Languages weakly recognized by a Monte Carlo 2-way finite  automaton with n states are proved to be strongly recognized by a Monte  Carlo 2-way finite automaton with n  O(n)  states. This improves dramatically  over the previously known result by M.Karpinski and R.Verbeek  [10] which is also nontrivial since these languages can be nonregular [5].  For tally languages the increase in the number of states is proved to be  only polynomial, and these languages are regular