Question

at a drug rehab center 35% experience depression and 25% experience weight gain. 18% 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?
yes
no

Explanation:

Step1: Define events and probabilities

Let $A$ be the event of depression, $P(A)=0.35$; $B$ be the event of weight - gain, $P(B)=0.25$; $P(A\cap B)=0.18$.

Step2: Find $P(A\cup B)$

Use the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. So $P(A\cup B)=0.35 + 0.25-0.18=0.42$.

Step3: Find the probability of neither depression nor weight - gain

The probability of neither $A$ nor $B$ is $P(\overline{A\cup B})=1 - P(A\cup B)$. So $P(\overline{A\cup B})=1 - 0.42 = 0.5800$.

Step4: Find $P(A|B)$

Use the formula for conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}$. So $P(A|B)=\frac{0.18}{0.25}=0.7200$.

Step5: Find $P(B|A)$

Use the formula for conditional probability $P(B|A)=\frac{P(A\cap B)}{P(A)}$. So $P(B|A)=\frac{0.18}{0.35}\approx0.5143$.

Step6: Check mutual - exclusivity

Two events are mutually exclusive if $P(A\cap B)=0$. Since $P(A\cap B)=0.18
eq0$, the answer is no.

Step7: Check independence

Two events are independent if $P(A\cap B)=P(A)\times P(B)$. Since $P(A)\times P(B)=0.35\times0.25 = 0.0875
eq0.18$, the answer is no.

Answer:

a. $0.5800$
b. $0.7200$
c. $0.5143$
d. no
e. no