12.05 Probability with Venn diagrams and two-way tables

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.