The Language of Argument

2 5 3

Bayes’s Theorem

bayeS’S THeorem

Although dice and cards provide nice, simple models for learning how

to calculate probabilities, real life is usually more complicated. One par-

ticularly interesting and important form of problem arises often in medi-

cine. Suppose that Wendy tests positive for colon cancer. The treatment

for colon cancer is painful and dangerous, so, before subjecting Wendy to

that treatment, her doctor wants to determine how likely it is that Wendy

really has colon cancer. After all, no test is perfect. Regarding the test

that was used on Wendy, previous studies have revealed the following

probabilities:

The probability that a person in the general population has colon cancer is

0.3 percent (or 0.003).

If a person has colon cancer, then the probability that the test is positive

is 90 percent (or 0.9).

If a person does not have colon cancer, then the probability that the test is

positive is 3 percent (or 0.03).

On these assumptions, what is the probability that Wendy actually has colon

cancer, given that she tested positive? Most people guess that this probabil-

ity is fairly high. Even most trained physicians would say that Wendy prob-

ably has colon cancer.^4

What is the correct answer? To calculate the probability that a person

who tests positive actually has colon cancer, we need to divide the number

of favorable outcomes by the number of total outcomes. The favorable

outcomes include everyone who tests positive and really has colon cancer.

This outcome is not “favorable” to Wendy, so we will describe this group as

true positives. The total outcomes include everyone who tests positive. This

1. You are presented with two bags, one containing two ham sandwiches
   and the other containing a ham sandwich and a cheese sandwich. You
   reach in one bag and draw out a ham sandwich. What is the probability
   that the other sandwich in the bag is also a ham sandwich?


2. You are presented with three bags: two contain a chicken-fat sandwich
   and one contains a cheese sandwich. You are asked to guess which bag
   contains the cheese sandwich. You do so, and the bag you selected is set
   aside. (You obviously have one chance in three of guessing correctly.)
   From the two remaining bags, one containing a chicken-fat sandwich
   is then removed. You are now given the opportunity to switch your
   selection to the remaining bag. Will such a switch increase, decrease, or
   leave unaffected your chances of correctly ending up with the bag with
   the cheese sandwich in it?

exercise V

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