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∩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∩W)=0.
Since P(D∩W)=0.13=0, depression and weight gain are not mutually exclusive.
\\(\boxed{\text{no}}\\)
e. Are depression and weight gain independent?
Check if P(D∩W)=P(D)⋅P(W).
Calculate P(D)⋅P(W)=0.32⋅0.30=0.096.
Since P(D∩W)=0.13=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}}\\)