Detailed fluctuation theorem bounds apparent violations of the second law

The second law of thermodynamics is a statement about the statistics of the entropy production, langle Σrangle geq 0. For small systems, it is known that the entropy production is a random variable and negative values (Σ< 0) might be observed in some experiments. This situation is sometimes called apparent violation of the second law. In this sense, how often is the second law violated? For a given average langle Σrangle, we show that the strong detailed fluctuation theorem implies a lower tight bound for the apparent violations of the second law. As applications, we verify that the bound is satisfied for the entropy produced in the heat exchange problem between two reservoirs mediated by a bosonic mode in the weak coupling approximation, a levitated nanoparticle and a classical particle in a box.