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 21% 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) 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. (b) 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.
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 21% 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) 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. (b) 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.
Chapter8: Sequences, Series,and Probability
Section: Chapter Questions
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Transcribed Image Text:**Educational Website Resource: Probability Application in Mental Health Statistics**
### Understanding Probabilities in Mental Health Contexts
Estimating probabilities can provide valuable insights into health behavior patterns within a population. Here, we'll explore an example scenario related to the use of therapy services and non-prescription antidepressants.
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#### Scenario
It is estimated that, during the past year:
- 28% of all adults visited a therapist.
- 44% of all adults used non-prescription antidepressants.
- 21% of all adults both visited a therapist and used a non-prescription antidepressant during the past year.
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### Questions
(a) **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.**
(b) **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.**
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### Explanation
To answer these questions, we'll utilize concepts from conditional probability.
1. **Conditional Probability Formula:**
- \( P(A|B) = \dfrac{P(A \cap B)}{P(B)} \)
2. Let's define the events:
- \( A \): The event that an adult visited a therapist.
- \( B \): The event that an adult used non-prescription antidepressants.
##### For Question (a):
- We need to find the probability \( P(A|B) \).
- Given:
- \( P(A) = 0.28 \)
- \( P(B) = 0.44 \)
- \( P(A \cap B) = 0.21 \)
- Using the formula:
- \( P(A|B) = \dfrac{P(A \cap B)}{P(B)} = \dfrac{0.21}{0.44} \approx 0.477 \)
- Rounded to 2 decimal places:
- \( P(A|B) \approx 0.48 \)
##### For Question (b):
- We need to find the probability \( P(B|A) \).
- Using the given probabilities and the same formula:
- \( P(B|A) = \dfrac{P(A \cap B)}{P(A)} = \dfrac{0.21
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