Probability
Grade 3-5Two-Way Tables
A two-way table organises data by two categories at once, and every row and column total is a clue that can be used to fill in a missing value or work out a probability. This lesson covers reading a two-way table, finding a missing value using row and column totals, and finding probabilities, including probabilities restricted to a specific group.
What you need to know
- A two-way table sorts data by two categories at once, and the totals must agree both ways.
- Fill in the totals first, then find the missing cells by subtraction.
- Read a probability straight from the table: the matching cell over the relevant total.
- Check every row and column adds to its total before you answer.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Reading a two-way table
A two-way table shows data split by two categories at once, one across the columns and one down the rows. Each cell shows the frequency for that specific combination, and the totals along the edges show the row and column totals, with the overall total in the bottom right corner.
In a table showing sport preferences by gender, the cell where the "Girls" row meets the "Football" column shows exactly how many girls prefer football.
Finding a missing value
Every row must add up to its row total, and every column must add up to its column total. Use whichever row or column contains the missing value, subtracting the known values from the total to find it.
In a row showing Boys: Football 12, Netball ?, Total 20, the missing value is found by 20 - 12 = 8.
Finding a probability from a two-way table
To find the probability of a specific combination, divide the frequency in that cell by the overall total in the table.
From a table of 40 students, 7 girls prefer football. The probability that a randomly selected student is a girl who prefers football is 7/40.
Finding a probability restricted to a group
When a question selects at random from a specific group, such as "from the boys", divide by that group's row or column total instead of the overall total, since the selection is limited to that group only.
From 20 boys, 8 prefer netball. The probability that a boy selected at random prefers netball is 8/20, which simplifies to 2/5, not 8/40.
Worked examples
Three exam-style questions, fully solved
The two-way table shows the sport preferences of 40 students. Boys: Football 12, Netball ?, Total 20. Find the missing value.
Easy- 1.Identify the row total: 20
- 2.Subtract the known value in the row: 20 - 12
Answer: 8
The two-way table shows the sport preferences of 40 students, including 7 girls who prefer football. A student is selected at random. Find the probability that they are a girl who prefers football.
Medium- 1.Find the frequency for girls who prefer football: 7
- 2.Find the overall total number of students: 40
- 3.Write the probability as a fraction: 7/40
Answer: 7/40
The two-way table shows the sport preferences of 40 students. There are 20 boys, 8 of whom prefer netball. A student is selected at random from the boys. Find the probability that they prefer netball.
Hard- 1.Recognise that the selection is restricted to boys only, so use the boys' total, not the overall total
- 2.Find the number of boys who prefer netball: 8
- 3.Write the probability using the boys' total: 8/20
Answer: 2/5
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
In a two-way table, 20 boys and 15 girls were surveyed. 12 of the boys said "yes". How many boys said "no"?
50 people were surveyed: 22 adults and 28 children. 16 adults said "yes". If 40 people said "yes" altogether, how many children said "yes"?
A two-way table records 30 people: 18 female and 12 male. 10 of the females wear glasses. If 16 people in total wear glasses, how many males wear glasses?
Using that table (12 males, 6 of whom wear glasses), how many males do NOT wear glasses?
In a survey of 80 people, 45 are men. 20 men and 15 women said they exercise daily. One person is chosen at random. Work out the probability they said they exercise daily.
Using that survey (80 people, 45 men, 15 women exercise daily), a woman is chosen at random. Work out the probability she exercises daily.
Avoid these
The mistakes students make most often
Subtracting from the wrong row or column total when finding a missing value, instead of using the total that contains the missing cell.
Using the overall total instead of a row or column subtotal when a question restricts the selection to a specific group, such as "from the boys".
Reading the wrong cell for a combined probability, mixing up which row and which column the question is asking about.
Leaving the resulting probability fraction unsimplified.
Not checking that a newly filled-in value makes both its row and its column add up correctly, missing an arithmetic slip.
FAQ
Questions students and parents ask
Before this topic, make sure you know
What to learn next
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