Question

A survey of senior citizens at a doctor’s office shows that 52% of the seniors take blood pressure

lowering medication and 40% take cholesterol lowering medication. 17% take both medications.
What is the probability that a senior citizen takes only one of these medications given that he or she
takes at least one of the medications?

Answer 1

The probability that a senior citizen takes only one of these medications given that he or she takes at least one of the medications is approximately 0.467.

Step-by-step explanation:

To find the probability that a senior citizen takes only one of these medications given that he or she takes at least one of the medications, we can use conditional probability. Let's denote the event that a senior takes blood pressure lowering medication as event A, and the event that a senior takes cholesterol lowering medication as event B.

We are given that 52% of the seniors take blood pressure lowering medication (P(A) = 0.52), 40% take cholesterol lowering medication (P(B) = 0.40), and 17% take both medications (P(A ∩ B) = 0.17).

The probability that a senior citizen takes at least one of the medications can be calculated as: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.52 + 0.40 - 0.17 = 0.75.

Now, to find the probability that a senior takes only one of these medications given that he or she takes at least one of the medications, we can use the formula for conditional probability:

P(A | A ∪ B) = P(A ∩ (A ∪ B)) / P(A ∪ B).

Since taking only one of the medications means not taking both medications, we can rewrite P(A | A ∪ B) as P(A ∩ B') / P(A ∪ B), where B' represents the complement of event B (i.e., the event that a senior does not take cholesterol lowering medication).

The probability that a senior takes only one of these medications given that he or she takes at least one of the medications is therefore: P(A ∩ B') / P(A ∪ B) = (P(A) - P(A ∩ B)) / P(A ∪ B) = (0.52 - 0.17) / 0.75 = 0.35 / 0.75 ≈ 0.467 (rounded to three decimal places).