Resolving Some Puzzles of Conditional Probability
This essay develops a systematic resolution to a number of familiar puzzles of probability, puzzles (like the famous Monty Hall problem) that hinge on conflicting estimates of how the probability of some possibility should be updated after receipt of supplementary information. My proposed
resolution hinges on a key observation – one fairly obvious in retrospect – the fact that the one piece of information can have different impacts upon the probability at issue, depending on how it was provided. So there should be no surprise at all in the fact that the estimates
differ, since they routinely presume the updating information to have been supplied by different processes. In consequence, a standard interpretation of conditional probability has to be abandoned. For the probability to be assigned to some outcome X after outcome Y has been
ascertained is often not p(X & Y)/p(Y). It routinely takes this value if the signalling process is one that verifies Y-ness every time that Y-ness has in fact been achieved, but this formula can fail with other, less persistent, sources
of information. In the more general situation (I argue, defending the observation central to my resolution.), we need to update our probability of X to something subtly different, viz. p(X&Y*)/p(Y*), where Y* is now an outcome
of the signalling process, that outcome which informs that Y has been achieved. My resolution of the puzzles is thus systematic. It depends on a general rule, and makes no ad hoc use of idiosyncratic features of individual problems.
Document Type: Research Article
Publication date: 01 December 2008
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