If A and B are two independent events such that `P(overline(A))=(7)/(10), P(overline(B))=alpha " and" P(A cup B)=(8)/(10)`, then `alpha`, is

Options

A. `2//7`
B. `5//7`
C. `1`
D. none of these

Explanation

To solve the problem, we need to find the value of \( \alpha \) given the probabilities of independent events \( A \) and \( B \).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We have \( P(\overline{A}) = \frac{7}{10} \)
- We have \( P(\overline{B}) = \alpha \)
- We have \( P(A \cup B) = \frac{8}{10} \)

2. **Finding \( P(A) \) and \( P(B) \)**:
- Since \( P(\overline{A}) = \frac{7}{10} \), we can find \( P(A) \):
\[ P(A) = 1 - P(\overline{A}) = 1 - \frac{7}{10} = \frac{3}{10} \]
- For \( P(B) \), since \( P(\overline{B}) = \alpha \):
\[ P(B) = 1 - P(\overline{B}) = 1 - \alpha \]

3. **Using the Formula for the Union of Two Independent Events**:
- For independent events, the probability of the union is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A)P(B) \]
- Substituting the known values:
\[ \frac{8}{10} = P(A) + P(B) - P(A)P(B) \]
- Substitute \( P(A) = \frac{3}{10} \) and \( P(B) = 1 - \alpha \):
\[ \frac{8}{10} = \frac{3}{10} + (1 - \alpha) - \left(\frac{3}{10}(1 - \alpha)\right) \]

4. **Simplifying the Equation**:
- Rewrite the equation:
\[ \frac{8}{10} = \frac{3}{10} + 1 - \alpha - \frac{3}{10} + \frac{3\alpha}{10} \]
- Combine like terms:
\[ \frac{8}{10} = 1 - \alpha + \frac{3\alpha}{10} \]
- Rearranging gives:
\[ \frac{8}{10} = 1 - \alpha + \frac{3\alpha}{10} \]
- Multiply everything by 10 to eliminate the fraction:
\[ 8 = 10 - 10\alpha + 3\alpha \]
- This simplifies to:
\[ 8 = 10 - 7\alpha \]

5. **Solving for \( \alpha \)**:
- Rearranging gives:
\[ 7\alpha = 10 - 8 \]
\[ 7\alpha = 2 \]
\[ \alpha = \frac{2}{7} \]

### Final Answer:
Thus, the value of \( \alpha \) is \( \frac{2}{7} \).