  Conditional probabilities in quantum systems which have both initial and
final boundary conditions are commonly evaluated using the
Aharonov-Bergmann-Lebowitz rule. In this short note we present a seemingly
disturbing paradox that appears when applying the rule to systems with slightly
broken degeneracies. In these cases we encounter a singular limit - the
probability "jumps" when going from perfect degeneracy to negligibly broken
one. We trace the origin of the paradox and solve it from both traditional and
modern perspectives in order to highlight the physics behind it: the necessity
to take into account the finite resolution of the measuring device. As a
practical example, we study the application of the rule to the Zeeman effect.
The analysis presented here may stress the general need to first consider the
governing physical principles before heading to the mathematical formalism, in
particular when exploring puzzling quantum phenomena.
