finite math help

Question

A pharmaceutical company has surveyed a group of college students about mental illness and the following data is collected:

* 53% know someone being treated for alcoholism

* 45% know someone being treated for anxiety

* 36% know someone being treated for depression

* 18% know someone being treated for both alcoholism and anxiety

* 15% know someone being treated for both alcoholism and depression

* 19% know someone being treated for both anxiety and depression

* 8% know of nobody being treated for any of these conditions

What percent of the students know someone being treated for all three mental illnesses?

Answer 1

Please draw a Venn diagram with three intersecting sets:

A is the set of students who know somebody treated for alcoholism,

X is the set of students who know somebody treated for anxiety,

D is the set of students who know somebody treated for depression.

We are given the sizes, denoted |...|, of various sets, their unions and intersections.

|A| = 53

|X| = 45

|D| = 36

|A ∩ X| = 18

|A ∩ D| = 15

|X ∩ D| = 19

|A ∪ X ∪ D| = 100 - 8 = 92

We are to calculate

|A ∩ X ∩ D|

Suppose we form the sum

|A| + |X| + |D| = 134

In that sum

we have counted three times those who know someone with all three conditions, i.e.,

those in A ∩ X ∩ D,

we have counted twice those who know someone with two conditions, i.e.,

those in A ∩ X, in A ∩ D, and in X ∩ D not belonging to A ∩ X ∩ D,

we have counted just once those who know someone with exactly one condition.

Suppose we form the sum

|A ∪ X ∪ D| + |A ∩ X| + |A ∩ D| + |X ∩ D| = 144

That sum counts everybody as many times as the previous sum |A| + |X| + |D|,

except those in A ∩ X ∩ D are counted four times (not three times).

Therefore |A ∩ X ∩ D| is the difference 10.