2 5 7ba y e s ’ s T h e o r e mSimilarly, we can fill out the third column by dividing the total number of
people without colon cancer into those who test positive and those who do
not test positive. Here we use the conditional probability of a positive test,
given that a person does not have colon cancer [Pr(e|~h)]. This probability
was given as 0.03, and 0.03 3 99,700 5 2,991. This number means that, out of
a normal population of 99,700 without colon cancer, 2,991 will test positive.
Since the figures in this column must add up to a total of 99,700, the remain-
ing figure is 99,700 – 2,991 5 96,709:Colon Cancer Not Colon Cancer Total
Test Positive 270 2,991
Do Not Test Positive 30 96,709
Total 300 99,700 100,000Finally, we can fill out the fourth column by calculating total numbers of
people who test positive or do not test positive. Simply add across the rows:Colon Cancer Not Colon Cancer Total
Test Positive 270 2,991 3,261Do Not Test Positive 30 96,709 96,739
Total 300 99,700 100,000Check your calculations by adding the right column: 3,261 1 96,739 5 100,000.
Now that our population is divided up, the solution is staring you
in the face. This table shows us that, in a normal population of 100,000
tested people distributed according to the given probabilities, a total of
3,261 will test positive. Out of those, 270 will have colon cancer. Thus,
the probability that the patient has colon cancer, given that this patient
tested positive, is 270/3,261, which is about 0.083 or 8.3 percent, just as
before.
You can also read off other conditional probabilities. If you want to know
the conditional probability of not having colon cancer, given that your test
did not come out positive, then you need to look at the row for those who
do not test positive. The figure at the right of this row tells you that a total
of 96,739 out of the total population do not test positive. The column under
“Not Colon Cancer” then tells you that 96,709 of these do not have colon
cancer. Thus, the conditional probability of not having colon cancer given
your test did not come out positive is 96,709/96,739 or about 0.9997. That
means that, if you test negative, the odds are extremely high that you do not
have colon cancer.97364_ch11_ptg01_239-262.indd 257 15/11/13 10:58 AM
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