Question

at a drug rehab center 40% experience depression and 26% experience weight gain. 16% 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?
no
yes
e. are depression and weight gain independent?
yes
no

Explanation:

Step1: Define probabilities

Let $P(D) = 0.4$ be the probability of depression, $P(W)=0.26$ be the probability of weight - gain, and $P(D\cap W)=0.16$.

Step2: Find $P(D\cup W)$

Use the formula $P(D\cup W)=P(D)+P(W)-P(D\cap W)$.
$P(D\cup W)=0.4 + 0.26-0.16=0.5$.

Step3: Find probability of neither

The probability of neither depression nor weight - gain is $P((D\cup W)^C)=1 - P(D\cup W)$.
$P((D\cup W)^C)=1 - 0.5 = 0.5000$.

Step4: Find conditional probability $P(D|W)$

Use the formula for conditional probability $P(D|W)=\frac{P(D\cap W)}{P(W)}$.
$P(D|W)=\frac{0.16}{0.26}\approx0.6154$.

Step5: Find conditional probability $P(W|D)$

Use the formula for conditional probability $P(W|D)=\frac{P(D\cap W)}{P(D)}$.
$P(W|D)=\frac{0.16}{0.4}=0.4000$.

Step6: Check mutual - exclusivity

Two events are mutually exclusive if $P(D\cap W) = 0$. Since $P(D\cap W)=0.16
eq0$, the answer is no.

Step7: Check independence

Two events are independent if $P(D\cap W)=P(D)\times P(W)$. Since $P(D)\times P(W)=0.4\times0.26 = 0.104
eq0.16$, the answer is no.

Answer:

a. $0.5000$
b. $0.6154$
c. $0.4000$
d. no
e. no