If A and B are two given events, then `P(A cap B)` is

Options

A. not less than `P(A) + P(B) -1`
B. not greater than `P(A) + P(B)`
C. equal to `P(A) + P(B) - P(A uuB)`
D. equal to `P(A) + P(B) + P(A uu B)`

Answer

To solve the problem, we need to find the expression for \( P(A \cap B) \) using the properties of probability and the addition theorem. Let's go through the solution step by step.

### Step-by-Step Solution

1. **Understanding the Addition Theorem**:
The addition theorem states that for any two events A and B:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

2. **Using the Property of Probability**:
We know that the probability of the union of two events cannot exceed 1:
\[ P(A \cup B) \leq 1 \]

3. **Substituting the Addition Theorem into the Inequality**:
By substituting the addition theorem into the inequality, we have:
\[ P(A) + P(B) - P(A \cap B) \leq 1 \]

4. **Rearranging the Inequality**:
Rearranging the above inequality gives us:
\[ P(A) + P(B) - 1 \leq P(A \cap B) \]

5. **Conclusion**:
This means that:
\[ P(A \cap B) \geq P(A) + P(B) - 1 \]

Thus, we have derived that \( P(A \cap B) \) is greater than or equal to \( P(A) + P(B) - 1 \).

### Final Result
The correct expression for \( P(A \cap B) \) is:
\[ P(A \cap B) \geq P(A) + P(B) - 1 \]