Certainly! Let's dissect the problem step by step:
The problem states that a pet store sells mice, reptiles, and birds.
We have the following definitions for events:
- Event [tex]\( A = \)[/tex] A customer buys a mouse.
- Event [tex]\( B = \)[/tex] A customer buys a bird.
We are given that [tex]\( P(A \text{ or } B) = 0.15 \)[/tex].
In probability theory, [tex]\( P(A \text{ or } B) \)[/tex] represents the probability that event [tex]\( A \)[/tex] or event [tex]\( B \)[/tex] (or both) occurs. This is also known as the union of events [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Given [tex]\( P(A \text{ or } B) = 0.15 \)[/tex], we interpret this as follows: The probability that a customer buys either a mouse or a bird (i.e., at least one of these two types of pets) is 15%.
Now, let's evaluate the options given:
A. "The probability that a customer buys both a mouse and a bird is [tex]\( 15 \% \)[/tex]." This does not relate to [tex]\( P(A \text{ or } B) \)[/tex]. It would rather be [tex]\( P(A \text{ and } B) \)[/tex].
B. "The probability that a customer buys either a mouse or a bird is [tex]\( 15 \% \)[/tex]." This correctly interprets [tex]\( P(A \text{ or } B) \)[/tex].
C. "The probability that a customer buys neither a mouse nor a bird is [tex]\( 15 \% \)[/tex]." This would instead describe [tex]\( P(\text{neither } A \text{ nor } B) \)[/tex], or [tex]\( P(A' \text{ and } B') \)[/tex], which is different from [tex]\( P(A \text{ or } B) \)[/tex].
D. "Buying a mouse and buying a bird are mutually exclusive events." This statement would mean that [tex]\( P(A \text{ and } B) = 0 \)[/tex], which is unrelated to the given [tex]\( P(A \text{ or } B) \)[/tex].
Based on this step-by-step evaluation, the correct interpretation of [tex]\( P(A \text{ or } B) = 0.15 \)[/tex] in the context of this problem is:
B. The probability that a customer buys either a mouse or a bird is [tex]\( 15 \% \)[/tex].
