# Two Way Tables

Lesson

## Two Way Tables

Like Venn Diagrams, two way tables are a visual way of representing information.

The two way table below represents how many numbers between 2-20 are even or a multiple of 3.

It's called a two way table because we can read information from it in two directions. If read across each row, we can tell how many numbers are even or not. If we read down each column, we can tell how many numbers are multiples of 3 or not.

Where a particular row and column overlap, these are how many numbers between 2 and 20 satisfy both categories. For example, there are 7 numbers that are even but not a multiple of 3.

Notice that the number in the bottom right cell (19) is how many numbers there are altogether between 2 and 20.

#### Example

John surveyed his classmates at school and found out that 56 students own a mobile phone and 40 of those students do not own an MP3 player. There are 8 students that do not own a mobile phone, but own an MP3 player. 6 students do not own either device. Construct a two way table summarising the data.

##### Step 1 - Create the Table With Appropriate Headings

Each student can be sorted by having an MP3 player or not, and having a mobile phone or not.

##### STEP 2 - add in the values you can read off straight from the question

We are told that:

• 40 students own a mobile phone but no MP3 player
• 8 students own an MP3 player but not a phone
• 6 students own neither
• In total, there are 56 students who own a mobile phone

##### STEP 3 - problem solve using addition and subtraction to complete the table

Adding across the second row, we can find the total number of students who do not own a mobile phone.

Subtracting across the first row, we can find the number of students who own both a mobile phone and an MP3 player.

Lastly, we can add down the columns to find the total number of students who own an MP3 player.

Notice that the bottom right cell contains the total number of students surveyed.

From here we could answer a wide variety of questions such as:

a) What percentage of students own a mobile phone?

To calculate a percentage we need two pieces of information.  The amount we are interested in (56 students own a mobile phone), and the total amount (70 students).

The percentage is:$\frac{\text{amount we want }}{\text{total }}=\frac{56}{70}$amount we want total =5670$=$=$0.8=80%$0.8=80%

b) What is the probability that a student selected from the group has both a mobile phone and an MP3 player?

To calculate a probability we need two pieces of information.  The number of favourable outcomes (16 students own both a mobile phone and an MP3 player), and the total number of outcomes (70 students).

The probability is $\frac{\text{total favourable outcomes }}{\text{total outcomes }}=\frac{16}{70}$total favourable outcomes total outcomes =1670$=$= $\frac{8}{35}$835

c) What is the probability that a student has neither a mobile phone, nor a MP3 player?

To calculate a probability we need two pieces of information.  The number of favourable outcomes (6 students own neither a mobile phone or an MP3 player), and the total number of outcomes (70 students).

The probability is $\frac{\text{total favourable outcomes }}{\text{total outcomes }}=\frac{6}{70}$total favourable outcomes total outcomes =670$=$=$\frac{3}{35}$335

#### Worked Examples

##### QUESTION 1

This table describes the departures of trains out of a train station for the months of March and April.

Month Departed on time Delayed
March $148$148 $38$38
April $140$140 $20$20
1. How many trains departed during March and April?

2. What percentage of the trains in April were delayed? Write your answer as a percentage to 1 decimal place.

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

4. What is the probability that a train selected at random in April would have departed on time?