Bag A contains $2$ white, $1$ black and $3$ red balls and bag B contains $3$ black, $2$ red and $n$ white balls. One bag is chosen at random and $2$ balls drawn from it at random are found to be $1$ red and $1$ black. If the probability that both balls come from Bag A is $\dfrac{1}{3}$, then $n$ is equal to: