Question

A study of the US clinical population found that 24.3% are diagnosed with a mental disorder, 13.4% 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. .275 Incorrect: Your answer is incorrect. (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) 0.299

b) 0.165

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.

4% are diagnosed with both disorders.

This means that
`A \cap B = 0.04`

13.4% are diagnosed with an alcohol-related disorder

This means that
`B = 0.134`

So

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

`0.134 = b + 0.04`

`b = 0.094`

24.3% are diagnosed with a mental disorder

This means that
`A = 0.243`

So

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

`0.243 = a + 0.04`

`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. So
`A \cap B`. So
`D = 0.04`

Total outcomes:

Alcohol-related disorder, which is
`B`. So
`T = 0.134`

Probability:

`P = (0.04)/(0.134) = 0.299`

(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. So
`A \cap B`. So
`D = 0.04`

Total outcomes:

Mental health disorder, which is
`A`. So
`T = 0.243`

Probability

`P = (0.04)/(0.243) = 0.165`