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Middle Years

12.05 Probability with Venn diagrams and two-way tables


Probability calculations

Many situations in probability can be organised into Venn diagrams or two-way tables to determine the size of different groups and do calculations.

Worked example

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=8034=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=12474=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=5046=4,

$\text{Number of secondary students}=124-80=44$Number of secondary students=12480=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.

Practice questions

Question 1

In their class $5$5 people play both football and tennis, $13$13 people in total play tennis, and $11$11 in total play football.


  1. How many people only play football?

  2. How many people play only one sport?

  3. If a random student is chosen from the group what is the probability that the student only plays tennis?

Question 2

In a study, some people were asked whether they were musicians or not.

$25$25 responders said they were a musician, of which $10$10 were children. $25$25 children said they were not musicians, and $13$13 adults said they are not musicians.

  1. How many people were in the study?

  2. What proportion of responders are musicians?

  3. What proportion of adults are musicians?


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