Question

the probability that a randomly selected adult in a particular community is a smoker is 18%. the probability that a randomly selected adult in the community is a smoker, given that the adult earns more than $75,000 per year, is 18%. are the events “is a smoker” and “earns more than $75,000 per year” independent? explain. choose the correct answer below. a. yes, because it is possible for an adult in the community to be a smoker and earn over $75,000 per year, the events are independent. b. no, because it is possible for an adult in the community to be a smoker and earn over $75,000 per year, the events are not independent. c. no, because the probability of smoking is the same, even when the income condition is added, the two events are not independent. d. yes, because the probability of smoking is the same, even when the income condition is added, the two events are independent.

Explanation:

Step1: Recall the definition of independent events

Two events $A$ and $B$ are independent if $P(A|B)=P(A)$. Let $A$ be the event "is a smoker" and $B$ be the event "earns more than $75000$ per year". We are given that $P(A) = 0.18$ and $P(A|B)=0.18$.

Step2: Determine independence

Since $P(A|B)=P(A) = 0.18$, the occurrence of event $B$ (earning more than $75000$ per year) does not affect the probability of event $A$ (being a smoker). So the two events are independent.

Answer:

D. Yes, because the probability of smoking is the same, even when the income condition is added, the two events are independent.