How many households have both treadmill and exercise bike? (Venn Diagram problem)

Question

Full question: Suppose that among 1000 households surveyed, 30 have neither an exercise bicycle nor a treadmill, 50 have only an exercise bicycle, and 60 have only a treadmill. How many households have both?

Textbook answer: 860

I am trying to figure out a formula for this kind of problem...

I know that the answer must have come from adding 30 + 50 + 60 and then subtracting this from 1000. Is this because 110 only have one or the other and 30 have neither, so the difference between the universe and those cases must be the case of having both?

Answer 1

A good formula is this: $$|A\cup B|=|A\setminus B|+|B\setminus A|+|A\cap B|$$

In this case, you know that $|A\cup B|=1000-30=970$, and $|A\setminus B|=50$, and $|B\setminus A|=60$. That leaves only one term unknown.

This works, because $A\cup B$ is the disjoint union of the other three sets in the formula.

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Another, related formula, equivalent to the above one, that also comes up in such problems, is this one:

$$|A\cup B| = |A| + |B|- |A\cap B|$$

Whether you use one or the other just depends on which information you start out with.

Answer 2

Your reasoning is correct. I suspect the instructor wanted you to draw a "pretty picture" Venn diagram, but I don't see the need in this case, given how clear your reasoning was.