If $P(A) = \frac{3}{5}$, $P(B) = \frac{1}{2}$ and $P(A \cap B) = \frac{1}{4}$, then $P(\overline{A} | \overline{B})$ is

To find the conditional probability $P(\overline{A} | \overline{B})$, we use the definition of conditional probability and De Morgan's Laws.

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

Find the denominator $P(\overline{B})$:

The probability of the complement of $B$ is:

$P(\overline{B}) = 1 - P(B)$

$P(\overline{B}) = 1 - \frac{1}{2} = \frac{1}{2}$

Find the numerator $P(\overline{A} \cap \overline{B})$ using De Morgan's Law:

According to De Morgan's Law, $\overline{A} \cap \overline{B} = \overline{A \cup B}$.

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

First, calculate $P(A \cup B)$ using the Addition Rule:

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

$P(A \cup B) = \frac{3}{5} + \frac{1}{2} - \frac{1}{4}$

To solve, use a common denominator of 20:

$P(A \cup B) = \frac{12}{20} + \frac{10}{20} - \frac{5}{20} = \frac{17}{20}$

Now find the complement:

$P(\overline{A} \cap \overline{B}) = 1 - \frac{17}{20} = \frac{3}{20}$

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Substitute the values back into the conditional probability formula:

$P(\overline{A} | \overline{B}) = \frac{3/20}{1/2}$

$P(\overline{A} | \overline{B}) = \frac{3}{20} \cdot \frac{2}{1} = \frac{6}{20}$

$P(\overline{A} | \overline{B}) = \frac{3}{10}$