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 the events.
Let \[ A=\text{event that \(P\) applies} \] and \[ B=\text{event that \(Q\) applies} \] Given, \[ P(A)=\frac{1}{4} \] Also, \[ P(A|B)=\frac{1}{2} \] and \[ P(B|A)=\frac{1}{3} \]

Step 2: Find \(P(A\cap B)\).
Using conditional probability, \[ P(B|A)=\frac{P(A\cap B)}{P(A)} \] Substituting the values, \[ \frac{1}{3} = \frac{P(A\cap B)}{1/4} \] Therefore, \[ P(A\cap B)=\frac{1}{3}\times \frac{1}{4} \] Hence, \[ P(A\cap B)=\frac{1}{12} \]

Step 3: Find \(P(B)\).
Using \[ P(A|B)=\frac{P(A\cap B)}{P(B)} \] Substituting the values, \[ \frac{1}{2} = \frac{1/12}{P(B)} \] Therefore, \[ P(B)=\frac{1}{6} \]

Step 4: Find the required probability.
We need \[ P(A'|B') \] Using conditional probability, \[ P(A'|B') = \frac{P(A'\cap B')}{P(B')} \] Now, \[ P(B')=1-P(B) \] So, \[ P(B')=1-\frac{1}{6} = \frac{5}{6} \] Also, \[ P(A'\cap B') = 1-P(A\cup B) \] Using \[ P(A\cup B)=P(A)+P(B)-P(A\cap B), \] we get \[ P(A\cup B) = \frac{1}{4}+\frac{1}{6}-\frac{1}{12} \] Taking LCM, \[ = \frac{3+2-1}{12} = \frac{4}{12} = \frac{1}{3} \] Therefore, \[ P(A'\cap B') = 1-\frac{1}{3} = \frac{2}{3} \] Hence, \[ P(A'|B') = \frac{2/3}{5/6} \] \[ = \frac{2}{3}\times \frac{6}{5} \] \[ = \frac{4}{5} \]

Step 5: Final conclusion.
Therefore, \[ \boxed{\frac{4}{5}} \]