If A and B are two independent events such that `P(A) = 7/10`, `P(B')=alpha`, `P(AcupB) = 8/10`, then `alpha` =

Options

A. `2/7`
B. `5/7`
C. `2/3`
D. `3/7`

Answer

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

- \( P(A) = \frac{7}{10} \)
- \( P(B') = \alpha \) (where \( B' \) is the complement of \( B \))
- \( P(A \cup B) = \frac{8}{10} \)

### Step-by-Step Solution:

1. **Determine \( P(B) \)**:
Since \( P(B') = \alpha \), we can find \( P(B) \) using the complement rule:
\[ P(B) = 1 - P(B') = 1 - \alpha \]

**Hint**: Remember that the probability of an event and its complement always add up to 1.

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

3. **Substituting the known values**:
We can substitute the known values into the union formula:
\[ \frac{8}{10} = P(A) + P(B) - P(A \cap B) \]
Substituting \( P(A) = \frac{7}{10} \) and \( P(B) = 1 - \alpha \):
\[ \frac{8}{10} = \frac{7}{10} + (1 - \alpha) - \left(\frac{7}{10} \cdot (1 - \alpha)\right) \]

4. **Simplifying the equation**:
Let's simplify the equation step by step:
\[ \frac{8}{10} = \frac{7}{10} + 1 - \alpha - \left(\frac{7}{10} - \frac{7\alpha}{10}\right) \]
Combining the terms:
\[ \frac{8}{10} = \frac{7}{10} + 1 - \alpha - \frac{7}{10} + \frac{7\alpha}{10} \]
This simplifies to:
\[ \frac{8}{10} = 1 - \alpha + \frac{7\alpha}{10} \]

5. **Rearranging the equation**:
Multiply through by 10 to eliminate the fraction:
\[ 8 = 10 - 10\alpha + 7\alpha \]
Rearranging gives:
\[ 8 = 10 - 3\alpha \]
Thus,
\[ 3\alpha = 10 - 8 \]
\[ 3\alpha = 2 \]

6. **Solving for \( \alpha \)**:
Dividing both sides by 3:
\[ \alpha = \frac{2}{3} \]

### Final Answer:
Thus, the value of \( \alpha \) is \( \frac{2}{3} \).