If A and B are two events such that $P(A)=\frac{1}{2}, P(B)=\frac{1}{3}, P(\frac{A}{B})=\frac{1}{4}$, then $P(\overline{A}\cap\overline{B})$ is equal to:

Question

Accepted Answer

To solve the problem and determine why Option 3 is the correct answer, we need to find $ P(\overline{A} \cap \overline{B}) $, which represents the probability that neither event $ A $ nor event $ B $ occurs.

Given:
- $ P(A) = \frac{1}{2} $
- $ P(B) = \frac{1}{3} $
- $ P\left(\frac{A}{B}\right) = \frac{1}{4} $

First, let's clarify the notation $ P\left(\frac{A}{B}\right) $. This is typically interpreted as the conditional probability $ P(A \mid B) $, which is the probability of $ A $ occurring given that $ B $ has occurred. Therefore, we have:

$$ P(A \mid B) = \frac{1}{4} $$

The formula for conditional probability is:

$$ P(A \mid B) = \frac{P(A \cap B)}{P(B)} $$

Substituting the given values:

$$ \frac{1}{4} = \frac{P(A \cap B)}{\frac{1}{3}} $$

Solving for $ P(A \cap B) $:

$$ P(A \cap B) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12} $$

Now, we use the formula for the probability of the union of two events:

$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$

Substitute the known values:

$$ P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{12} $$

To add these fractions, find a common denominator, which is 12:

$$ P(A \cup B) = \frac{6}{12} + \frac{4}{12} - \frac{1}{12} = \frac{9}{12} = \frac{3}{4} $$

Finally, the probability that neither $ A $ nor $ B $ occurs is the complement of $ P(A \cup B) $:

$$ P(\overline{A} \cap \overline{B}) = 1 - P(A \cup B) = 1 - \frac{3}{4} = \frac{1}{4} $$

Thus, the correct answer is Option 3: $\frac{1}{4}$.

**Explanation for Incorrect Options:**

1. **Option 1: $\frac{1}{12}$** - This value represents $ P(A \cap B) $, not $ P(\overline{A} \cap \overline{B}) $.

2. **Option 2: $\frac{3}{4}$** - This value is $ P(A \cup B) $, the probability that at least one of the events $ A $ or $ B $ occurs, not the probability that neither occurs.

3. **Option 4: $\frac{3}{16}$** - This value does not correspond to any relevant probability in this context and is not derived from the given information.

Therefore, Option 3 is the correct answer, as it accurately represents the probability that neither event $ A $ nor event $ B $ occurs.