Step-by-step Solution:

We are tasked with finding the probability \( P((A \cap B^c) \cup (A^c \cap B)) \), where \( A^c \) is the complement of \( A \), and \( B^c \) is the complement of \( B \). \[\] Step 1: Interpret the Event \[\] The expression \( (A \cap B^c) \cup (A^c \cap B) \) represents the union of two disjoint events: \[\] - \( A \cap B^c \): This is the event where \( A \) occurs and \( B \) does not occur. \[\] - \( A^c \cap B \): This is the event where \( B \) occurs and \( A \) does not occur. Thus, \( (A \cap B^c) \cup (A^c \cap B) \) represents the event where either \( A \) occurs and \( B \) does not, or \( B \) occurs and \( A \) does not. This is essentially the event that \( A \) and \( B \) occur exclusively — in other words, the event where \( A \) and \( B \) are not simultaneously true. \[\] Step 2: Use the Probability Formula \[\] Since \( A \cap B^c \) and \( A^c \cap B \) are disjoint events (they cannot both occur at the same time), the probability of their union is the sum of their individual probabilities: \[ P((A \cap B^c) \cup (A^c \cap B)) = P(A \cap B^c) + P(A^c \cap B) \] \[\] Step 3: Express Each Probability \[\] - \( P(A \cap B^c) \) is the probability that \( A \) occurs and \( B \) does not occur. This can be written as: \[ P(A \cap B^c) = P(A) - P(A \cap B) \] because \( A = (A \cap B) \cup (A \cap B^c) \), and \( A \cap B^c \) is the part of \( A \) where \( B \) does not occur. \[\] - \( P(A^c \cap B) \) is the probability that \( A \) does not occur and \( B \) occurs. This can be written as: \[ P(A^c \cap B) = P(B) - P(A \cap B) \] because \( B = (A \cap B) \cup (A^c \cap B) \), and \( A^c \cap B \) is the part of \( B \) where \( A \) does not occur. \[\] Step 4: Combine the Results Now, we combine these results: \[ P((A \cap B^c) \cup (A^c \cap B)) = (P(A) - P(A \cap B)) + (P(B) - P(A \cap B)) \] Simplifying: \[ P((A \cap B^c) \cup (A^c \cap B)) = P(A) + P(B) - 2P(A \cap B) \] Final Answer: The probability \( P((A \cap B^c) \cup (A^c \cap B)) \) is: \[ \boxed{P(A) + P(B) - 2P(A \cap B)} \]