Question

juanita has a storage closet at her shop with extra bottles of lotion and shower gel. some are scented and some are unscented. if she reaches into the closet and grabs a bottle without looking, she has a 42% chance of grabbing a bottle of shower gel. for the events \shower gel\ and \scented\ to be independent, what must be shown to be true? o p(lotion) = 42% o p(scented) = 42% o p(shower gel | scented) = 42% o p(scented | shower gel) = 42%

Explanation:

Step1: Recall independence formula

Two events \(A\) and \(B\) are independent if \(P(A|B)=P(A)\) and \(P(B|A) = P(B)\). Here, let \(A\) be the event of getting a shower - gel and \(B\) be the event of getting a scented product. Given \(P(A)=42\%\).

Step2: Apply independence condition

For events “shower gel” and “scented” to be independent, the conditional probability \(P(\text{shower gel}|\text{scented})\) must be equal to the marginal probability \(P(\text{shower gel})\). Since \(P(\text{shower gel}) = 42\%\), we must have \(P(\text{shower gel}|\text{scented})=42\%\).

Answer:

C. \(P(\text{shower gel}|\text{scented}) = 42\%\)