Probability from two-way tables is a staple question on GCSE Maths papers at Foundation and Higher tier for AQA, Edexcel and OCR. At Foundation level you calculate straightforward probabilities by reading values from the table. At Higher level, you face "given that" (conditional) probability questions where the denominator changes. This guide covers both types clearly, with worked examples that mirror real exam questions. For a full overview of every topic, see our complete GCSE Maths topics list.
What Is Probability from a Two-Way Table?
A two-way table displays frequencies for two categorical variables. You can calculate probabilities by dividing relevant cell frequencies by appropriate totals.
- Simple probability — divide the cell frequency by the grand total.
- Conditional probability ("given that") — divide the cell frequency by the row or column total that matches the given condition.
Key Formulas
The conditional probability restricts the sample space to only those items satisfying condition B.
Step-by-Step Method
- Read the table — identify what each row and column represents.
- Identify the event — what outcome does the question ask about?
- Decide the denominator:
- For simple probability, use the grand total.
- For "given that" questions, use the row or column total for the given condition.
- Find the numerator — the frequency of the specific cell or cells that satisfy the event.
- Calculate and simplify the fraction.
Worked Example 1 — Foundation Level
Question: 100 students were surveyed about their favourite subject.
| Maths | English | Science | Total | |
|---|---|---|---|---|
| Boys | 22 | 12 | 16 | 50 |
| Girls | 18 | 20 | 12 | 50 |
| Total | 40 | 32 | 28 | 100 |
A student is chosen at random. Find (a) P(Maths), (b) P(girl who prefers English).
Working:
(a) Maths total = 40. Grand total = 100. P(Maths) = 40/100 = 2/5.
(b) Girls who prefer English = 20. P(girl and English) = 20/100 = 1/5.
Answer: (a) 2/5 (b) 1/5.
Worked Example 2 — Higher Level
Question: Using the table above, find the probability that a randomly chosen student prefers Science, given that the student is a boy.
Working:
The condition is "given that the student is a boy." The denominator is the boys total = 50.
Boys who prefer Science = 16.
P(Science | boy) = 16/50 = 8/25.
Answer: 8/25.
Worked Example 3 — Exam Style
Question: 160 employees are classified by department and contract type.
| Sales | IT | Admin | Total | |
|---|---|---|---|---|
| Full-time | 30 | 24 | 36 | 90 |
| Part-time | 20 | 16 | 34 | 70 |
| Total | 50 | 40 | 70 | 160 |
(a) Find the probability that a randomly chosen employee works in IT. (b) Find the probability that an employee is full-time, given that they work in Admin. (c) Find the probability that a part-time employee works in Sales.
Working:
(a) P(IT) = 40/160 = 1/4.
(b) Condition: works in Admin. Admin total = 70. Full-time Admin = 36. P(full-time | Admin) = 36/70 = 18/35.
(c) Condition: part-time. Part-time total = 70. Part-time Sales = 20. P(Sales | part-time) = 20/70 = 2/7.
Answer: (a) 1/4 (b) 18/35 (c) 2/7.
Common Mistakes
- Using the grand total as the denominator for conditional probability. When the question says "given that" or restricts to a specific group, the denominator is the row or column total for that group, not the grand total.
- Misidentifying which row or column to use. Read "given that" carefully — the word after "given that" tells you which total to use as the denominator.
- Not simplifying fractions. Always simplify your probability to its lowest terms.
Exam Tips
- The phrase "given that" always signals conditional probability — change your denominator.
- Phrases like "of the boys" or "among the full-time workers" also signal conditional probability even without using the exact words "given that."
- At Foundation level, the denominator is almost always the grand total. At Higher level, watch for the conditional twist.
- For basic two-way table skills, see two-way tables. For more on conditional probability, see conditional probability.
Practice Questions
Q1 (Foundation): From the employees table above, find the probability that a randomly selected employee works in Sales.
Q2 (Foundation): Find the probability that a randomly chosen employee is part-time and works in IT.
Q3 (Higher): Given that an employee works in Sales, find the probability that they are part-time.
Practise probability from two-way tables for free on GCSEMathsAI.
Related Topics
Summary
To find a simple probability from a two-way table, divide the relevant frequency by the grand total. For conditional probability ("given that"), divide by the row or column total matching the condition — this restricts your sample space. The key skill is identifying the correct denominator. At Foundation level, the denominator is usually the grand total. At Higher level, look for "given that", "of the", or "among the" to signal conditional probability. Always simplify your final fraction.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
Further reading from leading academic institutions — free and open-access.
Probability investigations and games from Cambridge.
University of Cambridge · Free · Open AccessTree diagrams, Venn diagrams, and conditional probability.
Corbett Maths · Free · Open AccessMIT introduction to probability theory.
Massachusetts Institute of Technology · Free · Open Access