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:

Identify the given probability

Using the Independent Events knowledge point
The problem states that the probability of grabbing a bottle of shower gel is \(42\%\):

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

Apply the definition of independent events

Using the Independent Events and Conditional Probability knowledge points
For two events \(A\) and \(B\) to be independent, the occurrence of event \(B\) does not affect the probability of event \(A\). Mathematically, this is written as:

$$ P(A \mid B) = P(A) $$

Letting \(A = \text{shower gel}\) and \(B = \text{scented}\), independence requires:

$$ P(\text{shower gel} \mid \text{scented}) = P(\text{shower gel}) $$

Determine the required condition

Using the Independent Events knowledge point
Substituting the given value \(P(\text{shower gel}) = 42\%\) into the independence condition yields:

$$ P(\text{shower gel} \mid \text{scented}) = 42\% $$

This matches the third option.

Answer:

• P(lotion) = 42%
• P(scented) = 42%
• P(shower gel | scented) = 42% (Correct answer)
• P(scented | shower gel) = 42%