7.13 Venn diagrams and two-way tables - word problems
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).
State the value of $w$w in the diagram.
State the value of $x$x in the diagram.
State the value of $y$y in the diagram.
State the value of $z$z in the diagram.
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(T∣S)=$\frac{5}{12}$512
- $P\left(S\cap T'\right)=$P(S∩T′)=$\frac{35}{87}$3587
- $P\left(T\right)=$P(T)=$\frac{14}{29}$1429
Find the value of $n\left(S\cap T\right)$n(S∩T).
Use $Y=n\left(S\cap T\right)$Y=n(S∩T) and $X=n\left(S\cap T'\right)$X=n(S∩T′) to help you in your calculations.
Calculate $P\left(S'\cap T\right)$P(S′∩T).
Calculate $P\left(S\mid T\right)$P(S∣T).
Calculate $P\left(T\mid S'\right)$P(T∣S′).
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.
How many people only watch $MKR$MKR?
Of the people who watch $MC$MC, what proportion also watch $MKR$MKR?
Of those who don’t watch $MC$MC, what proportion watch neither?