Let A and B be two events such that `P(A)=0.3` and `P(AcupB)= 0.8`. Find `P(B), if P(Acap B) = P(A) P(B)`.

Answer

To solve the problem, we need to find \( P(B) \) given the following information:

- \( P(A) = 0.3 \)
- \( P(A \cup B) = 0.8 \)
- \( P(A \cap B) = P(A) \cdot P(B) \)

### Step-by-Step Solution:

1. **Use the formula for the union of two events**:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ 0.8 = 0.3 + P(B) - P(A \cap B) \]

2. **Substitute \( P(A \cap B) \)**:
Since \( P(A \cap B) = P(A) \cdot P(B) \), we can substitute this into the equation:
\[ 0.8 = 0.3 + P(B) - (0.3 \cdot P(B)) \]

3. **Factor out \( P(B) \)**:
Rearranging the equation gives:
\[ 0.8 = 0.3 + P(B)(1 - 0.3) \]
Simplifying further:
\[ 0.8 = 0.3 + 0.7 P(B) \]

4. **Isolate \( P(B) \)**:
Subtract \( 0.3 \) from both sides:
\[ 0.8 - 0.3 = 0.7 P(B) \]
\[ 0.5 = 0.7 P(B) \]

5. **Solve for \( P(B) \)**:
Divide both sides by \( 0.7 \):
\[ P(B) = \frac{0.5}{0.7} = \frac{5}{7} \]

### Final Answer:
Thus, the probability \( P(B) \) is:
\[ P(B) = \frac{5}{7} \]