If A and B are two events, then P (neither A nor B) equals

Options

A. `1-P(A cup B)`
B. `P(overline(A))+P(overline(B))`
C. `1-P(A)-P(B)`
D. none of these

Answer

To find the probability of neither event A nor event B occurring, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Events**: We have two events, A and B. We need to find the probability that neither A nor B occurs.

2. **Define the Complement Events**: The complement of event A is denoted as A', which represents the event that A does not occur. Similarly, the complement of event B is denoted as B', which represents the event that B does not occur.

3. **Identify the Required Probability**: The probability of neither A nor B occurring can be expressed as the intersection of their complements: P(A' ∩ B'). This means we are looking for the probability that both A and B do not happen.

4. **Use the Complement Rule**: The probability of neither A nor B can also be calculated using the complement of the union of A and B. According to the complement rule:
\[ P(A' ∩ B') = 1 - P(A ∪ B) \]
This means that to find the probability of neither A nor B, we subtract the probability of either A or B occurring from 1.

5. **Final Expression**: Thus, we can conclude that:
\[ P(\text{neither A nor B}) = 1 - P(A ∪ B) \]

### Conclusion:
The probability of neither A nor B occurring is given by the formula:
\[ P(\text{neither A nor B}) = 1 - P(A ∪ B) \]