Answer:
a) 0.299
b) 0.165
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:
![A = a + (A \cap B)](https://tex.z-dn.net/?f=A%20%3D%20a%20%2B%20%28A%20%5Ccap%20B%29)
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:
![B = b + (A \cap B)](https://tex.z-dn.net/?f=B%20%3D%20b%20%2B%20%28A%20%5Ccap%20B%29)
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](https://tex.z-dn.net/?f=A%20%5Ccap%20B%20%3D%200.04)
13.4% are diagnosed with an alcohol-related disorder
This means that ![B = 0.134](https://tex.z-dn.net/?f=B%20%3D%200.134)
So
![B = b + (A \cap B)](https://tex.z-dn.net/?f=B%20%3D%20b%20%2B%20%28A%20%5Ccap%20B%29)
![0.134 = b + 0.04](https://tex.z-dn.net/?f=0.134%20%3D%20b%20%2B%200.04)
![b = 0.094](https://tex.z-dn.net/?f=b%20%3D%200.094)
24.3% are diagnosed with a mental disorder
This means that ![A = 0.243](https://tex.z-dn.net/?f=A%20%3D%200.243)
So
![A = a + (A \cap B)](https://tex.z-dn.net/?f=A%20%3D%20a%20%2B%20%28A%20%5Ccap%20B%29)
![0.243 = a + 0.04](https://tex.z-dn.net/?f=0.243%20%3D%20a%20%2B%200.04)
![a = 0.203](https://tex.z-dn.net/?f=a%20%3D%200.203)
(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
. So ![D = 0.04](https://tex.z-dn.net/?f=D%20%3D%200.04)
Total outcomes:
Alcohol-related disorder, which is
. So ![T = 0.134](https://tex.z-dn.net/?f=T%20%3D%200.134)
Probability:
![P = \frac{0.04}{0.134} = 0.299](https://tex.z-dn.net/?f=P%20%3D%20%5Cfrac%7B0.04%7D%7B0.134%7D%20%3D%200.299)
(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
. So ![D = 0.04](https://tex.z-dn.net/?f=D%20%3D%200.04)
Total outcomes:
Mental health disorder, which is
. So ![T = 0.243](https://tex.z-dn.net/?f=T%20%3D%200.243)
Probability
![P = \frac{0.04}{0.243} = 0.165](https://tex.z-dn.net/?f=P%20%3D%20%5Cfrac%7B0.04%7D%7B0.243%7D%20%3D%200.165)