Let A be the event that P applies for the job.

Let B be the event that Q applies for the job.

We are given the following probabilities:

• The probability that P applies for the job: \( P(A) = \frac{1}{4} \)
• The probability that P applies for the job given that Q applies: \( P(A|B) = \frac{1}{2} \)
• The probability that Q applies for the job given that P applies: \( P(B|A) = \frac{1}{3} \)

We need to find the probability that P does not apply for the job given that Q does not apply. This can be written as \( P(A'|B') \), where A' is the complement of A (P does not apply) and B' is the complement of B (Q does not apply).

Calculating Probability Values

To find \( P(A'|B') \), we first need to determine the values of \( P(B) \), \( P(A \cap B) \), \( P(A') \), \( P(B') \), and \( P(A' \cap B') \).

Step 1: Find the probability that Q applies, \( P(B) \)

We can use the formula for conditional probability, \( P(B|A) = \frac{P(A \cap B)}{P(A)} \), and Bayes' Theorem \( P(A|B) = \frac{P(B|A)P(A)}{P(B)} \). Rearranging Bayes' Theorem to solve for \( P(B) \):

\( P(B) = \frac{P(B|A) P(A)}{P(A|B)} \)

Substituting the given values:

\( P(B) = \frac{(\frac{1}{3}) \times (\frac{1}{4})}{(\frac{1}{2})} \)

\( P(B) = \frac{\frac{1}{12}}{\frac{1}{2}} \)

\( P(B) = \frac{1}{12} \times 2 = \frac{2}{12} = \frac{1}{6} \)

Step 2: Find the probability that P does not apply, \( P(A') \)

Using the complement rule:

\( P(A') = 1 - P(A) \)

\( P(A') = 1 - \frac{1}{4} = \frac{3}{4} \)

Step 3: Find the probability that Q does not apply, \( P(B') \)

Using the complement rule:

\( P(B') = 1 - P(B) \)

\( P(B') = 1 - \frac{1}{6} = \frac{5}{6} \)

Step 4: Find the probability that both P and Q apply, \( P(A \cap B) \)

From the conditional probability formula \( P(B|A) = \frac{P(A \cap B)}{P(A)} \):

\( P(A \cap B) = P(B|A) \times P(A) \)

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

Step 5: Find the probability that neither P nor Q applies, \( P(A' \cap B') \)

We know that \( P(A' \cap B') = P((A \cup B)') \). First, we find \( P(A \cup B) \).

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

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

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

\( P(A \cup B) = \frac{3}{12} + \frac{2}{12} - \frac{1}{12} = \frac{3+2-1}{12} = \frac{4}{12} = \frac{1}{3} \)

Now, we can find \( P(A' \cap B') \):

\( P(A' \cap B') = 1 - P(A \cup B) \)

\( P(A' \cap B') = 1 - \frac{1}{3} = \frac{2}{3} \)

Calculating the Final Conditional Probability

Finally, we calculate the probability that P does not apply given that Q does not apply, \( P(A'|B') \).

Using the formula for conditional probability:

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

Substitute the values we found:

\( P(A'|B') = \frac{\frac{2}{3}}{\frac{5}{6}} \)

\( P(A'|B') = \frac{2}{3} \times \frac{6}{5} \)

\( P(A'|B') = \frac{12}{15} \)

\( P(A'|B') = \frac{4}{5} \)

Therefore, the probability that P does not apply for the job given that Q does not apply is \( \frac{4}{5} \).