1. **Problem Statement:** Show that for any two events $A$ and $B$, the probability of their union is given by: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ 2. **Formula and Explanation:** The formula expresses the probability of either event $A$ or event $B$ occurring. When we add $P(A)$ and $P(B)$, the intersection $P(A \cap B)$ is counted twice, so we subtract it once to correct the count. 3. **Step-by-step Proof:** - Start with the union of two events: $$A \cup B = A + B - (A \cap B)$$ - By the axioms of probability, the probability of the union is: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ 4. **Intuitive Explanation:** If you count the probabilities of $A$ and $B$ separately, the overlap where both happen is counted twice. Subtracting $P(A \cap B)$ once removes this double counting. 5. **Final Answer:** $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$