  We analytically derive universal bounds that describe the trade-off between
thermodynamic cost and precision in a sequence of events related to some
internal changes of an otherwise hidden physical system. The precision is
quantified by the fluctuations in either the number of events counted over time
or the times between successive events. Our results are valid for the same
broad class of nonequilibrium driven systems considered by the thermodynamic
uncertainty relation, but they extend to both time-symmetric and asymmetric
observables. We show how optimal precision saturating the bounds can be
achieved. For waiting time fluctuations of asymmetric observables, a phase
transition in the optimal configuration arises, where higher precision can be
achieved by combining several signals.
