If `A and B` are two independent events such that `P(B) = 2/7, , P(A uu B) = 0.8` then `P(A)=`

Options

A. 0.1
B. 0.2
C. 0.3
D. 0.4

Explanation

To solve the problem step by step, we will use the given information about the independent events A and B.

### Step 1: Understand the given information
We know:
- \( P(B) = \frac{2}{7} \)
- \( P(A \cup B') = 0.8 \) (where \( B' \) is the complement of event B)

### Step 2: Use the formula for the union of events
The formula for the probability of the union of two events A and \( B' \) is given by:
\[ P(A \cup B') = P(A) + P(B') - P(A \cap B') \]

### Step 3: Calculate \( P(B') \)
Since \( P(B) = \frac{2}{7} \), we can find \( P(B') \):
\[ P(B') = 1 - P(B) = 1 - \frac{2}{7} = \frac{5}{7} \]

### Step 4: Substitute \( P(B') \) into the union formula
Now we can substitute \( P(B') \) into the union formula:
\[ 0.8 = P(A) + \frac{5}{7} - P(A \cap B') \]

### Step 5: Express \( P(A \cap B') \)
Since A and B are independent, we can express \( P(A \cap B') \) as:
\[ P(A \cap B') = P(A) \cdot P(B') \]
Substituting \( P(B') \):
\[ P(A \cap B') = P(A) \cdot \frac{5}{7} \]

### Step 6: Substitute \( P(A \cap B') \) back into the equation
Now we substitute this back into our equation:
\[ 0.8 = P(A) + \frac{5}{7} - P(A) \cdot \frac{5}{7} \]

### Step 7: Simplify the equation
Rearranging gives:
\[ 0.8 = P(A) \left(1 - \frac{5}{7}\right) + \frac{5}{7} \]
Calculating \( 1 - \frac{5}{7} \):
\[ 1 - \frac{5}{7} = \frac{2}{7} \]
So we have:
\[ 0.8 = P(A) \cdot \frac{2}{7} + \frac{5}{7} \]

### Step 8: Isolate \( P(A) \)
Now, we can isolate \( P(A) \):
\[ 0.8 - \frac{5}{7} = P(A) \cdot \frac{2}{7} \]
Converting \( 0.8 \) to a fraction:
\[ 0.8 = \frac{8}{10} = \frac{4}{5} \]
Finding a common denominator for \( \frac{4}{5} \) and \( \frac{5}{7} \):
\[ \frac{4}{5} = \frac{28}{35}, \quad \frac{5}{7} = \frac{25}{35} \]
So:
\[ \frac{28}{35} - \frac{25}{35} = P(A) \cdot \frac{2}{7} \]
This simplifies to:
\[ \frac{3}{35} = P(A) \cdot \frac{2}{7} \]

### Step 9: Solve for \( P(A) \)
Now, multiply both sides by \( \frac{7}{2} \):
\[ P(A) = \frac{3}{35} \cdot \frac{7}{2} = \frac{21}{70} = \frac{3}{10} = 0.3 \]

### Final Answer
Thus, the probability \( P(A) \) is:
\[ \boxed{0.3} \]