Let one of the quantities be x. Thus the other is 2n−x.
Then product will be greatest when they are equal i.e., each is n in which case the product is n2.
By the given condition
x(2n−x)≥34n2
⇒8nx−4x2≥3n2⇒4x2−8nx+3n2≤0
⇒(2x−3n)(2x−n)≤0⇒n2≤x≤32n.
∴ favourable number of cases =3n2−n2
Hence required probability =n2n=12