The complement of P(A∣B) is
a. P(AIB).
b. P(A C∣B).
c. P(B∣A).
d. P(A∣B C).
The Correct Answer and Explanation is:
To solve the problem of finding the complement of ( P(A|B) ), let’s first clarify what ( P(A|B) ) represents. The notation ( P(A|B) ) stands for the conditional probability of event ( A ) occurring given that event ( B ) has occurred. Mathematically, it is defined as:
[
P(A|B) = \frac{P(A \cap B)}{P(B)}
]
The complement of an event refers to the probability that the event does not occur. Therefore, the complement of ( P(A|B) ), denoted as ( P(A^c|B) ), is the probability that event ( A ) does not happen given that ( B ) has occurred. This can be mathematically expressed as:
[
P(A^c|B) = 1 – P(A|B)
]
Now, we need to analyze the provided options to identify which one represents ( P(A^c|B) ):
a. ( P(A|B) ): This is the probability of ( A ) given ( B ), not its complement.
b. ( P(A^c|B) ): This is indeed the correct representation of the complement of ( P(A|B) ), as it directly refers to the probability of ( A ) not occurring given ( B ).
c. ( P(B|A) ): This is the conditional probability of ( B ) given ( A ), which does not relate to the complement of ( A ) given ( B ).
d. ( P(A|B^c) ): This denotes the probability of ( A ) given that ( B ) has not occurred, which is also unrelated to the complement of ( P(A|B) ).
Thus, the correct answer is (b) ( P(A^c|B) ).
In conclusion, when we consider the complement of the conditional probability ( P(A|B) ), we are interested in ( P(A^c|B) ), which reflects the likelihood that ( A ) does not occur under the condition that ( B ) is true. This distinction is vital in probability theory and has implications in various fields, such as statistics, risk assessment, and decision-making. Understanding how to navigate these probabilities allows for more informed interpretations and applications in real-world scenarios.