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.

a) 0.336

b) 0.207

Step-by-step 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:

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

By the same logic, we have that:

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

5% are diagnosed with both disorders.

This means that

14.9% are diagnosed with an alcohol-related disorder

This means that

So

24.1% are diagnosed with a mental disorder

This means that

So

(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 . So

Total outcomes:

Alcohol-related disorder, which is B. So

Probability:

(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 . So

Total outcomes:

Mental disorder, which is B. So

Probability:

let m = monomial

let t = trinomial

so, m times t = t.

monomial times a trinomial = another trinomial.