Explanation
The expression given in the problem is P((A∩BC)∪(AC∩B)). This represents the probability of the symmetric difference between events A and B, which means the probability that exactly one of the events occurs.
Step 1: Check for Mutually Exclusive Events
Let us analyze the two components inside the union:
Event 1: A∩BC (Event A occurs, but B does not)
Event 2: AC∩B (Event B occurs, but A does not)
Since an element cannot simultaneously be in A (and not B) and in B (and not A), these two events are completely mutually exclusive (disjoint). Their intersection is an empty set:
(A∩BC)∩(AC∩B)=∅
Step 2: Apply the Axiom of Additivity
For any two mutually exclusive events, the probability of their union is simply the sum of their individual probabilities:
P((A∩BC)∪(AC∩B))=P(A∩BC)+P(AC∩B)
Step 3: Express Components in Terms of Standard Probabilities
Using standard set theory rules and probability axioms:
Step 4: Substitute and Simplify
Now, substitute these two expressions back into our equation from Step 2:
P((A∩BC)∪(AC∩B))=[P(A)−P(A∩B)]+[P(B)−P(A∩B)]
Combine the like terms:
P((A∩BC)∪(AC∩B))=P(A)+P(B)−2P(A∩B)
