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At a drug rehab center 40% experience depression and 28% experience weight gain. 11% experience both. If a patient from the center is randomly selected, find the probability that the patient (Round all answers to four decimal places where possible.)

a. experiences neither depression nor weight gain.

b. experiences depression given that the patient experiences weight gain.

c. experiences weight gain given that the patient experiences depression.
(round to 4 decimal places)

d. Are depression and weight gain mutually exclusive?
yes
no

e. Are depression and weight gain independent?
no
yes

Respuesta :

Using probability concepts, it is found that:

a) 0.43 = 43% probability that the patient experiences neither depression nor weight gain.

b) 0.3929 = 39.29% probability that the patient experiences depression given that the patient experiences weight gain.

c) 0.275 = 27.5% probability that the patient experiences weight gain given that the patient experiences depression.

d) No.

e) No.

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  • Conditional probability is an important part of the exercise.
  • The formula is:

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]  

  • P(B|A) is the probability of event B happening, given that A happened.
  • [tex]\mathbf{P(A \cap B)}[/tex] is the probability of both A and B happening.
  • P(A) is the probability of A happening.

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  • 40% experience depression, thus [tex]P(A) = 0.4[/tex]
  • 28% experience weight gain, thus [tex]P(B) = 0.28[/tex]
  • 11% experience both, thus: [tex]P(A \cap B) = 0.11[/tex]

Item a:

  • The probability of experiencing at least one of them is:

[tex]P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.4 + 0.28 - 0.11 = 0.57[/tex]

  • The probability of none is:

[tex]p = 1 - P(A \cup B) = 1 - 0.57 = 0.43[/tex]

0.43 = 43% probability that the patient experiences neither depression nor weight gain.

Item b:

  • This probability is: [tex]P(A|B)[/tex].
  • Using conditional probability:

[tex]P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.11}{0.28} = 0.3929[/tex]

0.3929 = 39.29% probability that the patient experiences depression given that the patient experiences weight gain.

Item c:

  • Similar to b, thus:

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.11}{0.4} = 0.275[/tex]

0.275 = 27.5% probability that the patient experiences weight gain given that the patient experiences depression.

Item d:

[tex]P(A \cap B) \neq 0[/tex], thus, the events are not mutually exclusive.

Item e:

  • We have that: [tex]P(A \cap B) = 0.11[/tex]
  • Also: [tex]P(A)P(B) = 0.4(0.28) = 0.112[/tex]
  • [tex]P(A \cap B) = P(A)P(B)[/tex], thus, the are not independent.

A similar problem is given at https://brainly.com/question/21421475

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