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(\text{lotion}) = 42% )
○ ( p(\text{scented}) = 42% )
○ ( p(\text{shower gel} mid \text{scented}) = 42% )
○ ( p(\text{scented} mid \text{shower gel}) = 42% )

Explanation:

Step1: Recall Independence of Events

Two events \( A \) and \( B \) are independent if \( P(A|B)=P(A) \) (or equivalently \( P(B|A)=P(B) \)). Here, let \( A \) be "shower gel" and \( B \) be "scented". We know \( P(A) = 42\% \).

Step2: Analyze Each Option

• Option 1: \( P(\text{lotion}) = 42\% \) is about lotion, not related to independence of "shower gel" and "scented". Eliminate.
• Option 2: \( P(\text{scented}) = 42\% \) – we don't know if \( P(\text{shower gel}) = P(\text{scented}) \), and this doesn't use the conditional probability for independence. Eliminate.
• Option 3: \( P(\text{shower gel} | \text{scented}) = 42\% \). By independence, \( P(A|B)=P(A) \), so \( P(\text{shower gel} | \text{scented}) = P(\text{shower gel}) = 42\% \), which satisfies the independence condition.
• Option 4: \( P(\text{scented} | \text{shower gel}) = 42\% \) – this would mean \( P(\text{scented}) = 42\% \) if independent, but we don't know that; the given \( 42\% \) is \( P(\text{shower gel}) \), not \( P(\text{scented}) \). Eliminate.

Answer:

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