Probability

NZ Level 8 (NZC) Level 3 (NCEA) [In development]

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.

$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.

$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`.

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?

$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`′).

$531$531 people are asked whether they watch My Kitchen Rules ($MKR$`M``K``R`) or Masterchef ($MC$`M``C`).

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

How many people only watch $MKR$

`M``K``R`?Of the people who watch $MC$

`M``C`, what proportion also watch $MKR$`M``K``R`?Of those who don’t watch $MC$

`M``C`, what proportion watch neither?

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

Apply probability concepts in solving problems