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%

Hi, I was able to find the full context for this one from another source:

"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 "shower gel" and "scented" to be independent given the situation above, we need to show that P(A | B) = P(A). You can get this equation from the definition of "independence" where P(A ∩ B) = P(A)*P(B) and the formula for conditional probability P(A | B).

We only have the given P(shower gel) = 42% therefore the event "shower gel" must be the variable A.

ANSWER: To show independence of the two events, P(shower gel | scented) = 42% must be true.