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The Monty Hall puzzle has been solved and dissected in many ways, but alwaysusing probabilistic arguments, so it is considered a probability puzzle. Inthis paper the puzzle is set up as an orthodox statistical problem involving anunknown parameter, a probability model and an observation. This means we cancompute a likelihood function, and the decision to switch corresponds tochoosing the maximum likelihood solution. One advantage of the likelihood-basedsolution is that the reasoning applies to a single game, unaffected by thefuture plan of the host. I also describe an earlier version of the puzzle interms of three prisoners: two to be executed and one released. Unlike the goatsand the car, these prisoners have consciousness, so they can think aboutexchanging 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 statisticalthinking, so they are excellent teaching topics.

Likelihood-based solution to the Monty Hall puzzle and a related 3-prisoner paradox