This population size represents the total number of people who are tested. We have no information about the ones who are not tested, so they cannot figure into our calculations.
The bottom row can now be filled in by dividing the total population into those who have colon cancer and those who do not have colon cancer. Just multiply the population size by the probability of colon cancer in the general population [Pr(h)] to get a number for the second box on the bottom row. This figure represents the total number of people with colon cancer in this popula- tion. Then subtract that product from the population size and put the remain- der in the remaining box. This represents the total number of people without colon cancer in this population. Since these two groups exhaust the popula- tion, they must add up to the total population size. In our case, we were given that the probability that a person in the general population has colon cancer is 0.003. On this basis, we can fill in the bottom row of the table:
Colon Cancer Not Colon Cancer Total
Test Positive
Do Not Test Positive
Total 300 99,700 100,000
295
Bayes ’ s Theorem
Next, fill out the second column by dividing the total number of people with colon cancer into those who test positive and those who do not test positive. These numbers can be calculated with the given conditional proba- bility of testing positive, given colon cancer [Pr(e|h)]. In our example, if a person has colon cancer, the probability that the test is positive is 0.9. Thus, 270 (=0.9 × 300) of the people in the colon cancer column will test positive and the rest (300 – 270 = 30) will not, so we get these figures:
Colon Cancer Not Colon Cancer Total
Test Positive 270
Do Not Test Positive 30
Total 300 99,700 100,000
Similarly, 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 × 99,700 = 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 = 96,709:
Colon Cancer Not Colon Cancer Total
Test Positive 270 2,991
Do Not Test Positive 30 96,709
Total 300 99,700 100,000
Colon Cancer Not Colon Cancer Total
Test Positive 270 2,991 3,261
Do Not Test Positive 30 96,709 96,739
Total 300 99,700 100,000
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