  We consider a finite-dimensional quantum system, making a transition between
known initial and final states. The outcomes of several accurate measurements,
which {\it could be} made in the interim, define virtual paths, each endowed
with a probability amplitude. If the measurements are {\it actually made}, the
paths, which may now be called "real", acquire also the probabilities, related
to the frequencies, with which a path is seen to be travelled in a series of
identical trials. Different sets of measurements, made on the same system, can
produce different, or incompatible, statistical ensembles, whose conflicting
attributes may, although by no means should, appear "paradoxical". We describe
in detail the ensembles, resulting from intermediate measurements of mutually
commuting, or non-commuting, operators, in terms of the real paths produced. In
the same manner, we analyse the Hardy's and the "three box" paradoxes, the
photon's past in an interferometer, the "quantum Cheshire cat" experiment, an
the closely related subject of "interaction-free measurements". It is shown
that, in all these cases, inaccurate "weak measurements" produce no real paths,
and yield only limited information about the virtual paths' probability
amplitudes
