Question

it is estimated that, during the past year, 28% of all adults visited a therapist and 44% of all adults used non-prescription antidepressants. it is also estimated that 20% of all adults both visited a therapist and used a non-prescription antidepressant during the past year.

answer the questions below. (if necessary, consult a list of formulas.)

(a) an adult who visited a therapist during the past year is randomly selected. what is the probability this adult used non-prescription antidepressants? round your answer to 2 decimal places.

(b) what is the probability that an adult visited a therapist during the past year, given that he or she used non-prescription antidepressants? round your answer to 2 decimal places.

Explanation:

Define events and given probabilities

Using the Probability of Events knowledge point, we define the events and extract their probabilities from the problem statement:

• Let \(T\) be the event that a randomly selected adult visited a therapist during the past year.
• Let \(A\) be the event that a randomly selected adult used non-prescription antidepressants during the past year.

From the text, we have:

• \(P(T) = 0.28\)
• \(P(A) = 0.44\)
• \(P(T \cap A) = 0.20\)

Calculate conditional probability for part (a)

Using the Conditional Probability knowledge point, we find the probability that a randomly selected adult used non-prescription antidepressants given that they visited a therapist:

$$ P(A \mid T) = \frac{P(A \cap T)}{P(T)} $$

Substituting the known values:

$$ P(A \mid T) = \frac{0.20}{0.28} = \frac{5}{7} \approx 0.7143 $$

Rounding to 2 decimal places gives \(0.71\).

Calculate conditional probability for part (b)

Using the Conditional Probability knowledge point, we find the probability that a randomly selected adult visited a therapist given that they used non-prescription antidepressants:

$$ P(T \mid A) = \frac{P(T \cap A)}{P(A)} $$

Substituting the known values:

$$ P(T \mid A) = \frac{0.20}{0.44} = \frac{5}{11} \approx 0.4545 $$

Rounding to 2 decimal places gives \(0.45\).

Answer:

Question (a)

0.71

Question (b)

0.45