Question

P and Q are considering to apply for a job. The probability that P applies for the job is $\frac{1}{4}.$ The probability that P applies for the job given that Q applies for the job is $\frac{1}{2},$ and the probability that Q applies for the job given that P applies for the job is $\frac{1}{3}.$ Then the probability that P does not apply for the job given that Q does not apply for the job is

• A. $\frac{4}{5}$
• B. $\frac{5}{6}$
• C. $\frac{7}{8}$
• D. $\frac{11}{12}$

Answer 1

The correct option is A $\frac{4}{5}$

A = {P applies for job},
B = {Q applies for job}

Then $P\left(A\right)=\frac{1}{4},P\left(A/B\right)=\frac{1}{2}$ ...(1)
& $P\left(B/A\right)=\frac{1}{3}$ ...(2)

Using (2)
$\frac{P(B\cap A)}{P\left(A\right)}=\frac{1}{3}$

$\Rightarrow P(B\cap A)=\frac{1}{12}$

Using (1)

$\frac{P(B\cap A)}{P\left(B\right)}=\frac{1}{2}$

$\Rightarrow P\left(B\right)=\frac{1}{6}$

$\text{So}P\left(\frac{\overline{A}}{\overline{B}}\right)=\frac{P(\overline{A}\cap \overline{B})}{P\left(\overline{B}\right)}=\frac{1-P(A\cup B)}{1-P\left(B\right)}$

$=\frac{1-(\frac{1}{4}+\frac{1}{6}-\frac{1}{12})}{1-\frac{1}{6}}=\frac{2}{3}\times \frac{6}{5}=\frac{4}{5}$