Question

A study of the US clinical population found that 24.1% are diagnosed with a mental disorder, 14.9% are diagnosed with an alcohol-related disorder, and 5% 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

a) 0.336

b) 0.207

Explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

We solve this problem building the Venn's diagram of these probabilities.

I am going to say that:

A is the probability that a person from the clinical population is diagnosed with mental disorder.

B is the probability that a person from the clinical population is diagnosed with alcohol related disorder.

We have that:

`A = a + (A \cap B)`

In which a is the probability that a person is diagnosed with mental disorder but not alcohol related disorder and
`A \cap B` is the probability that a person is diagnosed with both of these disorders.

By the same logic, we have that:

`B = b + (A \cap B)`

We find the values of a,b and the intersection, starting from the intersection.

5% are diagnosed with both disorders.

This means that
`A \cap B = 0.05`

14.9% are diagnosed with an alcohol-related disorder

This means that
`B = 0.149`

So

`B = b + (A \cap B)`

`0.149 = b + 0.05`

`b = 0.099`

24.1% are diagnosed with a mental disorder

This means that
`A = 0.241`

So

`A = a + (A \cap B)`

`0.241 = a + 0.05`

`a = 0.203`

(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?

Desired outcomes:

Mental and alcohol-related disorders, which
`A \cap B`. So
`D = 0.05`

Total outcomes:

Alcohol-related disorder, which is B. So
`T = 0.149`

Probability:

`P = (0.05)/(0.149) = 0.336`

(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?

Desired outcomes:

Mental and alcohol-related disorders, which
`A \cap B`. So
`D = 0.05`

Total outcomes:

Mental disorder, which is B. So
`T = 0.241`

Probability:

`P = (0.05)/(0.241) = 0.207`