Certainly! Let's delve into the problem to understand it step-by-step, making sure we define what [tex]\( P(B \mid A) \)[/tex] represents.
We start with the given events:
- Event [tex]\( A \)[/tex]: The first dart hits the center.
- Event [tex]\( B \)[/tex]: The second dart hits the center.
When we see the notation [tex]\( P(B \mid A) \)[/tex], it refers to a conditional probability. Specifically, it represents the probability that event [tex]\( B \)[/tex] occurs given that event [tex]\( A \)[/tex] has already occurred.
To break it down:
1. [tex]\( P(B \)[/tex] represents the probability of the second dart hitting the center under normal circumstances (without any other conditions).
2. [tex]\( P(B \mid A) \)[/tex] changes this context and instead asks: What is the probability that the second dart hits the center assuming that we already know the first dart has hit the center?
In everyday terms, it's like saying we've already seen the first dart hit the center, and now we want to know the likelihood of the second dart doing the same.
Given the potential answer choices:
A. The probability that the second dart hits the center given that the first dart hits the center.
- This directly matches our interpretation of [tex]\( P(B \mid A) \)[/tex].
B. The probability that either the first dart or the second dart hits the center.
- This would be denoted as [tex]\( P(A \cup B) \)[/tex], which is not what [tex]\( P(B \mid A) \)[/tex] represents.
C. The probability that the second dart doesn't hit the center.
- This is [tex]\( P(B') \)[/tex], which is the complement of [tex]\( P(B) \)[/tex], thus not equivalent to [tex]\( P(B \mid A) \)[/tex].
D. The probability that the first dart hits the center given that the second dart hits the center.
- This would be expressed as [tex]\( P(A \mid B) \)[/tex], the reverse of our conditional probability.
Therefore, based on this analysis, [tex]\( P(B \mid A) \)[/tex] clearly represents:
The probability that the second dart hits the center given that the first dart hits the center.
The correct answer is:
A. The probability that the second dart hits the center given that the first dart hits the center.
