Question

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 1

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.

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})=(P(M\ and\ A))/(P(A))`

`P(\text{Mental disorder}\mid\text{alcohol-related disorder})=(0.04)/(0.14)=0.286`

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})=(P(M\ and\ A))/(P(M))`

`P(\text{alcohol-related disorder}\mid\text{Mental disorder})=(0.04)/(0.245)=0.163`

Hence, the required probability is 0.163.