Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!mnetor!uunet!munnari!mulga!lee From: lee@mulga.oz (Lee Naish) Newsgroups: sci.philosophy.tech Subject: Re: The nature of knowledge (probabilities) Message-ID: <2099@mulga.oz> Date: Fri, 17-Jul-87 01:59:35 EDT Article-I.D.: mulga.2099 Posted: Fri Jul 17 01:59:35 1987 Date-Received: Sat, 18-Jul-87 13:52:22 EDT References: <3587e521.44e6@apollo.uucp> <680@gargoyle.UChicago.EDU> <121@cavell.UUCP> <4865@milano.UUCP> <2400@hoptoad.uucp> Reply-To: lee@mulga.UUCP (Lee Naish) Distribution: world Organization: Comp Sci, Melbourne Uni, Australia Lines: 35 Keywords: knowledge belief truth certainty Summary: Probabilities may not add up In article <2400@hoptoad.uucp> laura@hoptoad.uucp (Laura Creighton) writes: >In article <4865@milano.UUCP> wex@milano.UUCP writes: >>If we ask him "Do you believe there >>is a typo on page of this book?" for all 350 possible values of >>, he will say "no" each time. >>However, if we ask "Do you believe there is a typo somewhere in the >>350 pages of this book?" he will answer "yes." Inconsistent? Yes. >> >>The best answer I could give him was that his beliefs were not a >>matter of simple truth/falsity, but were a matter of degree. Thus, >>the correct questions should have been "Do you believe that there is a >>one-in-three-hundred-fifty chance that there is a typo on page of >>this book?" To this, I claimed, he would have answered "yes." This >>makes consistent his reply of "yes" to the final question. Suppose each page of the book was simply a list of 100 numbers which (should) add up to 1000. Suppose also that the book source was on-line and with the appropriate tools all the numbers added by the computer and the result was 349999. The probability of there being an error is extremely high (say 0.999). What do you believe is the probability of an error on any given page? If you say 1/350 then the probability of an error in the book should be, according to simple probability theory, 1-(349/350)^350 = 0.63. If you say 10/350 (or whatever is needed to get the 0.999 figure) then the expected number of errors greatly increases (which I think is unreasonable). How can this paradox be resolved without admitting inconsistent beliefs? Lee Naish lee@mulga.oz.au lee@munnari.oz.au munnari!lee@seismo.css.gov {seismo,mcvax,ukc,ubc-vision}!munnari!lee