Let A and B be two events.if `P(A)=0.3, P(B)=0.4, P(A' nn B')=0.4` then A and B are independent or not ?

Explanation

To determine whether the events A and B are independent, we can follow these steps:

### Step 1: Understand the given probabilities
We are given:
- \( P(A) = 0.3 \)
- \( P(B) = 0.4 \)
- \( P(A' \cap B') = 0.4 \)

### Step 2: Use the complement rule
We know that:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
Thus, we can find \( P(A \cup B) \):
\[ P(A \cup B) = 1 - P(A' \cap B') = 1 - 0.4 = 0.6 \]

### Step 3: Use the formula for the union of two events
The probability of the union of two events can also be expressed as:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ 0.6 = P(A) + P(B) - P(A \cap B) \]
\[ 0.6 = 0.3 + 0.4 - P(A \cap B) \]

### Step 4: Solve for \( P(A \cap B) \)
Rearranging the equation gives:
\[ P(A \cap B) = 0.3 + 0.4 - 0.6 \]
\[ P(A \cap B) = 0.7 - 0.6 = 0.1 \]

### Step 5: Check for independence
Two events A and B are independent if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Calculating \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = 0.3 \cdot 0.4 = 0.12 \]

### Step 6: Compare the results
We found:
- \( P(A \cap B) = 0.1 \)
- \( P(A) \cdot P(B) = 0.12 \)

Since \( P(A \cap B) \neq P(A) \cdot P(B) \), we conclude that events A and B are not independent.

### Final Conclusion
A and B are not independent events.

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