A class has 12 boys and 8 girls. What is the total number of students?
Add boys and girls.
12 + 8 = 20 students.
Two Way Tables
Two way tables organise data into rows and columns to show relationships between variables. They are useful for calculating probabilities and are often used in conditional probability questions, building on basic probability.
Overview
A two way table organises information using rows and columns.
It is useful when data is split into two categories at the same time.
You need to be able to read values, complete missing values, and use the table to find probabilities.
What you should understand after this topic
- Understand how a two way table is structured
- Calculate row totals and column totals
- Find missing values in a table
- Answer probability questions using the table
- Avoid double counting
Key Definitions
Two Way Table
A table that sorts data by two different categories.
Row
A horizontal line of values in the table.
Column
A vertical line of values in the table.
Row Total
The total for one full row.
Column Total
The total for one full column.
Grand Total
The total number of all values in the table.
Key Rules
Add across rows
Use row values to find the row total.
Add down columns
Use column values to find the column total.
Use subtraction for missing values
Missing value = total − known values.
Probability from the table
\( \frac{\text{number wanted}}{\text{total}} \)
Quick Reminder
Check the headings
Make sure you are reading the correct row and column.
Do not double count
Each person or object belongs in one correct place.
Totals must match
Row totals and column totals should agree with the grand total.
Use the full table carefully
Probability questions often come directly from one part of the table.
How to Solve
Step 1: Understand two-way tables
A two-way table organises information using two categories at the same time.
Example: students grouped by gender and by whether they walk to school.
Exam tip: Always read the row and column headings first.
Step 2: Read rows and columns
Each cell belongs to both its row and its column.
Row totals are found by adding across.
Column totals are found by adding down.
Grand total is usually in the bottom-right corner.
Step 3: Find missing values
Use totals and subtraction to complete missing parts.
Missing value = total − known parts.
Exam thinking: Complete the table before answering probability questions.
Step 4: Use two-way tables for probability
Choose the correct value from the table, then divide by the total.
\( P(\text{event}) = \frac{\text{wanted value}}{\text{total}} \)
Common in conditional probability questions.
Step 5: Choose the correct table value
Single category
Use a row total or column total.
Combined category
Use the cell where row and column meet.
Missing value
Use totals and subtraction.
Probability
Wanted value over total outcomes.
Step 6: Exam method summary
See probability basics for writing probabilities.
- Read the row and column headings.
- Complete any missing totals.
- Use subtraction to find missing cells.
- Identify whether the question asks for a row, column or combined value.
- For probability, put the wanted value over the total.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on completing and interpreting two-way tables.
Edexcel
The table shows information about 40 students.
| Play sport | Do not play sport | Total | |
|---|---|---|---|
| Boy | 12 | 20 | |
| Girl | 10 | 20 | |
| Total | 18 | 40 |
Complete the two-way table.
Edexcel
The table shows information about students in a class.
| Like Maths | Do not like Maths | Total | |
|---|---|---|---|
| Boy | 8 | 7 | 15 |
| Girl | 10 | 5 | 15 |
| Total | 18 | 12 | 30 |
Find the probability that a randomly chosen student likes Maths.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on probability from two-way tables.
AQA
The table shows information about 50 people.
| Own a car | Do not own a car | Total | |
|---|---|---|---|
| Over 18 | 20 | 5 | 25 |
| Under 18 | 10 | 15 | 25 |
| Total | 30 | 20 | 50 |
Find the probability that a randomly chosen person owns a car.
AQA
Using the table above, find the probability that a person owns a car given that they are over 18.
AQA
A student says, "The probability of choosing someone under 18 is 0.6."
Tick one box. Correct ☐ Incorrect ☐
Give a reason for your answer.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising conditional probability from two-way tables.
OCR
The table shows information about 60 students.
| Study French | Do not study French | Total | |
|---|---|---|---|
| Boy | 15 | 10 | 25 |
| Girl | 20 | 15 | 35 |
| Total | 35 | 25 | 60 |
Find the probability that a randomly chosen student studies French.
OCR
Find the probability that a student studies French given that the student is a girl.
Exam Checklist
Step 1
Read the row and column headings carefully.
Step 2
Use addition to find totals.
Step 3
Use subtraction to fill in missing values.
Step 4
For probability, use wanted value over total value.
Most common exam mistakes
Heading mistake
Using the wrong category from the table.
Total mistake
Adding or subtracting incorrectly.
Probability mistake
Using the wrong denominator.
Cell mistake
Using a row total when the question asks for one cell only.
Common Mistakes
These are common mistakes students make when working with two-way tables in GCSE Maths.
Reading the wrong row or column
Incorrect
A student selects values from the incorrect part of the table.
Correct
Carefully match the row and column labels to the question before choosing a value.
Adding totals incorrectly
Incorrect
A student makes errors when calculating row or column totals.
Correct
Add values carefully and check that row and column totals are consistent.
Using the wrong total in probability
Incorrect
A student uses a row or column total instead of the overall total.
Correct
For overall probability, use the grand total unless the question restricts the sample space.
Using totals instead of individual values
Incorrect
A student uses a row or column total when a single cell value is required.
Correct
Check whether the question is asking for a specific group or a total before selecting the value.
Not checking consistency
Incorrect
A student does not verify that totals match.
Correct
Row totals and column totals should agree with the grand total. Always check for consistency.
Try It Yourself
Practise interpreting and completing two-way tables.
Foundation
Foundation Practice
Read and complete two-way tables.
In a table, there are 15 students who like football and 10 who like tennis. How many students are there in total?
Add the two groups.
15 + 10 = 25.
A table shows 7 boys like maths and 5 boys do not. How many boys are there in total?
Add both groups.
7 + 5 = 12 boys.
There are 20 students in total. 8 are girls. How many are boys?
Subtract girls from total.
20 − 8 = 12 boys.
A table shows 10 students like pizza and 15 like pasta. What is the total?
Add both values.
10 + 15 = 25.
There are 18 students. 7 like chocolate. How many do not?
Subtract from total.
18 − 7 = 11.
A student forgets to include the total row. What is the problem?
Totals are important.
Totals are needed to understand the data.
There are 30 students. 12 are boys. How many are girls?
Subtract boys from total.
30 − 12 = 18 girls.
A table has 9 in one cell and 6 in another. What is their combined total?
Add the numbers.
9 + 6 = 15.
There are 25 students in total. 10 are in one category. How many are in the other?
Subtract from total.
25 − 10 = 15.
Higher
Higher Practice
Complete two-way tables and use them to calculate probabilities.
A table shows 20 students. 8 are boys who like maths. 5 boys do not like maths. How many boys are there?
Add boys who like and do not like maths.
8 + 5 = 13 boys.
There are 40 students. 15 are boys. 10 girls like maths. How many girls are there?
Subtract boys from total.
40 − 15 = 25 girls.
A table shows 30 students. 12 like football. What is the probability a student chosen at random likes football?
Simplify 12/30.
12/30 = 2/5.
A table shows 50 students. 20 are boys. What is the probability of selecting a boy?
Probability = favourable ÷ total.
20/50 = 2/5.
A table shows 40 students. 18 are girls. What is the probability of selecting a girl?
Simplify the fraction.
18/40 = 9/20.
A table shows 60 students. 24 like maths. Find the probability of not liking maths.
Find those who do not like maths.
60 − 24 = 36 → 36/60 = 3/5.
A student uses column totals instead of overall total when calculating probability. What is wrong?
Use the overall total.
Probability must use total number of outcomes.
A table shows 45 students. 15 are boys who like sport. What is the probability of selecting one of these boys?
15 out of 45.
15/45 = 1/3.
A table has 80 students. 32 are girls who like music. What is the probability of selecting one?
Simplify 32/80.
32/80 = 2/5.
A table shows 100 people. 60 like tea. Find the probability of not liking tea.
Find those who do not like tea.
100 − 60 = 40 → 40/100 = 2/5.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What are two-way tables?
Tables showing two variables.
What do totals help with?
Checking calculations.
What can you find?
Probabilities and frequencies.