If A and B are independent events such that `P(A)= (1)/(5)P(A uu B)= ( 7)/(10)` then what is `P(barB)` equal to?

Options

A. `2//7`
B. `3//7`
C. `3//8`
D. `7//9`

Answer

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

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

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

### Step 3: Substitute the values into the formula
Let \( P(B) = x \). Then, we can express \( P(A \cap B) \) as:
\[ P(A \cap B) = P(A) \cdot P(B) = \frac{1}{5} \cdot x \]
Now substituting into the union formula:
\[ \frac{7}{10} = \frac{1}{5} + x - \frac{1}{5} \cdot x \]

### Step 4: Simplify the equation
First, we can rewrite \( \frac{1}{5} \) as \( \frac{2}{10} \) to have a common denominator:
\[ \frac{7}{10} = \frac{2}{10} + x - \frac{x}{5} \]
Now, convert \( \frac{x}{5} \) to have a denominator of 10:
\[ \frac{x}{5} = \frac{2x}{10} \]
So the equation becomes:
\[ \frac{7}{10} = \frac{2}{10} + x - \frac{2x}{10} \]
Combining the terms gives:
\[ \frac{7}{10} = \frac{2}{10} + \frac{10x - 2x}{10} \]
\[ \frac{7}{10} = \frac{2}{10} + \frac{8x}{10} \]

### Step 5: Clear the denominators
Multiply through by 10 to eliminate the denominators:
\[ 7 = 2 + 8x \]

### Step 6: Solve for \( x \)
Subtract 2 from both sides:
\[ 5 = 8x \]
Now divide by 8:
\[ x = \frac{5}{8} \]

### Step 7: Find \( P(\bar{B}) \)
The probability of the complement of B is given by:
\[ P(\bar{B}) = 1 - P(B) = 1 - \frac{5}{8} = \frac{3}{8} \]

### Final Answer
Thus, \( P(\bar{B}) = \frac{3}{8} \).

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