A and B are two independent events such that `P(A)=(1)/(5), P(A cup B)=(7)/(10). " Then ", P(overline(B))`=

Options

A. `3//8`
B. `2//7`
C. `7//9`
D. none of these

Answer

To solve the problem step by step, we need to find \( P(\overline{B}) \) given that \( P(A) = \frac{1}{5} \) and \( P(A \cup B) = \frac{7}{10} \), and that A and B are independent events.

### Step 1: Write down the known probabilities
We have:
- \( P(A) = \frac{1}{5} \)
- \( P(A \cup B) = \frac{7}{10} \)

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

### Step 3: Substitute the known values into the formula
Substituting the known values into the union formula:
\[ \frac{7}{10} = \frac{1}{5} + P(B) - P(A) \cdot P(B) \]
This can be rewritten as:
\[ \frac{7}{10} = \frac{1}{5} + P(B) - \frac{1}{5} \cdot P(B) \]

### Step 4: Simplify the equation
First, convert \( \frac{1}{5} \) to have a common denominator with \( \frac{7}{10} \):
\[ \frac{1}{5} = \frac{2}{10} \]
Now, substitute this back into the equation:
\[ \frac{7}{10} = \frac{2}{10} + P(B) - \frac{1}{5} \cdot P(B) \]
This simplifies to:
\[ \frac{7}{10} = \frac{2}{10} + P(B) - \frac{1}{5} P(B) \]

### Step 5: Combine like terms
Let \( P(B) = x \):
\[ \frac{7}{10} = \frac{2}{10} + x - \frac{1}{5}x \]
Convert \( \frac{1}{5}x \) to have a common denominator:
\[ \frac{1}{5}x = \frac{2}{10}x \]
Now, substitute:
\[ \frac{7}{10} = \frac{2}{10} + x - \frac{2}{10}x \]
This simplifies to:
\[ \frac{7}{10} = \frac{2}{10} + \frac{10}{10}x - \frac{2}{10}x \]
\[ \frac{7}{10} = \frac{2}{10} + \frac{8}{10}x \]

### Step 6: Isolate \( x \)
Subtract \( \frac{2}{10} \) from both sides:
\[ \frac{7}{10} - \frac{2}{10} = \frac{8}{10}x \]
\[ \frac{5}{10} = \frac{8}{10}x \]
Multiply both sides by \( \frac{10}{8} \):
\[ x = \frac{5}{8} \]
Thus, \( P(B) = \frac{5}{8} \).

### Step 7: Find \( P(\overline{B}) \)
Now, we need to find \( P(\overline{B}) \):
\[ P(\overline{B}) = 1 - P(B) = 1 - \frac{5}{8} = \frac{3}{8} \]

### Final Answer
Thus, the probability \( P(\overline{B}) \) is:
\[ \boxed{\frac{3}{8}} \]