If `A and B` are two events, the probability that at most one of these events occurs is

Options

A. `1-P(A cap B)`
B. `P(overline(A))+P(overline(B))-P(overline(A) cap overline(B))`
C. `P(overline(A))+P(overline(B))+P(A cup B)-1`
D. all the above

Explanation

To find the probability that at most one of the events A and B occurs, we can break down the problem into steps.

### Step-by-Step Solution:

1. **Understanding "At Most One"**:
- The phrase "at most one of these events occurs" means either none of the events occur or only one of them occurs. This can be mathematically expressed as:
\[ P(\text{at most one of A or B}) = P(A^c \cap B^c) + P(A \cap B^c) + P(A^c \cap B) \]

2. **Using Complementary Events**:
- We can use the property of complementary events to express the probabilities:
- \( P(A^c) \) is the probability that event A does not occur.
- \( P(B^c) \) is the probability that event B does not occur.
- Therefore, we can rewrite the expression:
\[ P(A^c \cap B^c) = 1 - P(A \cup B) \]

3. **Applying the Addition Rule**:
- The probability of the union of two events can be expressed as:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Thus, we can substitute this into our earlier expression:
\[ P(A^c \cap B^c) = 1 - (P(A) + P(B) - P(A \cap B)) \]

4. **Combining the Probabilities**:
- Now, substituting back into our original equation:
\[ P(\text{at most one of A or B}) = (1 - (P(A) + P(B) - P(A \cap B))) + P(A \cap B^c) + P(A^c \cap B) \]
- We know that:
- \( P(A \cap B^c) = P(A) - P(A \cap B) \)
- \( P(A^c \cap B) = P(B) - P(A \cap B) \)
- Therefore, we can rewrite:
\[ P(\text{at most one of A or B}) = 1 - (P(A) + P(B) - P(A \cap B)) + (P(A) - P(A \cap B)) + (P(B) - P(A \cap B)) \]

5. **Simplifying the Expression**:
- Simplifying this gives:
\[ P(\text{at most one of A or B}) = 1 - P(A \cap B) + P(A) + P(B) - 2P(A \cap B) \]
- Thus:
\[ P(\text{at most one of A or B}) = 1 - P(A \cap B) + P(A) + P(B) - 2P(A \cap B) \]

6. **Final Expression**:
- The final expression for the probability that at most one of the events A or B occurs is:
\[ P(A^c \cap B^c) + P(A \cap B^c) + P(A^c \cap B) = 1 - P(A \cap B) \]