If A and B are mutually exclusive events, then

Options

A. `P(B) le P(barA) `
B. `P(B) ge P(barA)`
C. `P(B) =P(barA)`
D. none of these

Answer

To solve the problem regarding mutually exclusive events A and B, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding Mutually Exclusive Events**:
- By definition, two events A and B are said to be mutually exclusive if they cannot occur at the same time. This means that the intersection of A and B is empty.
- Mathematically, this is expressed as:
\[ P(A \cap B) = 0 \]

2. **Using the Formula for Union of Two Events**:
- The probability of the union of two events A and B is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Since we know that \( P(A \cap B) = 0 \), we can simplify this to:
\[ P(A \cup B) = P(A) + P(B) \]

3. **Applying the Probability Limit**:
- The probability of any event must be less than or equal to 1:
\[ P(A \cup B) \leq 1 \]
- Therefore, we can write:
\[ P(A) + P(B) \leq 1 \]

4. **Rearranging the Inequality**:
- To find a relationship between P(B) and P(A), we can rearrange the inequality:
\[ P(B) \leq 1 - P(A) \]

5. **Expressing in Terms of Complement**:
- The term \( 1 - P(A) \) can be expressed as the probability of the complement of A:
\[ P(B) \leq P(A^c) \]
- Here, \( A^c \) denotes the complement of event A.

6. **Conclusion**:
- Thus, we conclude that:
\[ P(B) \leq P(A^c) \]
- This means that the probability of event B is less than or equal to the probability of the complement of event A.