Probabilities in the (<Emphasis Type="Italic">k</Emphasis>, ?) hook

We consider particular (k, ℓ)-hook probability measures on the space of the infinite standard Young tableaux, and calculate the probability that the entry at the (1, 2) cell is odd. As n goes to infinity, this, approximately, is the corresponding probability on tableaux of size n in the (k, ℓ) hook. In few cases of small k and ℓ we find exact formulas for the corresponding numbers of such standard tableaux.