  We present an analogy between the classic Gambler's Ruin problem and the
thermally-activated dynamics in periodic Brownian ratchets. By considering each
periodic unit of the ratchet as a site chain, we calculated the transition
probabilities and mean first passage time for transitions between energy minima
of adjacent units. We consider the specific case of Brownian ratchets driven by
Markov dichotomous noise. The explicit solution for the current is derived for
any arbitrary temperature, and is verified numerically by Langevin simulations.
The conditions for vanishing current and current reversal in the ratchet are
obtained and discussed.
