Answer:
Part (A) The probability that someone from the clinical population is diagnosed with a mental disorder, knowing that the person is diagnosed with an alcohol-related disorder is 0.286
Part (B) The probability that someone from the clinical population is diagnosed with an alcohol-related disorder, knowing that the person is diagnosed with a mental disorder is 0.163.
Step-by-step explanation:
Consider the provided information.
A study of the US clinical population found that 24.5% are diagnosed with a mental disorder, 14% are diagnosed with an alcohol-related disorder, and 4% are diagnosed with both disorders.
P(M) = P(Mental disorder) = 24.5% = 0.245
P(A) =P(Alcohol-related disorder) = 14% = 0.14
P(M and A) =P(Both disorder) = 4% = 0.04
Part (A) What is the probability that someone from the clinical population is diagnosed with a mental disorder, knowing that the person is diagnosed with an alcohol-related disorder?
![P(\text{Mental disorder}\mid\text{alcohol-related disorder})=\frac{P(M\ and\ A)}{P(A)}](https://tex.z-dn.net/?f=P%28%5Ctext%7BMental%20disorder%7D%5Cmid%5Ctext%7Balcohol-related%20disorder%7D%29%3D%5Cfrac%7BP%28M%5C%20and%5C%20A%29%7D%7BP%28A%29%7D)
![P(\text{Mental disorder}\mid\text{alcohol-related disorder})=\frac{0.04}{0.14}=0.286](https://tex.z-dn.net/?f=P%28%5Ctext%7BMental%20disorder%7D%5Cmid%5Ctext%7Balcohol-related%20disorder%7D%29%3D%5Cfrac%7B0.04%7D%7B0.14%7D%3D0.286)
Hence, the required probability is 0.286.
Part (b) What is the probability that someone from the clinical population is diagnosed with an alcohol-related disorder, knowing that the person is diagnosed with a mental disorder.
![P(\text{alcohol-related disorder}\mid\text{Mental disorder})=\frac{P(M\ and\ A)}{P(M)}](https://tex.z-dn.net/?f=P%28%5Ctext%7Balcohol-related%20disorder%7D%5Cmid%5Ctext%7BMental%20disorder%7D%29%3D%5Cfrac%7BP%28M%5C%20and%5C%20A%29%7D%7BP%28M%29%7D)
![P(\text{alcohol-related disorder}\mid\text{Mental disorder})=\frac{0.04}{0.245}=0.163](https://tex.z-dn.net/?f=P%28%5Ctext%7Balcohol-related%20disorder%7D%5Cmid%5Ctext%7BMental%20disorder%7D%29%3D%5Cfrac%7B0.04%7D%7B0.245%7D%3D0.163)
Hence, the required probability is 0.163.