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? p(lotion) = 42% p(scented) = 42% p(shower gel | scented) = 42% p(scented | shower gel) = 42%

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 “shower gel” and \(B\) be the event “scented”

We know that \(P(A) = 42\%\)

If \(A\) and \(B\) are independent, then \(P(A|B)=P(A)\)

Step2: Substitute the events

Since \(A\) is “shower gel” and \(B\) is “scented”

\(P(\text{shower gel}|\text{scented})=P(\text{shower gel})\)

And \(P(\text{shower gel}) = 42\%\)

Answer:

\(P(\text{shower gel}|\text{scented}) = 42\%\) (the third option)