A and B are two independent events such that P(A) = 0.8 and `P(A nn vec B ) = 0.3 ` Find P(B)

Answer

To solve the problem, we need to find the probability of event B, given the probabilities of event A and the intersection of A and the complement of B.

### Step-by-Step Solution:

1. **Identify Given Values**:
- \( P(A) = 0.8 \)
- \( P(A \cap B') = 0.3 \) (where \( B' \) is the complement of B)

2. **Use the Independence of Events**:
Since A and B are independent, A and \( B' \) are also independent. Therefore, we can use the formula:
\[ P(A \cap B') = P(A) \cdot P(B') \]

3. **Substitute the Known Values**:
Substitute the known values into the equation:
\[ 0.3 = 0.8 \cdot P(B') \]

4. **Solve for \( P(B') \)**:
To find \( P(B') \), divide both sides of the equation by 0.8:
\[ P(B') = \frac{0.3}{0.8} = 0.375 \]

5. **Find \( P(B) \)**:
Since \( P(B) + P(B') = 1 \), we can find \( P(B) \):
\[ P(B) = 1 - P(B') = 1 - 0.375 = 0.625 \]

6. **Final Result**:
Therefore, the probability of event B is:
\[ P(B) = 0.625 \]

### Summary:
- \( P(B) = 0.625 \)