  As examples such as the Monty Hall puzzle show, applying conditioning to
update a probability distribution on a ``naive space'', which does not take
into account the protocol used, can often lead to counterintuitive results.
Here we examine why. A criterion known as CAR (``coarsening at random'') in the
statistical literature characterizes when ``naive'' conditioning in a naive
space works. We show that the CAR condition holds rather infrequently, and we
provide a procedural characterization of it, by giving a randomized algorithm
that generates all and only distributions for which CAR holds. This
substantially extends previous characterizations of CAR. We also consider more
generalized notions of update such as Jeffrey conditioning and minimizing
relative entropy (MRE). We give a generalization of the CAR condition that
characterizes when Jeffrey conditioning leads to appropriate answers, and show
that there exist some very simple settings in which MRE essentially never gives
the right results. This generalizes and interconnects previous results obtained
in the literature on CAR and MRE.
