Let,
$P =$ P applies for the job
$Q =$ Q applies for the job
$P(P) = \frac{1}{4} \to(1)$
$P\left({P}\mid{Q}\right) = \frac{1}{2} \to(2)$
$P\left({Q}\mid{P}\right) = \frac{1}{3} \to (3)$
Now, we need to find $P\left({P'}\mid{Q'}\right)$
From $(2)$
$P({P}\mid {Q}) =\frac{P(P\cap Q)}{P(Q)} = \frac{1}{2}\to (4)$
From $(1)$ and $(3),$
$P({Q}\mid {P}) = \frac{P(P\cap Q)}{P(P)} = \frac{P(P\cap Q)}{\frac{1}{4}} = \frac{1}{3}$
$\therefore$ $P(P\cap Q) = \frac{1}{12}\to (5)$
From $(4)$ and $(5),$
$P(Q) = \frac{1}{6} \to (6)$
Now, $P({P'}\mid {Q'}) = \frac{P(P' \cap Q')}{P(Q')} \to (7)$
From $(6)$
$P(Q') = 1 - 1/6 = 5/6 \to (8)$
Also, $P(P' \cap Q') = 1 - [ P(P \cup Q) ]$
$\quad = 1 - [ P(P) + P(Q) - P(P\cap Q) ]$
$\quad = 1 - [ 1/4 + 1/6 - 1/12 ]$
$\quad = 1 - [ 1/3 ]$
$\quad = 2/3 \to (9)$
Hence, from $(7),(8)$ and $(9)$
$P(P' \mid Q')= \frac{\frac{2}{3}}{\frac{5}{6}} = \frac{4}{5}.$
Correct Answer: $A$