A and B are said to be independent if `P(AcapB)=P(A).P(B).` If B is independent with every subset of sample space then the value of 6 P(B) is ________

Answer

To solve the problem, we need to determine the value of \(6P(B)\) given that event B is independent with every subset of the sample space.

### Step-by-Step Solution:

1. **Understanding Independence**:
- By definition, two events A and B are independent if \(P(A \cap B) = P(A) \cdot P(B)\).
- The problem states that B is independent with every subset of the sample space.

2. **Analyzing the Implications**:
- If B is independent with every subset of the sample space, it implies that the occurrence of B does not affect the occurrence of any other event in the sample space.
- This can only be true if \(P(B) = 1\) or \(P(B) = 0\). However, since B is independent with every subset, we conclude that \(P(B) = 1\).

3. **Calculating \(6P(B)\)**:
- Since we have established that \(P(B) = 1\), we can substitute this value into the expression \(6P(B)\).
- Therefore, \(6P(B) = 6 \cdot 1 = 6\).

4. **Final Answer**:
- The value of \(6P(B)\) is \(6\).

### Conclusion:
The final answer is \(6\).

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