  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.
