subjective or personal probabilities, meaning that they are a statement of person belief,
rather than having a frequentist or analytic basis as defined at the beginning of the chapter.
Bayes’ theorem will work perfectly well with any kind of probability, but it is most often
seen with subjective probabilities.
Let’s take a simple example that I have modified from Stefan Waner’s website at
http://people.hofstra.edu/Stefan_Waner/tutorialsf3/unit6_6.html. (That site has some other
examples that may be helpful if you want them.) Psychologists have become quite inter-
ested in sports medicine, and this example is actually something that is relevant. In addi-
tion it fits perfectly with the work on decision making.
Let’s assume that an unnamed bicyclist has just failed a test for banned steroids after
finishing his race. (Waner used rugby instead of racing, but we all know that rugby guys
are good guys and follow the rules, while we are beginning to have our doubts about cy-
clists.) Our cyclist argues that he is perfectly innocent and would never use performance
enhancing drugs. Our task is to determine a reasonable probability about the guilt or inno-
cence of our cyclist. We do have a few facts that we can work with. First of all, the drug
company that markets the test tells us that 95% of steroid users test positive. In other
words, if you use drugs the probability of a positive result is .95. That sounds impressive.
Drug companies like to look good, so they don’t bother to point out that 10% of nonusers
also test positive, but we coaxed it out of them. We also know one other thing, which is that
past experience has shown that 10% of this racing team uses steroids (and the other 90%
do not). We can put this information together Table 5.3.
One of the important pieces of information that we have is called the prior probability,
which is the probability that the person is a drug user beforewe acquire any further informa-
tion. This is shown in the table as p(user) 5 .10. What we want to determine is the posterior
probability,which is our new probability afterwe have been given data (in this case the data
that he failed the test).
Bayes’ theorem tells us that we can derive the posterior probability from the informa-
tion we have above. Specifically:where Ustands for the hypothesis that he did use steroids, represents that hypothesis
that he did notuse steroids, and Pstands for the new data (that he failed the test). From the
information in the above table we can calculate=
(.95)(.10)
(.95)(.10) 1 (.15)(.90)
=
.095
(.095 1 .135)
=.413
p(U|P)=p(P|U)*p(U)
p(P|U)*p(U) 1 p(P|NU)*p(NU)NU
p(U|P)=p(P|U)*p(U)
p(P|U)*p(U) 1 p(P|NU)*p(NU)124 Chapter 5 Basic Concepts of Probability
Table 5.3 Probabilities associated with steroid use
Source of
Knowns p information
p(cyclist is user) p(U) .10 10% of team is
p(cyclist not a user) p(NU) .90 90% of team is not
p(positive | user) p(P|U) .95 From drug company
p(positive | non-user) p(P|NU) .10 Also from drug company
p(user | positive test) p(U|P)? Our goalprior probability
posterior
probability