Question

A company with 500 employees wants to have its group health insurance premiums lowered. Here is one question from the questionnaire that every employee filled out and returned to the HR department: What have you done in the past year to improve your health? (circle all that apply) A. I have eaten healthier. B. I have exercised regularly. When the results were tabulated, the HR clerk reported that 140 people circled response A, 290 circled response B, and 50 people circled both responses A and B. Given that a randomly selected individual answered B, what is the probability that he or she also answered A?

Answer 1

Let us first get the four groups clear: A, B, both and none. The latter group is often forgotten.

Only `A=140-50=90` people
Only `B=290-50=240` people
Both `AandB=50` people
None: `500-90-240-50=120` people

290 B's, of which 50 are A+B, so the probability you asked for:

`P(B->AandB)=50/260=0.192=19.2%`

This could have been done shorter, but then this would only answer this one specific question about his population. You can now answer most other ones, like:
- How many have only one answer circled?
- How many have none?
Etc.