q-analog of tableau containment

We prove a $q$-analog of the following result due to McKay, Morse and Wilf: the probability that a random standard Young tableau of size $n$ contains a fixed standard Young tableau of shape $\lambda\vdash k$ tends to $f^{\lambda}/k!$ in the large $n$ limit, where $f^{\lambda}$ is the number of standard Young tableaux of shape $\lambda$. We also consider the probability that a random pair $(P,Q)$ of standard Young tableaux of the same shape contains a fixed pair $(A,B)$ of standard Young tableaux.