Respuesta :
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:
[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.
4% are diagnosed with both disorders.
This means that [tex]A \cap B = 0.04[/tex]
13.4% are diagnosed with an alcohol-related disorder
This means that [tex]B = 0.134[/tex]
So
[tex]B = b + (A \cap B)[/tex]
[tex]0.134 = b + 0.04[/tex]
[tex]b = 0.094[/tex]
24.3% are diagnosed with a mental disorder
This means that [tex]A = 0.243[/tex]
So
[tex]A = a + (A \cap B)[/tex]
[tex]0.243 = a + 0.04[/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. So [tex]A \cap B[/tex]. So [tex]D = 0.04[/tex]
Total outcomes:
Alcohol-related disorder, which is [tex]B[/tex]. So [tex]T = 0.134[/tex]
Probability:
[tex]P = \frac{0.04}{0.134} = 0.299[/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. So [tex]A \cap B[/tex]. So [tex]D = 0.04[/tex]
Total outcomes:
Mental health disorder, which is [tex]A[/tex]. So [tex]T = 0.243[/tex]
Probability
[tex]P = \frac{0.04}{0.243} = 0.165[/tex]
