If A and B be two events such that `P(A)=0*3 and P(A cupoverline(B))=0*8`. If A and B are independent events, then P(B)=_____

Answer

To solve the problem, we need to find \( P(B) \) given that \( P(A) = 0.3 \) and \( P(A \cup \overline{B}) = 0.8 \), with the condition that events A and B are independent.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- We know \( P(A) = 0.3 \).
- We also know \( P(A \cup \overline{B}) = 0.8 \).

2. **Use the Formula for the Union of Two Events:**
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since A and B are independent, we have:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Thus, we can rewrite the equation as:
\[ P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B) \]

3. **Express \( P(A \cup \overline{B}) \):**
We can express \( P(A \cup \overline{B}) \) using the complement:
\[ P(A \cup \overline{B}) = 1 - P(A^c \cap B) \]
where \( A^c \) is the complement of A.

4. **Relate \( P(A \cup \overline{B}) \) to \( P(A) \) and \( P(B) \):**
We can also express \( P(A \cup \overline{B}) \) as:
\[ P(A \cup \overline{B}) = P(A) + P(\overline{B}) - P(A \cap \overline{B}) \]
Since \( P(\overline{B}) = 1 - P(B) \) and \( P(A \cap \overline{B}) = P(A) \cdot P(\overline{B}) \) (because A and B are independent), we have:
\[ P(A \cup \overline{B}) = P(A) + (1 - P(B)) - P(A) \cdot (1 - P(B)) \]

5. **Substituting the Values:**
Substitute \( P(A) = 0.3 \) and \( P(A \cup \overline{B}) = 0.8 \):
\[ 0.8 = 0.3 + (1 - P(B)) - 0.3 \cdot (1 - P(B)) \]
Simplifying this gives:
\[ 0.8 = 0.3 + 1 - P(B) - 0.3 + 0.3P(B) \]
\[ 0.8 = 1 - P(B) + 0.3P(B) \]
\[ 0.8 = 1 - 0.7P(B) \]

6. **Solving for \( P(B) \):**
Rearranging the equation gives:
\[ 0.7P(B) = 1 - 0.8 \]
\[ 0.7P(B) = 0.2 \]
\[ P(B) = \frac{0.2}{0.7} = \frac{2}{7} \]

### Final Answer:
Thus, \( P(B) = \frac{2}{7} \).