  We derive a thermodynamic uncertainty relation (TUR) for first-passage times
(FPTs) on continuous time Markov chains. The TUR utilizes the entropy
production coming from bidirectional transitions, and the net flux coming from
unidirectional transitions, to provide a lower bound on FPT fluctuations. As
every bidirectional transition can also be seen as a pair of separate
unidirectional ones, our approach typically yields an ensemble of TURs. The
tightest bound on FPT fluctuations can then be obtained from this ensemble by a
simple and physically motivated optimization procedure. The results presented
herein are valid for arbitrary initial conditions, out-of-equilibrium dynamics,
and are therefore well suited to describe the inherently irreversible
first-passage event. They can thus be readily applied to a myriad of
first-passage problems that arise across a wide range of disciplines.
