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It is argued that there is a sensible way to define conditional probabilitiesin quantum mechanics, assuming only Bayes's theorem and standard quantumtheory. These probabilities are equivalent to the ``weak measurement''predictions due to Aharonov {\it et al.}, and hence describe the outcomes ofreal measurements made on subensembles. In particular, this approach is used toaddress the question of the history of a particle which has tunnelled across abarrier. A {\it gedankenexperiment} is presented to demonstrate the physicallytestable implications of the results of these calculations, along with graphsof the time-evolution of the conditional probability distribution for atunneling particle and for one undergoing allowed transmission. Numericalresults are also presented for the effects of loss in a bandgap medium ontransmission and on reflection, as a function of the position of the lossyregion; such loss should provide a feasible, though indirect, test of thepresent conclusions. It is argued that the effects of loss on the pulse {\itdelay time} are related to the imaginary value of the momentum of a tunnelingparticle, and it is suggested that this might help explain a small discrepancyin an earlier experiment.Comment: 11 pages, latex, 4 postscript figures separate (one w/ 3 parts

Conditional probabilities in quantum theory and the tunneling-time controversy