  We investigate the consistency of conditional quantum probabilities. This is
whether there is compatibility between the Kolmogorov-Bayes conditional
probabilities and the Born rule. We show that they are not compatible in the
sense that there are situations where there is no legitimate density matrix
that may reproduce the conditional statistics of the other observable via the
Born rule. This is to say that the Gleason theorem does not apply to
conditional probabilities. Moreover, we show that when this occurs the joint
statistics is nonclassical. We show that conditional probabilities are not
equivalent to state reduction, so these results do not affect the validity of
the L\"{u}ders expression.
