6.03 Two-way tables

A pet store sells three types of animals. Some of them have spots, some do not. This two-way table represents the animals they have for sale one day:

	Cat	Dog	Fish
Has spots	$7$7	$4$4	$11$11
No spots	$5$5	$10$10	$8$8

If the store owner chooses an animal at random, what is the probability that they will select a fish?

Think: We can use the following formula to find the probability:

$\text{Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Probability=Number of favourable outcomesTotal number of outcomes​

We will need to find the total number of animals and the number of fish to substitute into the formula.

Do: The number of spotted fish is the number in the top-right cell, and the number of fish without spots is the number in the bottom-right cell. Adding these together tells us that there are $11+8=19$11+8=19 fish altogether.

The total number of animals is all the numbers added together, $7+4+11+5+10+8=45$7+4+11+5+10+8=45.

This means the probability is $\frac{19}{45}$1945​.

Using this Venn diagram:

Create a two-way table representing the same information.

 

Think: Each circle represents a category, so each circle name should be a heading - one a row heading, and one a column heading.

The numbers outside the "Cut" circle are "Not cut", and the numbers outside the "Dyed" circle are "Not dyed", so they should be the other headings.

Do: The empty two way table will look like this:

	Cut	Not cut
Dyed
Not dyed

Using the Venn diagram we can fill in the table by looking at the corresponding regions. The top left cell is for the number of people who had their hair cut and dyed, which was $35$35 people. The bottom left cell is for the number of people who had their hair cut but not dyed, which was $12$12. Filling in the rest of the table like this gives us the final result:

	Cut	Not cut
Dyed	$35$35	$20$20
Not dyed	$12$12	$3$3

There are $124$124 students at a school, $74$74 of them attended the school sports carnival, of which $34$34 were primary students and $40$40 were senior students. There are a total of $80$80 primary students at school. How many senior students didn't attend the sports carnival?

Think: This information can be placed in a two-way table that includes a "Total" column and a "Total" row.

Do: Create a two-way table and fill in the values given by the question.

	Primary	Secondary	Total
Attended	$34$34	$40$40	$74$74
Didn't attend
Total	$80$80		$124$124

In the first and the last columns we have one piece of information missing, so we can find the values to go into those cells by using subtraction. The number of primary students who attended $\left(34\right)$(34) plus the number of primary students who didn't attend $\left(\text{blank}\right)$(blank) will be equal to the total number of primary students $\left(80\right)$(80), so

$\text{Number of primary student who didn't attend}=80-34=46$Number of primary student who didn't attend=80−34=46.

Similarly, looking at the "Total" column,

$\text{Number of students who didn't attend}=124-74=50$Number of students who didn't attend=124−74=50.

We write these values in the table:

	Primary	Secondary	Total
Attended	$34$34	$40$40	$74$74
Didn't attend	$46$46		$50$50
Total	$80$80		$124$124

Now we can use the numbers in the rows in a similar way to find the last two values:

$\text{Number of secondary students who didn't attend}=50-46=4$Number of secondary students who didn't attend=50−46=4,

$\text{Number of secondary students}=124-80=44$Number of secondary students=124−80=44.

Here is the completed table:

	Primary	Secondary	Total
Attended	$34$34	$40$40	$74$74
Didn't attend	$46$46	$4$4	$50$50
Total	$80$80	$44$44	$124$124

We can now answer the original question: There were $4$4 senior students who didn't attend the sports carnival.