  The Monty Hall puzzle has been solved and dissected in many ways, but always
using probabilistic arguments, so it is considered a probability puzzle. In
this paper the puzzle is set up as an orthodox statistical problem involving an
unknown parameter, a probability model and an observation. This means we can
compute a likelihood function, and the decision to switch corresponds to
choosing the maximum likelihood solution. One advantage of the likelihood-based
solution is that the reasoning applies to a single game, unaffected by the
future plan of the host. I also describe an earlier version of the puzzle in
terms of three prisoners: two to be executed and one released. Unlike the goats
and the car, these prisoners have consciousness, so they can think about
exchanging punishments. When two of them do that, however, we have a paradox,
where it is advantageous for both to exchange their punishment with each other.
Overall, the puzzle and the paradox are useful examples of statistical
thinking, so they are excellent teaching topics.
