Two events A and B will be independent if

Question

Accepted Answer

The correct answer is Option 1: $P(A'B')=[1-P(A)][1-P(B)]$.

### Explanation:

1. **Why Option 1 is Correct:**
   - Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. Mathematically, this is expressed as $P(AB) = P(A)P(B)$.
   - For the complements of A and B, denoted as A' and B', the events are also independent if $P(A'B') = P(A')P(B')$.
   - Since $P(A') = 1 - P(A)$ and $P(B') = 1 - P(B)$, substituting these into the independence condition gives $P(A'B') = [1-P(A)][1-P(B)]$.
   - Therefore, Option 1 correctly represents the condition for the independence of the complements of A and B.

2. **Why the Other Options are Incorrect:**

- **Option 2: $P(A)=P(B)$**
     - This condition implies that the probabilities of events A and B are equal, but it does not imply independence. Two events can have the same probability and still be dependent. Independence is specifically about the relationship between the probabilities of the events occurring together and separately, not about their individual probabilities being equal.

- **Option 3: $P(A)+P(B)=1$**
     - This condition suggests that the sum of the probabilities of A and B is 1, which implies that A and B are complementary events (i.e., one occurs if and only if the other does not). This is a condition for mutual exclusivity, not independence. Independent events can both occur simultaneously, so their probabilities do not need to sum to 1.

- **Option 4: $P(AB)
e P(A)P(B)$**
     - This condition directly contradicts the definition of independence. For two events to be independent, $P(AB)$ must equal $P(A)P(B)$. Therefore, stating that $P(AB) 
e P(A)P(B)$ indicates dependence, not independence.

In conclusion, Option 1 correctly describes the condition for the independence of the complements of two events, aligning with the fundamental principles of probability theory.