Discrete Mathematics for Computer Science

514 CHAPTER 8 Discrete Probability

One key feature of Bayes' Rule is that it can be used in situations where the underlying

probability distribution of the sample space is not known but estimates of the probabilities

for some of the events are available. The next example illustrates this. The designers of

computer systems that aid in medical diagnosis and other similar problems must deal with

uncertain information of this kind.

Example 5. Suppose it is estimated that 10% of a population has a certain disease. Tests

for this disease are being developed but are not yet perfect. In fact, an individual who has

the disease may test negative. Suppose experience with a particular test shows that 5%

of the results are actually false negatives-that is, the individual actually does have the

disease. Also, suppose that 8% of the tests done so far have been positive. What is the

probability that a sick person will receive a false-negative test result?

Solution. Let Qi consist of the population, A denote the subset of people who would test

positive if they took the test, and B denote the subset of people who have the disease.

Which people actually make up these subsets is not known: Not everybody may take the

test, and only time will tell which people actually have the disease. Nevertheless, we can

estimate that P(A) = 0.08, P(B) = 0.10, and P(B[LA) = 0.05. We are interested in the

probability that a person who is ill tests negative. Hence, we must compute P(AIB). By

Bayes' Rule,

P(AIB) = P(BIA)P - P(A) (B) - (0.05)(0.92)(0.10) 0.46 M

Because of the wide applicability of Bayes' Rule, we highlight it here.

Using Bayes' Rule

In many applications, the answer being sought can be expressed in the form of a con-

ditional probability P(BIA), and the other probabilities P(A I B), P(A), and P(B)

that are needed to apply Bayes' Rule can be estimated by experiment or experience:

P(A I B) .P(B)

P(BLA) P(AP (A)

8.5.4 Using Bayes' Rule with the Theorem of Total Probability

We devote this subsection to examples illustrating the power of using Bayes' Rule together

with the Theorem of Total Probability.

Example 6. (Communication Channel Reliability) Consider a noisy communication

channel over which a 0 or a 1 is to be sent. Suppose that the probability the bit to be sent

is a 0 is 0.4 and the probability that it is a 1 is 0.6. Also, suppose that due to noise, the

probability that a 0 is changed to a 1 during transmission is 0.2 and the probability that a

I is changed to a 0 is 0.1 (see Figure 8.5). Suppose a 1 is received. What is the probability

that a 1 was sent?