It is estimated that, during the past year, 27% of all adults visited a therapist and 45% of all adults used non-prescription antidepressants. It is also estimated that 21% of all adults both visited a therapist and used a non-prescription antidepressant during the past year.

(a) What is the probability that an adult visited a therapist during the past year, given that he or she used non-prescription antidepressants?

(b) What is the probability that a randomly selected adult who visited a therapist during the past year also used non-prescription antidepressants?

#### Solution

Let A denote the event where adult visit therapist

and B denote the event where adult used non-prescription antidepressants.

Then $A \cap B$ denote the event where adult visit therapist and used non-prescription antidepressants.

Given that $P(A) = 0.26$, $P(B) = 0.44$ and $P(A\cap B) = 0.22$

(a) The probability that an adult visited a therapist during the past year, given that he or she used non-prescription antidepressants is $P(A|B)$.

` $$ \begin{aligned} P(A|B) &= \frac{P(A\cap B)}{P(B)}\\ &= \frac{0.22}{0.44}\\ &= 0.5 \end{aligned} $$ `

(b) The probability that a randomly selected adult who visited a therapist during the past year also used non-prescription antidepressants $P(B|A)$.

` $$ \begin{aligned} P(B|A) &= \frac{P(B \cap A)}{P(A)}\\ &= \frac{0.22}{0.26}\\ &= 0.85 \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators