A two-way table sorts data by two categories at once, such as age group and activity, with row totals, column totals and a grand total. To complete a table, use the fact that every row and column adds up to its total, filling in whichever cell has only one unknown. Probabilities come straight from the table: the relevant cell or total divided by the grand total.
Two-way tables
- Each row adds up to its row total
- Each column adds up to its column total
- Row totals and column totals both add up to the grand total
- P(event) = number in the relevant cell or total ÷ grand total
Reading a two-way table
Each cell counts the items in one row category and one column category. The totals at the end of each row and the bottom of each column add up the cells, and the corner cell is the grand total. Read the row and column headings carefully before taking a value.
Completing a two-way table
Use the fact that each row and each column adds up to its total. Find a row or column with only one missing value, work it out by subtraction, and repeat. Information given in words, such as “17 adults chose bowling”, should be written into the table first.
Probabilities from a two-way table
To find a probability, divide the relevant number by the total. For P(a child who chose bowling), use the single cell for children and bowling. For P(an adult), use the adults' row total. Unless the question restricts the group, divide by the grand total.
Restricted groups
If the question says “given that the person is a child” or “one of the children is chosen”, divide by the number of children instead of the grand total. This is conditional probability, and two-way tables make it straightforward.
Checking your table
When the table is complete, check that the row totals add up to the grand total and the column totals do too. A mismatch shows an arithmetic slip before it spoils the later probability parts.
Worked examples
Example 1. 90 people chose cinema or bowling. Of the 45 who chose cinema, 21 were adults. 38 people were adults altogether. Complete the table.
- Cinema: children = 45 − 21 = 24.
- Bowling: total = 90 − 45 = 45; adults = 38 − 21 = 17; children = 45 − 17 = 28.
- Children total: 24 + 28 = 52.
Answer: Cinema: 21 adults, 24 children. Bowling: 17 adults, 28 children. Totals: 38 adults, 52 children.
Example 2. Using the table, one person is chosen at random. Find P(an adult).
- 38 adults out of 90.
Answer: 38/90 = 19/45.
Example 3. Find P(a child who chose bowling).
- The cell for children and bowling is 28.
Answer: 28/90 = 14/45.
Example 4. A child is chosen at random. Find the probability that the child chose cinema.
- The group is restricted to the 52 children.
- 24 of them chose cinema.
Answer: 24/52 = 6/13.
Example 5. A table has a row with entries 23 and x and a row total of 41. Find x.
- 23 + x = 41.
Answer: x = 18.
Common mistakes
- Dividing by the wrong total. Use the grand total unless the question restricts the group.
- Not using the totals to fill gaps. Each row and column must add up to its total.
- Mixing up rows and columns. Check the headings before reading a value.
- Ignoring information given in words. Write every fact into the table before calculating.
- Not checking the grand total. Rows and columns should both add up to it.
How two-way tables are examined
Two-way tables are Foundation content for both the UK GCSE (1MA1) and the International GCSE (4MA1), often combined with probability. Questions ask you to:
- read values from a two-way table
- complete a table from partial information
- find probabilities from the table
- find a probability within a restricted group (Higher)
The sample space and tables worksheet includes two-way table questions with answers, and the GCSE probability revision engine generates more.
Restricted-group questions lead to conditional probability. Data can also be shown in Venn diagrams.
Frequently asked questions
What is a two-way table?
A table that sorts data by two different categories at the same time, with totals for each row and column.
How do you complete a two-way table?
Use the row and column totals: find a row or column with one unknown and work it out by subtraction.
How do you find a probability from a two-way table?
Divide the relevant cell or total by the grand total, or by the restricted group's total if the question says “given that”.