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Conditional Probability

HigherHigher tier onlyAQAEdexcel

Revise Conditional Probability for GCSE Statistics with this free worksheet and full mark scheme — Higher tier exam-style questions with worked answers for AQA and Edexcel. Conditional probability is the probability of an event given that another event has already happened.

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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.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

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Topic overview

Conditional probability is the probability of one event given that another has already happened. This is a Higher-only topic.

It is written \(P(A \mid B)\), read as the probability of A given B. The formula is \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\).

The crucial point is that knowing B has happened restricts the possibilities to those within B, so B becomes the new denominator. Reading from a Venn diagram, \(P(A \mid B)\) is the number in the overlap divided by the number in B — not divided by the grand total, which is the most frequent error.

Revision notes

The notation and formula

\(P(A \mid B)\) means the probability of A given that B has happened.

\(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\). The vertical bar means 'given', not division.

Why B becomes the denominator

Knowing B has happened restricts the possibilities to those within B.

Outcomes outside B are no longer possible, so B replaces the grand total as the denominator. This is the whole idea of conditional probability.

Reading from diagrams

From a Venn diagram, \(P(A \mid B)\) is the number in the overlap divided by the total number in B.

From a two-way table, use the relevant row or column total as the denominator rather than the grand total.

Key points

  • Conditional probability assumes an event has happened.
  • \(P(A \mid B)\) means A given B.
  • \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\).
  • Knowing B restricts the possibilities to B.
  • B becomes the new denominator.
  • Do not divide by the grand total.

Worked examples

Example 1

A Venn diagram shows 6 in the overlap and 15 in set B altogether. Work out \(P(A \mid B)\). [2 marks]

Working

\(P(A \mid B) = \frac{\text{number in overlap}}{\text{number in B}} = \frac{6}{15}\)use B as the denominator
\(= \frac{2}{5}\)simplify the fraction

Example 2

\(P(A \cap B) = 0.2\) and \(P(B) = 0.5\). Work out \(P(A \mid B)\). [2 marks]

Working

\(P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.2}{0.5}\)substitute into the formula
\(= 0.4\)work out the conditional probability

Example 3

Explain why the denominator is \(P(B)\) rather than 1. [2 marks]

Working

Knowing that B has happened means outcomes outside B are no longer possiblestate what the condition does
so B becomes the new set of possibilities and replaces the total as the denominatorexplain the consequence

Common mistakes

  • Dividing by the grand total.

    The denominator is the total for the given event.

  • Reversing the condition.

    \(P(A \mid B)\) and \(P(B \mid A)\) are different and usually unequal.

  • Reading the bar as division.

    It means 'given', not divide.

  • Using the overlap as the denominator.

    The overlap is the numerator; the given event is the denominator.

Exam tips

  • Identify the given event and use it as the denominator.
  • Read the bar as 'given'.
  • Check the order — A given B is not B given A.
  • Remember this is a Higher-only topic.

Key terms

Conditional probability
The probability of A given that B has happened.
\(P(A \mid B)\)
Notation for the probability of A given B.
Given
The condition restricting the possible outcomes.
Intersection
\(A \cap B\), used as the numerator.

Related topics

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