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.

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:

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.

4% are diagnosed with both disorders.

This means that

13.4% are diagnosed with an alcohol-related disorder

This means that

So

24.3% 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. So . So

Total outcomes:

Alcohol-related disorder, which is . 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. So . So

Total outcomes:

Mental health disorder, which is . So

Probability

- 2 + 10x³ - 21x² + 19x - 4

step-by-step explanation:

each term in the second factor is multiplied by each term in the first factor

x²(- 2x² + 4x - 1) - 3x(- 2x² + 4x - 1) + 4(- 2x² + 4x - 1) ← distribute

= - 2 + 4x³ - x² + 6x³ - 12x² + 3x -8x² + 16x - 4

collect like terms

= - 2 + 10x³ - 21x² + 19x - 4 ← in standard form

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!