Conditional Probability
Master conditional probability for GCSE Maths with structured, exam-style practice. This Higher resource covers calculating probabilities and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Update the totals after the first event before finding the second probability.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR 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.
Topic overview
Conditional probability is the probability of an event given that another has already happened. It is written \(P(B \mid A)\), read as the probability of \(B\) given \(A\).
The key effect is that the total shrinks. If you know a counter drawn was red, you are no longer considering the whole bag — only the outcomes consistent with that knowledge.
Without replacement, both the favourable count and the total fall. From \(10\) counters with \(4\) red, a second red after a first has probability \(\frac{3}{9}\), because one red and one counter overall have been removed.
Revision notes
The shrinking total
Once an event is known to have happened, restrict attention to the outcomes consistent with it.
The denominator becomes the number of outcomes in that restricted set, not the original total.
Without replacement
Reduce both the count of the item and the overall total by one.
From \(4\) red in \(10\), a second red has probability \(\frac{3}{9}\), which simplifies to \(\frac{1}{3}\).
Using tables and Venn diagrams
In a two-way table, a conditional probability uses a row or column total as the denominator rather than the grand total.
If asked for the probability someone walks given they are a girl, divide the girls-who-walk cell by the total number of girls.
Key points
- Conditional probability assumes an event has happened.
- Written \(P(B \mid A)\).
- The total shrinks to the restricted set.
- Without replacement, both count and total fall.
- Use a row or column total in a two-way table.
- Multiply along branches on a tree diagram.
Worked examples
Example 1
A bag has \(5\) red of \(12\) counters. One red is drawn and not replaced. Find the probability the next is red.
Working
Example 2
Of \(30\) girls, \(18\) walk to school. Find the probability a girl walks.
Working
Example 3
A bag has \(4\) blue of \(9\). One blue is removed. Find the probability the next is blue.
Working
Common mistakes
Keeping the original total.
Conditional probability restricts the total to the outcomes consistent with what is known.
Reducing only the numerator.
Without replacement, the overall total falls too.
Using the grand total in a two-way table.
A conditional probability uses the relevant row or column total.
Confusing \(P(B \mid A)\) with \(P(A \mid B)\).
The order matters: the condition is what comes after the bar.
Exam tips
- Identify what is already known before choosing the denominator.
- Reduce both numbers when there is no replacement.
- Use the row or column total for conditional questions on tables.
- Read the wording carefully to see which event is the condition.
Key terms
- Conditional probability
- The probability of an event given another has occurred.
- Without replacement
- Not returning an item before the next draw.
- Restricted set
- The outcomes consistent with the known event.
- Given
- Indicating the condition that has already happened.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.