The Language of Argument

2 5 5

B a y e s ’ s T h e o r e m

Finally, we apply Rule 2G to item 4 and get:

BT: Pr 1 h 0 e 25

Pr 1 h 23 Pr 1 e 0 h 2

(^3) Pr 1 h 23 Pr 1 e 0 h 241 Pr 1 ~h 23 Pr 1 e 0 ~h 24
This is a simplified version of Bayes’s theorem.
This theorem enables us to calculate the desired probability in our original
example:
h 5 the patient has colon cancer
e 5 the patient tests positive for colon cancer
Pr(h) 5 0.003
Pr(~h) 5 1 – Pr(h) 5 0.997
Pr(e|h) 5 0.9
Pr(e|~h) 5 0.03
If we substitute these values into Bayes’s theorem, we get:
Pr 1 h 0 e 25

0.003 3 0.9

(^3) 0.003 3 0.9 (^413) 0.997 3 0.03 4
5 about 0.083
In this way, we can calculate the conditional probability of the hypothesis
given the evidence from its reverse, that is, from the conditional probability
of the evidence given the hypothesis. That is what makes Bayes’s theorem
so useful.
Many people find a different method more intuitive. The first step is to
set up a table. The two factors to be related are: (1) whether the patient has
colon cancer and (2) whether the patient tests positive for colon cancer. To
chart all possible combinations of these two factors, we need a table like this:
Colon Cancer Not Colon Cancer Total
Test Positive
Do Not Test Positive
Total
Next, we need to enter a population size in the lower right box. The prob-
abilities will not be affected by the population size, but it is cleaner to pick a
population that is large enough to get whole numbers when the population
is multiplied by the given probabilities. To determine the right size popula-
tion, add the number of places to the right of the decimal point in the two
most specific probabilities, then pick a population of 10 to the power of that
sum. In our example, the most specific probabilities are 0.003 and 0.03, and
3 1 2 5 5, so we can enter 10^5 :
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