If A and B are two events such that ` P(A) = (1)/(4) ,P(B) = (1)/(2) and P(A nnB) = (1)/(8) , ` then P ( not A and not B) =

Options

A. ` (3)/(8)`
B. ` ( 5)/(8)`
C. ` ( 7)/(8)`
D. None of these

Answer

To solve the problem step by step, we will use the given probabilities and apply the formula for the probability of the union of two events.

### Step 1: Identify the given probabilities
We are given:
- \( P(A) = \frac{1}{4} \)
- \( P(B) = \frac{1}{2} \)
- \( P(A \cap B) = \frac{1}{8} \)

### Step 2: Use the formula for the probability of the union of two events
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: Substitute the values into the formula
Now, we substitute the given values into the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
\[ P(A \cup B) = \frac{1}{4} + \frac{1}{2} - \frac{1}{8} \]

### Step 4: Find a common denominator
To perform the addition and subtraction, we need a common denominator. The least common multiple of 4, 2, and 8 is 8. We convert each fraction:
- \( P(A) = \frac{1}{4} = \frac{2}{8} \)
- \( P(B) = \frac{1}{2} = \frac{4}{8} \)
- \( P(A \cap B) = \frac{1}{8} \)

### Step 5: Calculate \( P(A \cup B) \)
Now substituting these values:
\[ P(A \cup B) = \frac{2}{8} + \frac{4}{8} - \frac{1}{8} \]
\[ P(A \cup B) = \frac{2 + 4 - 1}{8} = \frac{5}{8} \]

### Step 6: Calculate \( P(\text{not } A \text{ and not } B) \)
The probability of neither \( A \) nor \( B \) occurring is given by:
\[ P(\text{not } A \text{ and not } B) = P(A' \cap B') = 1 - P(A \cup B) \]
Substituting the value we found for \( P(A \cup B) \):
\[ P(\text{not } A \text{ and not } B) = 1 - \frac{5}{8} = \frac{3}{8} \]

### Final Answer
Thus, the probability \( P(\text{not } A \text{ and not } B) \) is:
\[ \boxed{\frac{3}{8}} \]