Respuesta :
Answer:
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:
[tex]A = a + (A \cap B)[/tex]
In which a is the probability that a person is diagnosed with mental disorder but not alcohol related disorder and [tex]A \cap B[/tex] is the probability that a person is diagnosed with both of these disorders.
By the same logic, we have that:
[tex]B = b + (A \cap B)[/tex]
We find the values of a,b and the intersection, starting from the intersection.
5% are diagnosed with both disorders.
This means that [tex]A \cap B = 0.05[/tex]
14.9% are diagnosed with an alcohol-related disorder
This means that [tex]B = 0.149[/tex]
So
[tex]B = b + (A \cap B)[/tex]
[tex]0.149 = b + 0.05[/tex]
[tex]b = 0.099[/tex]
24.1% are diagnosed with a mental disorder
This means that [tex]A = 0.241[/tex]
So
[tex]A = a + (A \cap B)[/tex]
[tex]0.241 = a + 0.05[/tex]
[tex]a = 0.203[/tex]
(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 [tex]A \cap B[/tex]. So [tex]D = 0.05[/tex]
Total outcomes:
Alcohol-related disorder, which is B. So [tex]T = 0.149[/tex]
Probability:
[tex]P = \frac{0.05}{0.149} = 0.336[/tex]
(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 [tex]A \cap B[/tex]. So [tex]D = 0.05[/tex]
Total outcomes:
Mental disorder, which is B. So [tex]T = 0.241[/tex]
Probability:
[tex]P = \frac{0.05}{0.241} = 0.207[/tex]
