A lower bound on work extraction probability prescribed by nonequilibrium work relation

In nonequilibrium processes, work extraction from a system is subject to random fluctuations associated with the statistical distribution prescribed by its environment. The probability of extracting work above a given arbitrary threshold can be a measure of restriction imposed by experimental circumstances. We present a lower bound for the probability when the work value lies in a finite range. For the case of unrestricted maximum work, the lower bound gets larger as the free energy difference between initial and final states becomes larger. We point out also that an upper bound previously reported in the literature is a direct consequence of the well-known second mean value theorem for definite integrals.