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.

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05-10-2019
Mathematics

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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.

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FelisFelis FelisFelis · Answer 1 · 2019-10-07T08:24:11+02:00

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?

[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.