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All of these refer to 40 year old women.

 * 1% who participate in routine screening have breast cancer.
 * 80% with breast cancer will get positive mammographies.
 * 9.6% without breast cancer will also get positive mammographies.
 * Given that a woman has a positive mammography in a routine screening,
   what is the probability that she actually has breast cancer?

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The same statements in formal language, using the notation
   cancer    = has breast cancer
   !cancer   = doesn't have breast cancer
   pos       = mammogram test is positive
   !po       = mammogram test is negative

 * P(cancer)      = 0.01
 * P(pos|cancer)  = 0.80
 * P(pos|!cancer) = 0.096
 * P(cancer|pos)  = ?

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First by counting the numbers.  Suppose N = 1000.  Then:

 * N(cancer) = P(cancer)*N = 10
   * 8 (i.e. 80% of 10) of these will test positive for cancer.
   * 2 of these will test negative.

 * N(!cancer) = 990
   * 95 (i.e. 0.096*990) of these test positive for cancer
   * 895 of these test negative.

The total number of women who test positive is 95 + 8 = 103,
and so P(cancer|positive) = 8/103 = 0.078 = 7.8%

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The joint probability distribution is

   P(cancer,  pos)  = 8/1000   = 0.008
   P(cancer,  !pos) = 2/1000   = 0.002
   P(!cancer, pos)  = 95/1000  = 0.095
   P(!cancer, !pos) = 895/1000 = 0.895

which lets us calculate anything directly from definitions:

   P(cancer)    = 0.008 + 0.002 = 0.01
   P(!cancer)   = 0.095 + 0.895 = 0.99

   P(pos)       = 0.008 + 0.095 = 0.103
   P(!pos)      = 0.002 + 0.895 = 0.897

   P(cancer|pos) = P(cancer,pos)/P(pos) = 0.008/0.103 = 0.078

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The Bayes formula is

 P(a|b) = P(b|a) * P(a) / P(b)
        = P(b|a) * P(a) / sum_x { P(b|x) * P(x) }

which in this case is

                   P(pos|cancer) * P(cancer)
 P(cancer|pos) =  ----------------------------------------------------------
                   P(pos|cancer) * P(cancer) + P(pose|!cancer) * P(!cancer)

                   0.80 * 0.01
               =  ----------------------------
                   0.80 * 0.01 + 0.096 * 0.99

               = 0.008/(0.008 + 0.095)

               = 0.008/0.103

               = 0.078

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