Let for two events `A` and `B`, `P(A)=p` and `P(B)=q`.
Statement-1 : The probability that exactly one of the event `A` and `B` occurs is `p+q-2pq`
Statement-2 : `P(AcupB)=P(A)+P(B)-P(AcapB)`.

Options

A. Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement -3
B. Statement-1 is true, Statement-2 is true, Statement-2 is Not a correct explanation for statement -3
C. Statement-1 is true, Statement-2 is False
D. Statement-1 is False, Statement-2 is true

Answer

To solve the given problem, we need to analyze both statements regarding the probabilities of events A and B.

### Step-by-Step Solution:

**Step 1: Analyze Statement 2**
- Statement 2 states that \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).
- This is a well-known property of probability, known as the inclusion-exclusion principle.
- Since this statement is a fundamental theorem in probability, it is **True**.

**Step 2: Analyze Statement 1**
- Statement 1 claims that the probability that exactly one of the events A and B occurs is given by \( P(A) + P(B) - 2P(A \cap B) \).
- The probability that exactly one of the events A or B occurs can be expressed as:
\[ P(\text{exactly one of A or B}) = P(A \cup B) - P(A \cap B) \]
- Using the inclusion-exclusion principle from Statement 2, we can substitute \( P(A \cup B) \):
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Therefore, we can rewrite the probability of exactly one of the events occurring as:
\[ P(\text{exactly one of A or B}) = [P(A) + P(B) - P(A \cap B)] - P(A \cap B) \]
\[ = P(A) + P(B) - 2P(A \cap B) \]
- Now, substituting \( P(A) = p \) and \( P(B) = q \):
\[ P(\text{exactly one of A or B}) = p + q - 2P(A \cap B) \]
- The statement claims this is equal to \( p + q - 2pq \). Thus, we need to check if \( P(A \cap B) = pq \) holds true in general.

**Step 3: Evaluate \( P(A \cap B) \)**
- The expression \( P(A \cap B) = pq \) is true only if A and B are independent events.
- However, without any information about the independence of A and B, we cannot assume this equality holds in all cases.

**Conclusion:**
- Since Statement 1 is only true under the condition that A and B are independent, and we have no such information, we conclude that Statement 1 is **False**.

### Final Results:
- **Statement 1**: False
- **Statement 2**: True