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. (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? Please use 3 decimal places. (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? Please use 3 decimal places.

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 0.286Part (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% = 0.04Part (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? [tex]P(\text{Mental disorder}\mid\text{alcohol-related disorder})=\frac{P(M\ and\ A)}{P(A)}[/tex][tex]P(\text{Mental disorder}\mid\text{alcohol-related disorder})=\frac{0.04}{0.14}=0.286[/tex]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.[tex]P(\text{alcohol-related disorder}\mid\text{Mental disorder})=\frac{P(M\ and\ A)}{P(M)}[/tex][tex]P(\text{alcohol-related disorder}\mid\text{Mental disorder})=\frac{0.04}{0.245}=0.163[/tex]Hence, the required probability is 0.163.