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%

Answer 1

Independence means P(shower gel | scented) = P(shower gel). Since P(shower gel) = 42%, you must show

P(shower gel | scented) = 42%.

(Equivalently you could show P(scented | shower gel) = P(scented), but that does not necessarily equal 42% unless P(scented) = 42%.)