Friday, December 12, 2003

Puzzle

Republicans have been shown to tell the truth 1/4th of the time, lying at random 3/4th of the time. George, one of these very dishonest Republicans, makes a statement. The probability that it is true is, by assumption, 1/4. Then Bill, another very dishonest Republican, backs him up, saying George's statement is true. Given that Bill supports it, what is the probability that George's statement is true now?

The answer isn't difficult: First we ask how probable it is that George utters a true statement and Bill makes a true statement of support. Since they both tell the truth 1/4 of the time, these events will both turn out to be true 1/16 of the time (1/4 x 1/4). Now we ask how probable it is that Bill will make a statement of support. Since Bill will utter his support when either both he and George tell the truth or when they both lie, the probability of this is 10/16 (1/4 x 1/4 + 3/4 x 3/4). Thus the probability that George is telling the truth given that Bill supports him is 1/10 ( the ratio of 1/16 to 10/16).

The moral: Confirmation of a very dishonest person's unreliable statement by another very dishonest person makes the statement even less reliable.



(Adapted from the writing of John Allen Paulos. Partisanized by me.)

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