Question

It is estimated that, during the past year, 27% of all adults visited a therapist and 43% of all adults used antidepressants. It is also estimated that 20% of all adults both visited a therapist and used a antidepressant during the past year.

(a)What is the probability that a randomly selected adult who visited a therapist during the past year also used antidepressants? Round your answer to the nearest hundredth.

(b)What is the probability that an adult visited a therapist during the past year, given that he or she used antidepressants? Round your answer to the nearest hundredth.

Answer 1

(a) To find the probability that a randomly selected adult who visited a therapist during the past year also used antidepressants, we use the formula:

P(A|B) = P(A and B) / P(B)

where A is the event that an adult used antidepressants, and B is the event that an adult visited a therapist. We are given:

P(A) = 0.43

P(B) = 0.27

P(A and B) = 0.20

Plugging these values into the formula, we get:

P(A|B) = 0.20 / 0.27 ≈ 0.74

Therefore, the probability that a randomly selected adult who visited a therapist during the past year also used antidepressants is approximately 0.74.

(b) To find the probability that an adult visited a therapist during the past year, given that he or she used antidepressants, we use Bayes' theorem:

P(B|A) = P(A|B) * P(B) / P(A)

where A and B are defined as before. We have already calculated P(A|B) and P(B), and we know:

P(A) = 0.43

Plugging these values into Bayes' theorem, we get:

P(B|A) = (0.74 * 0.27) / 0.43 ≈ 0.47

Therefore, the probability that an adult visited a therapist during the past year, given that he or she used antidepressants, is approximately 0.47.