75
5
.
b) Since we know the woman is not pregnant, we are limited to the 19 women in the second
row, of which 5 had a positive test.
P
(positive test result | not pregnant) =
263
.
0
19
5
The second result is what is usually called a false positive:
A positive result when the
woman is not actually pregnant.
Positive
test
Negative test
Total
Pregnant
70
4
74
Not Pregnant
5
14
19
Total
75
18
93

290
Bayes Theorem
In this section we concentrate on the more complex conditional probability problems we
began looking at in the last section.
Example 19
Suppose a certain disease has an incidence rate of 0.1% (that is, it afflicts 0.1% of the
population). A test has been devised to detect this disease. The test does not produce false
negatives (that is, anyone who has the disease will test positive for it), but the false positive
rate is 5% (that is, about 5% of people who take the test will test positive, even though they
do not have the disease). Suppose a randomly selected person takes the test and tests
positive. What is the probability that this person actually has the disease?
There are two ways to approach the solution to this problem. One involves an important
result in probability theory called Bayes' theorem. We will discuss this theorem a bit later,
but for now we will use an alternative and, we hope, much more intuitive approach.
Let's break down the information in the problem piece by piece.
Suppose a certain disease has an incidence rate of 0.1% (that is, it afflicts 0.1% of the
population).
The percentage 0.1% can be converted to a decimal number by moving the
decimal place two places to the left, to get 0.001. In turn, 0.001 can be rewritten as a
fraction: 1/1000. This tells us that about 1 in every 1000 people has the disease. (If we
wanted we could write
P
(disease)=0.001.)
A test has been devised to detect this disease. The test does not produce false negatives (that
is, anyone who has the disease will test positive for it).
This part is fairly straightforward:
everyone who has the disease will test positive, or alternatively everyone who tests negative
does not have the disease. (We could also say
P
(positive | disease)=1.)
The false positive rate is 5% (that is, about 5% of people who take the test will test positive,
even though they do not have the disease).
This is even more straightforward. Another way
of looking at it is that of every 100 people who are tested and do not have the disease, 5 will
test positive even though they do not have the disease. (We could also say that
P
(positive |
no disease)=0.05.)
Suppose a randomly selected person takes the test and tests positive. What is the probability
that this person actually has the disease?
Here we want to compute
P
(disease|positive). We
already know that
P
(positive|disease)=1, but remember that conditional probabilities are not
equal if the conditions are switched.
Rather than thinking in terms of all these probabilities we have developed, let's create a
hypothetical situation and apply the facts as set out above. First, suppose we randomly select
1000 people and administer the test. How many do we expect to have the disease? Since
about 1/1000 of all people are afflicted with the disease, 1/1000 of 1000 people is 1. (Now
you know why we chose 1000.) Only 1 of 1000 test subjects actually has the disease; the
other 999 do not.