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Thread: Probability

  1. #1 Probability 
    Forum Junior epidecus's Avatar
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    So I was presented with this elementary statistics problem...

    Quote Originally Posted by Problem
    Approximately 1% of women aged 40-50 have breast cancer.
    A woman with breast cancer has a 90% chance of a positive test from a mammogram,
    while a woman without breast cancer has a 10% percent chance of a false positive result.
    Using the simple statement of Baye's theorem, find the probability that a woman has breast cancer given she receive a positive result.
    Okay. So Baye's theorem states .

    Let denote a woman with breast cancer, a woman without, a positive result, and " - " negative.

    The problem information directly implies...







    Now, I have to find the probability of .



    So if a woman receives a positive result from the mammogram, there's a 9 in 108 chance that she actually has breast cancer, or approximately 8.33%. Apparently, there's also a 91.7% chance that she comes positive but actually doesn't have it (yes, I also computed as well as the other two outcomes). So two of the four outcome probabilities seem anti-intuitive, namely and . 8.3% seems... unreliable at best. But maybe the whole point of this problem is to show an unexpectancy. (Monty Hall, anyone?)

    Are my calculations correct? Are the statistics shaky or is a more detailed statement of Baye's theorem needed? Thanks in advance.


    Last edited by epidecus; January 4th, 2013 at 03:59 AM. Reason: Miscalculation
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  3. #2  
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    Your calculations sem correct. The results may sound paradoxal, but IMO, they are not.

    The probability to get a positive test is 10.8%, but, in most cases (9.9%), this result corresponds to an healthy person. So, in most cases, a positive result is false.
    But a positive test is a start point for further examinations.


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  4. #3  
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    Hello,

    Should not be written in the numerator 0.01 x 0.90, i.e. likelihood that a person is a women with cancer and likelihood that is a positive test given she really has cancer?
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  5. #4  
    Forum Junior epidecus's Avatar
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    Yes, ivanf. I had the right thing in mind and wrote the correct result, but typo'd the function. My bad.
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