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 f λ / k ! in the large n limit, where f λ 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.