Questions: At a drug rehab center 32% experience depression and 30% experience weight gain. 13% 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.) Tip 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. d. Are depression and weight gain mutually exclusive? O no O yes e. Are depression and weight gain independent? O yes O no Calculator Scratchwork Area

a. experiences neither depression nor weight gain.

Calculate the probability of experiencing neither depression nor weight gain.

Let $P(D) = 0.32$ (probability of depression), $P(W) = 0.30$ (probability of weight gain), and $P(D \cap W) = 0.13$ (probability of both).
The probability of experiencing neither is:
\[ P(\text{Neither}) = 1 - P(D) - P(W) + P(D \cap W) = 1 - 0.32 - 0.30 + 0.13 = 0.51
\]

\\(\boxed{0.5100}\\)

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

Calculate the conditional probability of depression given weight gain.

The conditional probability is:
\[ P(D \mid W) = \frac{P(D \cap W)}{P(W)} = \frac{0.13}{0.30} \approx 0.4333
\]

\\(\boxed{0.4333}\\)

c. experiences weight gain, given that the patient experiences depression.

Calculate the conditional probability of weight gain given depression.

The conditional probability is:
\[ P(W \mid D) = \frac{P(D \cap W)}{P(D)} = \frac{0.13}{0.32} \approx 0.4063
\]

\\(\boxed{0.4063}\\)

d. Are depression and weight gain mutually exclusive?

Check if $P(D \cap W) = 0$.

Since $P(D \cap W) = 0.13 \neq 0$, depression and weight gain are not mutually exclusive.

\\(\boxed{\text{no}}\\)

e. Are depression and weight gain independent?

Check if $P(D \cap W) = P(D) \cdot P(W)$.

Calculate $P(D) \cdot P(W) = 0.32 \cdot 0.30 = 0.096$.
Since $P(D \cap W) = 0.13 \neq 0.096$, depression and weight gain are not independent.

\\(\boxed{\text{no}}\\)

a. \\(\boxed{0.5100}\\)
b. \\(\boxed{0.4333}\\)
c. \\(\boxed{0.4063}\\)
d. \\(\boxed{\text{no}}\\)
e. \\(\boxed{\text{no}}\\)