Bhisham C. Gupta - Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP

Solution:To answer this question, we use Bayes's theorem ( 3.6.3) to find the posterior probability of a set's failure being due to PCD, after observing that the failure is diagnosed as being due to a faulty PCD. We let

F = event, set fails due to PCD

E = event, set failure is diagnosed as being due to PCD

and we wish to determine the posterior probability .

We are given that so that , and that and . Applying ( 3.6.3) gives

Notice that in light of the event E having occurred, the probability of F has changed from the prior probability of 25% to the posterior probability of 47.1%.

Formula ( 3.6.3) can be generalized to more complicated situations. Indeed Bayes stated his theorem for the more general situation, which appears below.

Figure 3.6.2 Venn diagram showing mutually exclusive events in S .

Theorem 3.6.1(Bayes's theorem) Suppose that are mutually exclusive events in S such that , and E is any other event in S. Then

(3.6.4)

We note that ( 3.6.3) is a special case of ( 3.6.4), with , , and . Bayes's theorem for k events has aroused much controversy. The reason for this is that in many situations, the prior probabilities are unknown. In practice, when not much is known about a priori, these have often been set equal to as advocated by Bayes himself. The setting of in what is called the “in‐ignorance” situation is the source of the controversy. Of course, when the 's are known or may be estimated on the basis of considerable past experience, ( 3.6.4) provides a way of incorporating prior knowledge about the to determine the conditional probabilities as given by ( 3.6.4). We illustrate ( 3.6.4) with the following example.

Example 3.6.2(Applying Bayes's theorem) David, Kevin, and Anita are three doctors in a clinic. Dr. David sees 40% of the patients, Dr. Anita sees 25% of the patients, and 35% of the patients are seen by Dr. Kevin. Further 10% of Dr. David's patients are on Medicare, while 15% of Dr. Anita's and 20% of Dr. Kevin's patients are on Medicare. It is found that a randomly selected patient is a Medicare patient. Find the probability that he/she is Dr. Kevin's patient.

Solution:Let

= Person is Dr. Kevin's patient

= Person is a Medicare patient

and let

= Person is Dr. Anita's patient

= Person is Dr. David's patient

We are given that , , while , and . We wish to find . Using ( 3.6.4) with , we have