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9 Probability
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Australia
Year 8

9.04 Two-way tables

Lesson
Worksheet
Practice
Lesson

Two-way table

Two-way tables represent data that is classified by two criteria.

RightLeft
Enjoys English 4 9
Doesn't enjoy English 2 15

If the first criterion was a student's main writing hand, and the other was whether or not they enjoy English, we could produce a table like this.

The columns show the writing hand, and the rows show their enjoyment of English.

To read a two-way table, look at the column and row that a number is in to find the right cell, the box where a single number is written. For example there are 9 students who are left-handed and enjoy English (top-right cell), and 2 students who are right handed that don't enjoy English (bottom-left cell). Tables will often include totals of each column, each row, and the total sum in the corner:

RightLeftTotal
Enjoys English4913
Doesn't enjoy English21517
Total62430

The categories should always be chosen so that each data point goes in exactly one of the cells.

Examples

Example 1

This table describes the departures of trains out of a train station for the months of May and June.

Departed on timeDeparted late
May11331
June108 33
a

How many trains departed during May and June?

Worked Solution
Create a strategy

Add all the numbers in the table

Apply the idea
\displaystyle \text{Number of trains}\displaystyle =\displaystyle 113+108+31+33Add the numbers
\displaystyle =\displaystyle 285Evaluate
b

What percentage of the trains in June were delayed? Write your answer as a percentage to one decimal place.

Worked Solution
Create a strategy

Find the number of delayed trains in June as a percentage of the total trains departing in June.

Apply the idea
\displaystyle \text{June trains}\displaystyle =\displaystyle 108+33Add all the June trains
\displaystyle =\displaystyle 141Evaluate

Now we can find the 33 delayed June trains as a percentage of the total June trains:

\displaystyle \text{Percentage delayed}\displaystyle =\displaystyle \dfrac{33}{141}\times 100\%Find 33 as a percentage of 141
\displaystyle \approx\displaystyle 23.4\%Evaluate and round
Reflect and check

This percentage also tells us that if a train in June was randomly selected, the probability that it would have been delayed is 23.4\%.

c

What fraction of the total number of trains during the 2 months were ones that departed on time in May?

Worked Solution
Create a strategy

Find the number of on time trains in May as a fraction of the total trains.

Apply the idea

From part (a) we found that there are 285 total trains. We can see from the table that 113 departed on time in May.

\displaystyle \text{Fraction}\displaystyle =\displaystyle \dfrac{113}{285}Write as a fraction
Reflect and check

This fraction also tells us that the probability that a train selected at random departed on time in May is \dfrac{113}{285}.

Idea summary

Two-way tables represent data that is classified by two criteria.

Two-way tables and venn diagrams

A two-way table often presents information that could also be presented with a Venn diagram. We can convert between a two-way table and a Venn diagram by matching up their different parts.

A two-way table with columns left and right and rows entered and didn't enter. Ask your teacher for more information.

This two-way table represents the handedness of students, and whether or not they entered the Talent Show.

A Venn diagram with 2 overlapping circles called left-handed and entered. Ask your teacher for more information.

To represent this information in a Venn diagram, we choose one row and one column to become circles. Here we chose the column marked "Left" and the row marked "Entered".

  • The number that is in both chosen categories (4) goes in the overlap of the two circles.

  • The other value in the "Entered" row (9) represents the "Entered and right-handed" students, and goes in the "Entered" circle outside the overlap.

  • The other value in the "Left" column (2) represents the "Didn't enter and left-handed" students, and goes in the "Left-handed" circle outside the overlap.

  • Any cells that are in neither the highlighted row nor the highlighted column (15) goes into the surrounding box, outside both circles.

Examples

Example 2

Students in Irene's class were asked if they owned a dog and asked if they owned a snake. The following two way table shows that information.

Owns a dogDoesn't own a dog
Owns a snake 2 3
Doesn't own a snake 13 11

Construct a Venn diagram that represents the information provided in the two-way table.

Worked Solution
Create a strategy

The Venn diagram should have two overlapping circles for each type of pet.

Apply the idea

  • The 2 that own a dog and a snake will go in the intersection.

  • The 3 that own a snake but not a dog will go in the snake circle outside the intersection.

  • The 13 that own a dog but not a snake will go in the dog circle outside the intersection.

  • The 11 that do not own a snake or a dog will go outside the circles.

A Venn diagram with 2 overlapping sets dog owners and snake owners. Ask your teacher for more information.
Idea summary

We can convert between a two-way table and a Venn diagram by matching up their different parts.

Outcomes

ACMSP205

Describe events using language of 'at least', exclusive 'or' (A or B but not both), inclusive 'or' (A or B or both) and 'and'.

ACMSP292

Represent events in two-way tables and Venn diagrams and solve related problems

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