Question

Two persons \(P\) and \(Q\) are considering to apply for a job. The probability that \(P\) applies for the job is \(1/4\), the probability that \(P\) applies for the job given that \(Q\) applies for the job is \(1/2\), and the probability that \(Q\) applies for the job given that \(P\) applies for the job is \(1/3\). Then the probability that \(P\) does not apply for the job given that \(Q\) does not apply for the job is

• A. \(\dfrac{4}{5}\)
• B. \(\dfrac{5}{6}\)
• C. \(\dfrac{7}{8}\)
• D. \(\dfrac{11}{12}\)

Hint:

Always use \[ P(A|B)=\frac{P(A\cap B)}{P(B)} \] to convert conditional probabilities into intersection probabilities. Then apply complement rules carefully for events like \(A'\) and \(B'\).

Answer 1

The Correct Option is A

Step 1: Define events and list given information.
Let $ A $ = event P applies, $ B $ = event Q applies. Given: $ P(A) = \dfrac{1}{4} $, $ P(A|B) = \dfrac{1}{2} $, $ P(B|A) = \dfrac{1}{3} $. We want $ P(A'|B') $.
Step 2: Find $ P(A \cap B) $ using $ P(B|A) $.
By definition, $ P(B|A) = \dfrac{P(A \cap B)}{P(A)} $. So $ P(A \cap B) = P(B|A) \cdot P(A) = \dfrac{1}{3} \times \dfrac{1}{4} = \dfrac{1}{12} $.
Step 3: Find $ P(B) $ using $ P(A|B) $.
By definition, $ P(A|B) = \dfrac{P(A \cap B)}{P(B)} $. So $ P(B) = \dfrac{P(A \cap B)}{P(A|B)} = \dfrac{1/12}{1/2} = \dfrac{1}{6} $.
Step 4: Find $ P(A \cup B) $.
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{1}{4} + \frac{1}{6} - \frac{1}{12} = \frac{3+2-1}{12} = \frac{4}{12} = \frac{1}{3} \]
Step 5: Find $ P(A' \cap B') $ and $ P(B') $.
$ P(A' \cap B') = P((A \cup B)') = 1 - P(A \cup B) = 1 - \dfrac{1}{3} = \dfrac{2}{3} $. Also $ P(B') = 1 - \dfrac{1}{6} = \dfrac{5}{6} $.
Step 6: Compute the required conditional probability.
\[ P(A'|B') = \frac{P(A' \cap B')}{P(B')} = \frac{2/3}{5/6} = \frac{2}{3} \times \frac{6}{5} = \frac{4}{5} \] \[ \boxed{\frac{4}{5}} \]