If `P(bar(A))=0.4,P(A uu B)=0.7` and A and B are given to be independent events, then P(B) = ___________.

Answer

To solve the problem step by step, we will use the given probabilities and the properties of independent events.

### Step 1: Understand the Given Information
We are given:
- \( P(\bar{A}) = 0.4 \)
- \( P(A \cup B) = 0.7 \)
- Events A and B are independent.

### Step 2: Find \( P(A) \)
Since \( P(\bar{A}) + P(A) = 1 \), we can find \( P(A) \):
\[ P(A) = 1 - P(\bar{A}) = 1 - 0.4 = 0.6 \]

### Step 3: Use the Formula for Union of Two Events
For two events A and B, the probability of their union is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since A and B are independent, we have:
\[ P(A \cap B) = P(A) \cdot P(B) \]

### Step 4: Substitute Known Values into the Union Formula
We can substitute the known values into the union formula:
\[ 0.7 = P(A) + P(B) - P(A) \cdot P(B) \]
Substituting \( P(A) = 0.6 \):
\[ 0.7 = 0.6 + P(B) - 0.6 \cdot P(B) \]

### Step 5: Simplify the Equation
Rearranging the equation gives:
\[ 0.7 = 0.6 + P(B)(1 - 0.6) \]
\[ 0.7 = 0.6 + 0.4P(B) \]

### Step 6: Isolate \( P(B) \)
Now, isolate \( P(B) \):
\[ 0.7 - 0.6 = 0.4P(B) \]
\[ 0.1 = 0.4P(B) \]
\[ P(B) = \frac{0.1}{0.4} = 0.25 \]

### Final Answer
Thus, the probability \( P(B) \) is:
\[ \boxed{0.25} \]