If `P(A)=(1)/(4), P(B)=(1)/(13) and P(AcapB)=(1)/(52)`, then the value of `P(overline(A)capoverline(B))`, is

A. `(3)/(13)`
B. `(5)/(13)`
C. `(7)/(13)`
D. `(9)/(13)`

Options

A. `(3)/(13)`
B. `(5)/(13)`
C. `(7)/(13)`
D. `(9)/(13)`

Answer

To find the value of \( P(\overline{A} \cap \overline{B}) \), we can use De Morgan's Law, which states that:

\[ P(\overline{A} \cap \overline{B}) = P(A \cup B)^{\overline{}} \]

This means we need to first calculate \( P(A \cup B) \).

### Step 1: Calculate \( P(A \cup B) \)

Using the formula for the union of two events:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Substituting the given values:

\[ P(A) = \frac{1}{4}, \quad P(B) = \frac{1}{13}, \quad P(A \cap B) = \frac{1}{52} \]

So we have:

\[ P(A \cup B) = \frac{1}{4} + \frac{1}{13} - \frac{1}{52} \]

### Step 2: Find a common denominator

The least common multiple (LCM) of 4, 13, and 52 is 52. We will convert each fraction:

\[ P(A) = \frac{1}{4} = \frac{13}{52}, \quad P(B) = \frac{1}{13} = \frac{4}{52}, \quad P(A \cap B) = \frac{1}{52} \]

### Step 3: Substitute and simplify

Now substituting these values into the equation:

\[ P(A \cup B) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} \]

Combining the fractions:

\[ P(A \cup B) = \frac{13 + 4 - 1}{52} = \frac{16}{52} \]

Simplifying \( \frac{16}{52} \):

\[ P(A \cup B) = \frac{4}{13} \]

### Step 4: Calculate \( P(\overline{A} \cap \overline{B}) \)

Now we can find \( P(\overline{A} \cap \overline{B}) \):

\[ P(\overline{A} \cap \overline{B}) = 1 - P(A \cup B) \]

Substituting the value we found:

\[ P(\overline{A} \cap \overline{B}) = 1 - \frac{4}{13} \]

### Step 5: Simplify the result

Calculating this gives:

\[ P(\overline{A} \cap \overline{B}) = \frac{13}{13} - \frac{4}{13} = \frac{9}{13} \]

### Final Answer

Thus, the value of \( P(\overline{A} \cap \overline{B}) \) is:

\[ \boxed{\frac{9}{13}} \]