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Math Help - Conditional Probablity Problem

  1. #1
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    Conditional Probablity Problem

    I was wondering if someone could help me with this...

    A test for the presence of E. coli in water detects the bacteria 97% of the time when the bacteria is present, but also gives a false positive test 2% of the time, wrongly indicating the presence of E. coli in uninfected water.
    If 10% of the water samples tested contain E. coli, what is the probability that a test result indicating the presence of the bacteria is accurate?

    Thanks for any help!
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  2. #2
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    Quote Originally Posted by ty2391 View Post
    I was wondering if someone could help me with this...

    A test for the presence of E. coli in water detects the bacteria 97% of the time when the bacteria is present, but also gives a false positive test 2% of the time, wrongly indicating the presence of E. coli in uninfected water.
    If 10% of the water samples tested contain E. coli, what is the probability that a test result indicating the presence of the bacteria is accurate?

    Thanks for any help!
    Draw a tree diagram!

    Pr(E) = 0.1 and Pr(E') = 0.9 are the first two branches.

    From E the two branches are Pr(+ve) = 0.97 and Pr(-ve) = 0.03. (Note that these probabilities are actually Pr(+ve | E) = 0.97 and Pr(-ve | E) = 0.03).

    From E' the two branches are Pr(+ve) = 0.02 and Pr(-ve) = 0.98. (Note that these probabilities are actually Pr(+ve | E') = 0.02 and Pr(-ve | E') = 0.98).

    Then Pr(E | +ve) = \frac{(0.1)(0.97)}{(0.1)(0.97) + (0.9)(0.02)} = \, ....
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  3. #3
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    Thanks for your reply!
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