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

Answer

To find \( P(\text{not } A \text{ and not } B) \), we can use the relationship between the complements of events and the union of events. Here’s a step-by-step solution:

### Step 1: Understand the relationship
We know that:
\[ P(\text{not } A \text{ and not } B) = P(A' \cap B') = P((A \cup B)') = 1 - P(A \cup B) \]
This means that to find \( P(\text{not } A \text{ and not } B) \), we first need to calculate \( P(A \cup B) \).

### Step 2: Use the formula for the union of two events
The probability of the union of two events can be calculated using the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Given:
- \( P(A) = \frac{1}{4} \)
- \( P(B) = \frac{1}{2} \)
- \( P(A \cap B) = \frac{1}{8} \)

### Step 3: Substitute the values into the formula
Now, substitute the values into the formula:
\[ P(A \cup B) = \frac{1}{4} + \frac{1}{2} - \frac{1}{8} \]

### 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. Rewrite the fractions:
- \( \frac{1}{4} = \frac{2}{8} \)
- \( \frac{1}{2} = \frac{4}{8} \)
- \( \frac{1}{8} = \frac{1}{8} \)

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

### Step 6: Calculate \( P(\text{not } A \text{ and not } B) \)
Now that we have \( P(A \cup B) \), we can find \( P(\text{not } A \text{ and not } B) \):
\[ P(\text{not } A \text{ and not } B) = 1 - P(A \cup B) = 1 - \frac{5}{8} = \frac{3}{8} \]

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