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 λ⊢k tends to fλ/k! in the large n limit, where fλ is the number of standard Young tableaux of shape λ. 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.