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10 Probability
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Stage 4

10.05 Probability calculations

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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.

Examples

Example 1

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

a

How many students only play football?

Worked Solution
Create a strategy

We can use a Venn diagram.

Apply the idea
A Venn diagram with 2 overlapping circles for tennis and football. Ask your teacher for more information.

Set up the Venn diagram.

The intersection of the circles has a 5 since 5 people do both sports.

All the numbers in the football region must add up to 11.

A Venn diagram with 2 overlapping circles. The second circle is shaded but not the intersection.

This region will be the football only players.

To find the number that goes here we need to find the number that when added to 5 gives 11.

\begin{aligned} 5+x &=11 \\ x &= 6 \end{aligned}

So 6 students only play football.

b

How many students play only one sport?

Worked Solution
Create a strategy

Use the Venn diagram.

Apply the idea
A Venn diagram with 2 overlapping circles for tennis and football. Ask your teacher for more information.

We now have a 6 in the Venn diagram in the football only region.

A Venn diagram with 2 overlapping circles. The circles are shaded except for the intersection.

The students that only play only one sport will be in this section of the Venn diagram.

Since we know that 13 students play tennis, to find the number of students that only play tennis we subtract the 5 that do both from 13.

\begin{aligned} \text{Only tennis} &=13-5 \\ &=8 \end{aligned}

Now we can add this with the students that only play football:

\displaystyle \text{Students who play one sport}\displaystyle =\displaystyle \text{Only football} + \text{Only tennis}
\displaystyle =\displaystyle 6+8Add the values
\displaystyle =\displaystyle 14Evaluate
c

If a random student is chosen from the group, find the probability that the student only plays tennis.

Worked Solution
Create a strategy

We can use the formula: \text{Probability}=\dfrac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}

Apply the idea

In part (b) we found that 8 students only play tennis. To find the total students we add the students that do only 1 sport with the students that do both sports.

\displaystyle \text{Total students}\displaystyle =\displaystyle 14+5Add the students
\displaystyle =\displaystyle 19Evaluate
\displaystyle \text{Probability}\displaystyle =\displaystyle \dfrac{8}{19}Substitute the values

Example 2

In a study, some people were asked whether they were musicians or not. 25 responders said they were a musician, of which 10 were children. 25 children said they were not musicians, and 13 adults said they were not musicians.

a

How many people were in the study?

Worked Solution
Create a strategy

We can create a two-way table to find the number of people in the study.

Apply the idea
Not a musicianMusician
Children25 10
Adults13 15

We can create this table from the information provided.

Now we can add all the values in the table to find the people in the study.

\displaystyle \text{Number of people}\displaystyle =\displaystyle 25+10+13+15Add the people
\displaystyle =\displaystyle 63Evaluate
b

State the proportion of responders that are musicians.

Worked Solution
Create a strategy

We can use the formula: \text{Probability}=\dfrac{\text{Number of musicians}}{\text{Total number of people}}

Apply the idea
\displaystyle \text{Probability}\displaystyle =\displaystyle \dfrac{10+15}{63}Add the musicians
\displaystyle =\displaystyle \dfrac{25}{63}Evaluate
c

State the proportion of adults that are musicians.

Worked Solution
Create a strategy

We can use the formula: \text{Probability}=\dfrac{\text{Number of adult musicians}}{\text{Total number of adults}}

Apply the idea
\displaystyle \text{Probability}\displaystyle =\displaystyle \dfrac{15}{13+15}Add the adults
\displaystyle =\displaystyle \dfrac{15}{28}Evaluate
Idea summary

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

Outcomes

MA4-21SP

represents probabilities of simple and compound events

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