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An interesting version of the conditional probability formula (3.5.1) comes from the work of the Reverend Thomas Bayes. Bayes's result was published posthumously in 1763.

Suppose that E and F are two events in a sample space S and such that . From the Venn diagram in Figure 3.6.1, we can see that the events and are disjoint and that their union is E, so that

(3.6.1)

Using the rule given by (3.5.6), we can rewrite equation (3.6.1) as

(3.6.2)

Figure 3.6.1 Venn diagram showing events and .

We can rewrite (3.5.1) in the form

(3.6.3)

The rule provided by (3.6.3) is known as Bayes's theorem for two events E and F; the probabilities and are sometimes referred to as the prior probabilities of events F and , respectively (note that ). The conditional probability as given by Bayes's theorem (3.6.3), is referred to as the posterior probability of F, given that the event E has occurred. An interpretation of (3.6.3) is that, posterior to observing that the event E has occurred, the probability of F changes from , the prior probability, to , the posterior probability.

Example 3.6.1 (Bayes's theorem in action) The Gimmick TV model A uses a printed circuit, and the company has a routine method for diagnosing defects in the circuitry when a set fails. Over the years, the experience with this routine diagnostic method yields the following pertinent information: the probability that a set that fails due to printed circuit defects (PCD) is correctly diagnosed as failing because of PCD is 80%. The probability that a set that fails due to causes other than PCD has been diagnosed incorrectly as failing because of PCD is 30%. Experience with printed circuits further shows that about 25% of all model A failures are due to PCD. Find the probability that the model A set's failure is due to PCD, given that it has been diagnosed as being due to PCD.

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

We note that the posterior probability of , given E, is 0.475, while the prior probability of was . We sometimes say that the prior information about has been updated in light of the information that E occurred to the posterior probability of , given E, through Bayes's theorem.