For the two events A and B, let `P(A)=0.5` and `P(B)=0.6`. The necessarily false statement(s) is/are

Options

A. `P(A nn B)=0.35`
B. `P(A nn B)=0.45`
C. `P(A nn B)=0.65`
D. `P(A nn B)=0.02`

Answer

To solve the problem, we need to analyze the probabilities of two events A and B, given that \( P(A) = 0.5 \) and \( P(B) = 0.6 \). We will determine which statements about the intersection of these events are necessarily false.

### Step-by-Step Solution:

1. **Understand the Probability of Union**:
The probability of the union of two events A and B is given by the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Here, \( P(A \cup B) \) represents the probability that either event A or event B or both occur.

2. **Substitute Known Values**:
We know:
- \( P(A) = 0.5 \)
- \( P(B) = 0.6 \)
Thus, we can substitute these values into the formula:
\[ P(A \cup B) = 0.5 + 0.6 - P(A \cap B) \]
This simplifies to:
\[ P(A \cup B) = 1.1 - P(A \cap B) \]

3. **Determine the Range of \( P(A \cap B) \)**:
Since the probability of any event must be between 0 and 1, we have:
\[ 0 \leq P(A \cup B) \leq 1 \]
Therefore, we can set up the following inequalities based on our equation:
\[ 0 \leq 1.1 - P(A \cap B) \leq 1 \]

4. **Solving the Inequalities**:
- From \( 1.1 - P(A \cap B) \geq 0 \):
\[ P(A \cap B) \leq 1.1 \]
- From \( 1.1 - P(A \cap B) \leq 1 \):
\[ P(A \cap B) \geq 0.1 \]

Thus, we conclude:
\[ 0.1 \leq P(A \cap B) \leq 1.1 \]

5. **Analyzing the Statements**:
We need to check the given options for \( P(A \cap B) \) to see which one is necessarily false. The valid range for \( P(A \cap B) \) is \( [0.1, 1] \). Any value outside this range would be false.

- **Option 1**: \( P(A \cap B) = 0.35 \) (Valid)
- **Option 2**: \( P(A \cap B) = 0.45 \) (Valid)
- **Option 3**: \( P(A \cap B) = 0.65 \) (Valid)
- **Option 4**: \( P(A \cap B) = 0.02 \) (Invalid, since 0.02 < 0.1)

### Conclusion:
The necessarily false statement is **Option 4**: \( P(A \cap B) = 0.02 \).