  Jarzynski's equality sets a strong bound on the probability of violating the
second law of thermodynamics by extracting work beyond the free energy
difference. We derive finite-time refinements to this bound for driven systems
in contact with a thermal Markovian environment, which can be expressed in
terms of the geometric notion of thermodynamic length. We show that finite-time
protocols converge to Jarzynski's bound at a rate slower than $1/\sqrt{\tau}$,
where $\tau$ is the total time of the work-extraction protocol. Our result
highlights a new application of minimal dissipation processes and demonstrates
a connection between thermodynamic geometry and the higher order statistical
properties of work.
