The main purpose of this paper is to show that we can ex- ploit the difference in the probability calculation between quantum and probabilistic computations to claim the difference in their space efficien- cies. It is shown that, for each n, there is a finite language L which contains sentences of length up to O(n^c+1) such that: (i) There is a one- way quantum finite automaton (qfa) of O(n^c+4) states which recognizes L. (ii) However, if we try to simulate this qfa by a probabilistic finite au tomaton (pfa) using the same algorithm, then it needs Ω(n^2c+4) states. It should be noted that we do not prove real lower bounds for pfa’s but show that if pfa’s and qfa’s use exactly the same algorithm, then qfa’s need much less states.