  The second law of thermodynamics is a statement about the statistics of the
entropy production, $\langle \Sigma \rangle \geq 0$. For small systems, it is
known that the entropy production is a random variable and negative values
($\Sigma < 0$) 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 \Sigma \rangle $, 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.
