If A and B are two independent events such that P(B) = `(2)/(7) , P(A cup B^(c)) = 0.8`, then find P(A).

Answer

To solve the problem, we need to find the probability of event A given that events A and B are independent, P(B) = \( \frac{2}{7} \), and P(A ∪ B^c) = 0.8.

### Step-by-Step Solution:

1. **Understand the given probabilities**:
- P(B) = \( \frac{2}{7} \)
- P(A ∪ B^c) = 0.8

2. **Use the formula for the union of two events**:
The formula for the probability of the union of two events A and B^c is:
\[ P(A ∪ B^c) = P(A) + P(B^c) - P(A ∩ B^c) \]

3. **Calculate P(B^c)**:
Since P(B) = \( \frac{2}{7} \), we can find P(B^c):
\[ P(B^c) = 1 - P(B) = 1 - \frac{2}{7} = \frac{5}{7} \]

4. **Substitute P(B^c) into the union formula**:
Now we substitute P(B^c) into the union formula:
\[ 0.8 = P(A) + \frac{5}{7} - P(A ∩ B^c) \]

5. **Find P(A ∩ B^c)**:
Since A and B are independent, we can express P(A ∩ B) as:
\[ P(A ∩ B) = P(A) \cdot P(B) \]
Therefore, P(A ∩ B^c) can be calculated as:
\[ P(A ∩ B^c) = P(A) - P(A ∩ B) = P(A) - P(A) \cdot P(B) = P(A)(1 - P(B)) = P(A) \cdot \frac{5}{7} \]

6. **Substitute P(A ∩ B^c) back into the equation**:
Now we can substitute this back into our equation:
\[ 0.8 = P(A) + \frac{5}{7} - P(A) \cdot \frac{5}{7} \]

7. **Rearranging the equation**:
Rearranging gives:
\[ 0.8 = P(A) \left(1 - \frac{5}{7}\right) + \frac{5}{7} \]
Simplifying \(1 - \frac{5}{7}\) gives \(\frac{2}{7}\):
\[ 0.8 = P(A) \cdot \frac{2}{7} + \frac{5}{7} \]

8. **Isolate P(A)**:
To isolate P(A), we multiply both sides by 7:
\[ 5.6 = 2P(A) + 5 \]
Subtracting 5 from both sides:
\[ 0.6 = 2P(A) \]
Dividing by 2:
\[ P(A) = 0.3 \]

### Final Answer:
Thus, the probability of event A is:
\[ P(A) = 0.3 \]