A study of the US clinical population found that 248 are dia

A study of the US clinical population found that 24.8% are diagnosed with a mental disorder, 14.2% are diagnosed with an alcohol-related disorder, and 6% 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.

Solution

Probability that a person is diagnosed with a mental disorder, P(M) = 0.248

Probability that a person is diagnosed with alcohol-related disorder, P(A) = 0.142

Probabiltiy that a person is diagonosed with both, P(A and M) = 0.06

P( A or M) = P(M) + P(A) - P(A and M) = 0.248 + 0.142 - 0.06 = 0.33

1) 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(M | A)

= P( M and A) / P(A) = 0.06/0.142 = 0.423

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2) 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(A | M)

= P( A and M) / P(M)

= 0.06 / 0.248

= 0.242