mutually exclusive events have probabilities 0.3 and 0.45. Work out the probability that one or the other occurs. [2 marks]

Working

The events are mutually exclusive, so add: \(0.3 + 0.45\)identify that no overlap exists
\(= 0.75\)work out the probability

Example 3

Explain why the overlap is subtracted in the addition rule. [2 marks]

Working

Outcomes satisfying both events are counted in \(P(A)\) and again in \(P(B)\)state the problem
so subtracting \(P(A \text{ and } B)\) once removes the double countingexplain the correction

Common mistakes

  • Adding without subtracting the overlap.

    This double counts outcomes satisfying both events.

  • Subtracting when the events are mutually exclusive.

    There is no overlap, so nothing needs subtracting.

  • Not checking whether events can both occur.

    That determines which form of the rule applies.

  • Confusing 'or' with 'and'.

    'Or' uses addition; 'and' uses multiplication.

Exam tips

  • Check whether the events can both happen.
  • Identify the overlap explicitly before subtracting.
  • Use addition for 'or' and multiplication for 'and'.
  • Show each probability separately before combining.

Key terms

Addition rule
The rule for finding \(P(A \text{ or } B)\).
Mutually exclusive
Events that cannot both happen.
Overlap
Outcomes satisfying both events.
Double counting
Including the same outcome twice.

Related topics

Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.