A and B are two independent events such that `P(A uu B')=0.8` and `P(A)=0.3`. Then `P(B)` is

Options

A. `2//7`
B. `2//3`
C. `3//8`
D. `1//8`

Answer

To solve the problem, we need to find \( P(B) \) given that \( P(A \cup B') = 0.8 \) and \( P(A) = 0.3 \).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We know that \( P(A \cup B') = 0.8 \) (where \( B' \) is the complement of event B).
- We know that \( P(A) = 0.3 \).

2. **Using the Formula for 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 can express \( P(A \cap B') \) as:
\[ P(A \cap B') = P(A) \cdot P(B') \]

3. **Substituting the Known Values**:
- Substitute \( P(A) = 0.3 \) into the union formula:
\[ 0.8 = 0.3 + P(B') - P(A) \cdot P(B') \]
- This simplifies to:
\[ 0.8 = 0.3 + P(B') - 0.3 \cdot P(B') \]

4. **Factoring Out \( P(B') \)**:
- Rearranging gives:
\[ 0.8 - 0.3 = P(B') - 0.3 \cdot P(B') \]
- Thus:
\[ 0.5 = P(B') (1 - 0.3) = P(B') \cdot 0.7 \]

5. **Solving for \( P(B') \)**:
- Now, divide both sides by 0.7:
\[ P(B') = \frac{0.5}{0.7} = \frac{5}{7} \]

6. **Finding \( P(B) \)**:
- Since \( P(B) + P(B') = 1 \), we can find \( P(B) \):
\[ P(B) = 1 - P(B') = 1 - \frac{5}{7} = \frac{2}{7} \]

### Final Answer:
Thus, the probability of event B is:
\[ P(B) = \frac{2}{7} \]