Question

question 3 of 10
at a store, the probability that a customer buys socks is 0.15. the probability that a customer buys socks given that the customer buys shoes is 0.20.
which statement is true?
a. the probability that a customer buys socks and shoes is 0.05.
b. buying socks and buying shoes are independent events.
c. buying socks and buying shoes are dependent events.
d. every customer who buys shoes also buys socks.

Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Let \(A\) be the event of buying socks (\(P(A) = 0.15\)) and \(B\) be the event of buying shoes (\(P(B)=0.20\)), and \(P(A\cap B) = 0.05\).

Step2: Calculate the conditional probability \(P(A|B)\)

Using the formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), substitute the values: \(P(A|B)=\frac{0.05}{0.20}=\frac{1}{4}=0.25\).

Step3: Check for independence

Two events \(A\) and \(B\) are independent if \(P(A\cap B)=P(A)\times P(B)\). Calculate \(P(A)\times P(B)=0.15\times0.20 = 0.03
eq0.05 = P(A\cap B)\). Since \(P(A\cap B)
eq P(A)\times P(B)\), the events are not independent. Also, \(P(A|B)=0.25
eq P(A) = 0.15\), which also shows dependence.

Step4: Analyze option C and D

Option C: Just because \(P(A\cap B)=0.05\) (non - zero) doesn't mean every customer who buys shoes buys socks. Option D is wrong as \(P(A\cap B)=0.05\) and \(P(B) = 0.20\), so there are customers who buy shoes but not socks (\(P(B)-P(A\cap B)=0.20 - 0.05=0.15\)).

Answer:

B. Buying socks and buying shoes are dependent events.