Two-Way Tables

What Are Two-Way Tables?

Two-way tables are a helpful way to organise and compare information about two different categories at once. For example:

• Gender and favourite sport
• Year group and language choice
• Class and preferred activity

Each row and column of the table corresponds to one category, and the numbers inside show how many people fall into both.

Example Context

A total of 90 students attend a weekend workshop. They are split into two groups (Group X and Group Y), and they choose between three activities: Coding, Robotics, and Game Design. You're told:

• 20 students in Group X chose Coding
• 14 students in Group Y chose Game Design
• 28 students in total chose Robotics
• 16 students in Group X chose Robotics
• 8 students in Group Y chose Coding
• 12 students in Group X chose Game Design

(a) Construct a Two-Way Table

Start by filling in what you know:

Now complete the table using row and column totals:

• Group X total = 20 + 16 + 12 = 48 (already given)
• Coding total = 20 + 8 = 28
• Game Design total = 12 + 14 = 26
• Robotics total already given = 28
• Group Y total = 90 - 48 = 42
• Group Y Robotics = 28 - 16 = 12

Final table:

(b) Find the Probability That a Randomly Selected Student:

(i) Chose Coding

There are 28 students who chose Coding out of 90 total:

$P(\text{Coding}) = \frac{28}{90} = \frac{14}{45}$

(ii) Is From Group Y and Chose Robotics

There are 12 such students:

$P(\text{Group Y and Robotics}) = \frac{12}{90} = \frac{2}{15}$

(c) A Student From Group X Is Chosen. What Is the Probability They Chose Game Design?

There are 12 students in Group X who chose Game Design, and 48 students in Group X:

$P(\text{Game Design | Group X}) = \frac{12}{48} = \frac{1}{4}$