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Math Help - bayes theorem question

  1. #1
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    bayes theorem question

    Over a long period of study, it has been established that a particular lie detector based on galvanic skin response accurately signals a positive response (a lie) when a subject is indeed lying in 99 % of cases. When a subject is not lying, the detector signals a negative response also in 99 % of cases.
    A man is murdered on a large housing estate and investigators are keen to establish who on the estate met the deceased in the local pub on the night of the murder. Based on some simple calculations, the police believe 0.5 % of the adult population of the estate are likely to have met him that night. Adult residents on the estate are randomly selected for interview and are requested to undertake the lie detector test.

    .......

    One young lady randomly selected for interview claims not to have seen him that but the detector signals positively indicating a lie.

    use Bayes theorem to calculate the probability that the young lady met the deceased on the night of the murder.

    Finding this quite difficult to get my head round, any help would be much appretiated
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by feage7 View Post
    Over a long period of study, it has been established that a particular lie detector based on galvanic skin response accurately signals a positive response (a lie) when a subject is indeed lying in 99 % of cases. When a subject is not lying, the detector signals a negative response also in 99 % of cases.
    A man is murdered on a large housing estate and investigators are keen to establish who on the estate met the deceased in the local pub on the night of the murder. Based on some simple calculations, the police believe 0.5 % of the adult population of the estate are likely to have met him that night. Adult residents on the estate are randomly selected for interview and are requested to undertake the lie detector test.

    .......

    One young lady randomly selected for interview claims not to have seen him that but the detector signals positively indicating a lie.

    use Bayes theorem to calculate the probability that the young lady met the deceased on the night of the murder.

    Finding this quite difficult to get my head round, any help would be much appretiated
    First assume that nobody is going to admit to having seen the guy.

    Then Bayes theorem tells us that:

    p(seen|positive)p(positive)=p(positive|seen)p(seen )

    we are told that:

    p(seen)=0.005
    p(positive)=p(not seen)p(false positive)+p(seen)*p(correct positive)=0.995*0.01+0.005*0.99=0.0149
    p(positive|seen)=p(correct positive)=0.99

    So

    p(seen|positive)=(0.99*0.005)/0.0149 =0.332

    or about 33%.

    CB
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