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)-P(A cup B)`
C. equal to `P(A)+P(B)+P(A cap B)`
D. equl to `P(A)+P(B)+P(A cup B)`

Explanation

To solve the problem, we need to determine the relationship of \( P(A \cap B) \) with respect to the probabilities of events A and B.

### Step-by-Step Solution:

1. **Understand the Formula for Union of Two Events**:
The probability of the union of two events A and B is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

2. **Apply the Property of Probability**:
We know that the probability of any event cannot exceed 1, so:
\[ P(A \cup B) \leq 1 \]
Substituting the formula for \( P(A \cup B) \) into this inequality gives:
\[ P(A) + P(B) - P(A \cap B) \leq 1 \]

3. **Rearranging the Inequality**:
Rearranging the above inequality to isolate \( P(A \cap B) \):
\[ P(A) + P(B) - 1 \leq P(A \cap B) \]
This means:
\[ P(A \cap B) \geq P(A) + P(B) - 1 \]

4. **Conclusion**:
From the above steps, we conclude that:
\[ P(A \cap B) \text{ is not less than } P(A) + P(B) - 1 \]
Therefore, the correct option is the first one.

### Final Answer:
The correct answer is:
\[ P(A \cap B) \geq P(A) + P(B) - 1 \]