Venn diagram question with approximate ranges

Question

Sorry if my title sounds vague and inaccurate. I can't think of a better way to put it lol. Anyway, I stumbled upon this problem today while preparing for Oxford tsa:

A survey of households in a town showed that (allowing for sampling errors) between 75% and 85% owned a dishwasher, between 35% and 40% owned a tumble dryer and less than 5% owned neither.

How many people own both a tumble dryer and a dishwasher?

In which the answer is: "between 10% and 30%"

I spent lots of time trying to figure out how to solve it but to no avail. The inclusion of the ranges in a typical Venn diagram question totally throws me off. Can anyone help me with this? Thousands of thanks in advance :)

Answer 1

The relevant groups are: $A$: People who have both a dryer and a dishwasher, $B$: People who have a tumble dryer but not a dishwasher, $C$: People who have a dishwasher but not a tumble dryer, and $D$: People who have neither a tumble dryer or a dishwasher. Note that every person (or household) belongs to exactly one of these groups, thus $A+B+C+D=100\%$. Express the rest of your givens in terms of these groups, and the answer should come easily.

Answer 2

If the numbers were certain, you could use $P(A \cap B)=P(A)+P(B)-P(A \cup B)$ But since the numbers are not certain, you have to look at "best case" and "worst case" scenarios. The higher $P(A)$ and $P(B)$ are, the higher the $P(A \cap B)$ will be. Since $P(A\cup B)$ is known (0.95), you have no room to move there. To make $P(A \cap B)$ large, assume that both the dishwasher and dryer ownership numbers are also high. So $$P_{high}=0.85+0.40-0.95=0.3$$ Similarly with the lower number.

Answer 3

Instead of a Venn Diagram, try and find the minimum and maximum working directly with numbers:

For example, suppose 5% owns neither. Then 95% has a dishwasher, dryer, or both. Then, if 75% has a dishwasher, and 35% a dryer (which is 110% total), you would have the difference (15%) own both. We can lower this by saying no one owns neither, for then the difference is only 10%. On the other hand, if we stick to 5% owning neither, but say 40% has a dryer, then it goes up to 20% difference. Upping the number of dishwasher owners to 85% will make the difference the largest: 30%. So that's the range.

Sometimes, you just have to play directly with the numbers and see what happens, and not directly try formulas (of course, for a mathematical proof, you may have to do that ... but first you need to get a 'feel' for the situation)