New Zealand
Level 8 - NCEA Level 3

# Venn Diagrams and Two-Way Tables - Word Problems

Lesson

We've seen Venn diagram and Two-Way table word problems before, but let's take a look at a few more challenging questions and how to solve them.

#### Example 1:

$200$200 people were questioned about whether they read the newspaper online or in paper form.

• $20$20% said they did both
• $\frac{3}{10}$310 of those surveyed said they read the paper version
• Of those who who didn't read the paper version, the probability that they read an online version was $50$50%

a)  Complete a two-way table containing this information

Think: To work out the first two pieces of information we can find $20$20% and $30$30% of $200$200 people respectively.

Do:

Paper Not Paper Total
Online 40
Not Online 20
Total 60 140 200

To use the third point we are best off using the formula for conditional probability.

Now we can fill in the rest of the table.

Paper Not Paper Total
Online 40 70 110
Not Online 20 70 90
Total 60 140 200

b)  Of those who read the paper online, what proportion also read a paper version?

Think: Notice that this is a conditional probability question.

Do: $\frac{40}{110}$40110

We'll now do a similar question, but using a Venn diagram.

#### Example 2:

$400$400 people were questioned about whether they make or buy their bread.

• $2$2% did neither
• $30$30% did both
• The probability that a person didn't make their own bread, given that they bought their bread was $\frac{5}{9}$59

Construct a Venn diagram with this information

Think:  We can easily fit the first two pieces of information into our Venn Diagram.

Do:

To use the third piece of information is a little more complicated. Not only will we again need to use the rule for conditional probability, but we'll also need to introduce $x$x.

#### Worked examples

##### Question 1

At a university there are $816$816 students studying first year engineering, $497$497 of whom are female (set $F$F). $237$237 of these women are studying Civil Engineering, and there are $348$348 students studying Civil Engineering altogether (set $C$C).

1. State the value of $w$w in the diagram.

2. State the value of $x$x in the diagram.

3. State the value of $y$y in the diagram.

4. State the value of $z$z in the diagram.

5. What is the probability that a randomly selected male student does not study Civil Engineering?

##### Question 2

$87$87 people are questioned about whether they own a tablet ($T$T) or a smartphone ($S$S). The probabilities shown in the list below were determined from the results.

• $P\left(T\mid S\right)=$P(TS)=$\frac{5}{12}$512
• $P\left(S\cap T'\right)=$P(ST)=$\frac{35}{87}$3587
• $P\left(T\right)=$P(T)=$\frac{14}{29}$1429
1. Find the value of $n\left(S\cap T\right)$n(ST).

Use $Y=n\left(S\cap T\right)$Y=n(ST) and $X=n\left(S\cap T'\right)$X=n(ST) to help you in your calculations.

2. Calculate $P\left(S'\cap T\right)$P(ST).

3. Calculate $P\left(S\mid T\right)$P(ST).

4. Calculate $P\left(T\mid S'\right)$P(TS).

##### Question 3

$531$531 people are asked whether they watch My Kitchen Rules ($MKR$MKR) or Masterchef ($MC$MC).

$177$177 people watch both and $65$65 watch neither. The number who watch $MKR$MKR is twice the number who watch both.

1. How many people only watch $MKR$MKR?

2. Of the people who watch $MC$MC, what proportion also watch $MKR$MKR?

3. Of those who don’t watch $MC$MC, what proportion watch neither?

### Outcomes

#### S8-4

Investigate situations that involve elements of chance: A calculating probabilities of independent, combined, and conditional events B calculating and interpreting expected values and standard deviations of discrete random variables C applying distributions such as the Poisson, binomial, and normal

#### 91585

Apply probability concepts in solving problems