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Math Help - Probability problem-urgent, need it for monday morning!

  1. #1
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    Probability problem-urgent, need it for monday morning!

    new analytical method to detect pollutants in water is
    being tested. This new method of chemical analysis is
    important because, if adopted, it could be used to detect
    three different contaminants: organic pollutants volatile
    solvents and chlorinated compounds, instead of having to
    use a different test for each pollutant.
    The makers of the test claim that it can detect high levels of
    organic pollutants with 99.35% accuracy, volatile solvents
    with 99.15% accuracy and chlorinated compounds with
    82.7% accuracy. If a pollutant is not present, the test does
    not signal. Samples are prepared for the calibration of the
    test and 50% of them are contaminated with organic
    pollutants, 32% with volatile solvents and 18% with traces
    of chlorinated compounds.
    A test sample is randomly chosen.
    (i) What is the probability that the test will signal?
    (ii) If a test signals, what is the probability that
    chlorinated compounds are present.
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    new analytical method to detect pollutants in water is
    being tested. This new method of chemical analysis is
    important because, if adopted, it could be used to detect
    three different contaminants: organic pollutants volatile
    solvents and chlorinated compounds, instead of having to
    use a different test for each pollutant.
    The makers of the test claim that it can detect high levels of
    organic pollutants with 99.35% accuracy, volatile solvents
    with 99.15% accuracy and chlorinated compounds with
    82.7% accuracy. If a pollutant is not present, the test does
    not signal. Samples are prepared for the calibration of the
    test and 50% of them are contaminated with organic
    pollutants, 32% with volatile solvents and 18% with traces
    of chlorinated compounds.
    A test sample is randomly chosen.
    (i) What is the probability that the test will signal?
    (ii) If a test signals, what is the probability that chlorinated compounds are present.
    Strewth, what a mouthfull. The first thing you absolutely have to do is break down all this gunk into clear mathematical statements:

    Test gunk:

    Pr(O.P. signal | O.P. present) = 0.9935

    Pr(V.S. signal | V.S. present) = 0.9915

    Pr(C.C. signal | C.C. present) = 0.827

    Pr(no signal | no pollutant present) = 1

    Calibration gunk:

    Pr(O.P. present) = 0.5

    Pr(V.S. present) = 0.32

    Pr(C.C.) = 0.18


    Now you absolutely have to give a clear mathematical statement of what you're trying to calculate:

    (i) Pr(signal) = ?

    (ii) Pr(C.C. present | signal) = ?


    Then you use what you know to solve for ?

    (i) Pr(signal) = Pr(O.P. signal | O.P. present) Pr(O.P. present)
    + Pr(V.S. signal | V.S. present) Pr(V.S. present) + Pr(C.C. signal | C.C. present) Pr(C.C. present) = ......

    I get 0.96289, that is, a 96.289% chance of a signal.

    -----------------------------------------------------------------------------------------------------------

    (ii) Pr(C.C. present | signal) = \frac{\text{Pr(C.C. present and C.C. signal)}}{\text{Pr(signal)}}

    Note that Pr(C.C. present and signal) = Pr(signal and C.C present) = Pr(signal | C.C. present) Pr(C.C. present) = .....

    I get 0.154597, to the stated accuracy, that is, a 15.4597% chance.

    But you should double check my arithmetic .....
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