We study the modal logic ML^r of the countable random frame, which is contained in and 'approximates' the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of ML^r and show that it is not nitely axiomatizable. Then we describe the finite frames of that logic and show that it has the finite frame property and its satisfiability problem is...