Consider a machine that produces items in sequence. Under

Suppose that k events B1,...., Bk form a partition of the sample space S. For i=1,...,k, let Pr(Bi) denote the prior probability of Bi. Also for each event A such that Pr(A) > 0, let (Bi|A) denote the posterior probability of Bi given that the event A has occured. Prove that if Pr(B1|A)<Pr(Bi|A)>Pr(Bi) for at least one value of i(i=2,...,k).

Consider again the conditions of Example 2.3.4 in this section, in which an item was selected at random from a batch of manufactured items and was found to be defective. For which values of i (i = 1, 2, 3) is the posterior probability that the item was produced by machine Mi larger than the prior probability that the item was produced by machine Mi?

Suppose that in Example 2.3.4 in this section, the item selected at random from the entire lot is found to be nondefective. Determine the posterior probability that it was produced by machine M2.

A new test has been devised for detecting a particular type of cancer. If the test is applied to a person who has this type of cancer, the probability that the person will have a positive reaction is 0.95 and the probability that the person will have a negative reaction is 0.05. If the test is applied to a person who does not have this type of cancer, the probability that the person will have a positive reaction is 0.05 and the probability that the person will have a negative reaction is 0.95. Suppose that in the general population, one person out of every 100,000 people has this type of cancer. If a person selected at random has a positive reaction to the test, what is the probability that he has this type of cancer?

In a certain city, 30 percent of the people are Conservatives, 50 percent are Liberals, and 20 percent are Independents. Records show that in a particular election, 65 percent of the Conservatives voted, 82 percent of the Liberals voted, and 50 percent of the Independents voted. If a person in the city is selected at random and it is learned that she did not vote in the last election, what is the probability that she is a Liberal?

Suppose that when a machine is adjusted properly, 50 percent of the items produced by it are of high quality and the other 50 percent are of medium quality. Suppose, however, that the machine is improperly adjusted during 10 percent of the time and that, under these conditions, 25 percent of the items produced by it are of high quality and 75 percent are of medium quality. a. Suppose that five items produced by the machine at a certain time are selected at random and inspected. If four of these items are of high quality and one item is of medium quality, what is the probability that the machine was adjusted properly at that time? b. Suppose that one additional item, which was produced by the machine at the same time as the other five items, is selected and found to be of medium quality. What is the new posterior probability that the machine was adjusted properly?

Suppose that a box contains five coins and that for each coin there is a different probability that a head will be obtained when the coin is tossed. Let pi denote the probability of a head when the ith coin is tossed (i =1,..., 5), and suppose that p1 = 0, p2 = 1/4, p3 = 1/2, p4 = 3/4, and p5 = 1. a. Suppose that one coin is selected at random from the box and when it is tossed once, a head is obtained. What is the posterior probability that the ith coin was selected (i = 1,..., 5)? b. If the same coin were tossed again, what would be the probability of obtaining another head? c. If a tail had been obtained on the first toss of the selected coin and the same coin were tossed again, what would be the probability of obtaining a head on the second toss?

Consider again the box containing the five different coins described in Exercise 7. Suppose that one coin is selected at random from the box and is tossed repeatedly until a head is obtained. a. If the first head is obtained on the fourth toss, what is the posterior probability that the ith coin was selected (i = 1,..., 5)? b. If we continue to toss the same coin until another head is obtained, what is the probability that exactly three additional tosses will be required?

Consider again the conditions of Exercise 14 in Sec. 2.1. Suppose that several parts will be observed and that the different parts are conditionally independent given each of the three states of repair of the machine. If seven parts are observed and exactly one is defective, compute the posterior probabilities of the three states of repair.

Consider again the conditions of Example 2.3.5, in which the phenotype of an individual was observed and found to be the dominant trait. For which values of i (i = 1,..., 6) is the posterior probability that the parents have the genotypes of event Bi smaller than the prior probability that the parents have the genotyes of event Bi?

Suppose that in Example 2.3.5 the observed individual has the recessive trait. Determine the posterior probability that the parents have the genotypes of event B4.

In the clinical trial in Examples 2.3.7 and 2.3.8, suppose that we have only observed the first five patients and three of the five had been successes. Use the two different sets of prior probabilities from Examples 2.3.7 and 2.3.8 to calculate two sets of posterior probabilities. Are these two sets of posterior probabilities as close to each other as were the two in Examples 2.3.7 and 2.3.8? Why or why not?

Suppose that a box contains one fair coin and one coin with a head on each side. Suppose that a coin is drawn at random from this box and that we begin to flip the coin. In Eqs. (2.3.4) and (2.3.5), we computed the conditional probability that the coin was fair given that the first two flips both produce heads.

Consider again the conditions of Exercise 23 in Sec. 2.2. Assume that Pr(B) = 0.4. Let A be the event that exactly 8 out of 11 programs compiled. Compute the conditional probability of B given A.

Use the prior probabilities in Example 2.3.8 for the events B1,...,B11. Let E1 be the event that the first patient is a success. Compute the probability of E1 and explain why it is so much less than the value computed in Example 2.3.7.

Consider a machine that produces items in sequence. Under normal operating conditions, the items are independent with probability 0.01 of being defective. However, it is possible for the machine to develop a memory in the following sense: After each defective item, and independent of anything that happened earlier, the probability that the next item is defective is 2/5. After each nondefective item, and independent of anything that happened earlier, the probability that the next item is defective is 1/165. Assume that the machine is either operating normally for the whole time we observe or has a memory for the whole time that we observe. Let B be the event that the machine is operating normally, and assume that Pr(B) = 2/3. Let Di be the event that the ith item inspected is defective. Assume that D1 is independent of B. a. Prove that Pr(Di) = 0.01 for all i. Hint: Use induction. b. Assume that we observe the first six items and the event that occurs is E = Dc 1 Dc 2 D3 D4 Dc 5 Dc 6. That is, the third and fourth items are defective, but the other four are not. Compute Pr(B|D).