Let A and B be two independent events such that `P(A)=(1)/(4)` and `P(B)=(1)/(2)`, find P(not A and not B).

Answer

To find \( P(\text{not } A \text{ and not } B) \), we can use the following steps:

### Step 1: Understand the Problem
We need to find the probability of both events A and B not occurring, denoted as \( P(A' \cap B') \) where \( A' \) is the complement of A and \( B' \) is the complement of B.

### Step 2: Use the Complement Rule
We know that:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
This means we can find the probability of \( A \cup B \) and subtract it from 1.

### Step 3: Calculate \( P(A \cup B) \)
Using the formula for the probability of the union of two independent events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since A and B are independent, we have:
\[ P(A \cap B) = P(A) \times P(B) \]
Given \( P(A) = \frac{1}{4} \) and \( P(B) = \frac{1}{2} \), we can calculate:
\[ P(A \cap B) = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \]

### Step 4: Substitute Values into the Union Formula
Now we substitute the values into the union formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
\[ P(A \cup B) = \frac{1}{4} + \frac{1}{2} - \frac{1}{8} \]
To perform the addition, we convert all fractions to have a common denominator (which is 8):
\[ P(A) = \frac{1}{4} = \frac{2}{8}, \quad P(B) = \frac{1}{2} = \frac{4}{8} \]
Thus,
\[ P(A \cup B) = \frac{2}{8} + \frac{4}{8} - \frac{1}{8} = \frac{5}{8} \]

### Step 5: Calculate \( P(A' \cap B') \)
Now we can find \( P(A' \cap B') \):
\[ P(A' \cap B') = 1 - P(A \cup B) = 1 - \frac{5}{8} = \frac{3}{8} \]

### Final Answer
Thus, the probability that neither A nor B occurs is:
\[ P(\text{not } A \text{ and not } B) = \frac{3}{8} \]

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