91Ó°ÊÓ

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 ofi(i=1,2,3) is the posterior probability that the item was produced by machineMilarger than the prior probability that the item was producedby machineMi?

If A and B are disjoint events and Pr(B) > 0, what is the value of Pr(A|B)?

If S is the sample space of an experiment and A is any event in that space, what is the value of \({\bf{Pr}}\left( {{\bf{A}}\left| {\bf{S}} \right.} \right)\)?

Suppose that in Example 2.3.4 in this section, the item selected at random from the entire lot is found to be non-defective. Determine the posterior probability that it was produced by machine\({{\bf{M}}_{\bf{2}}}\).

Each time a shopper purchases a tube of toothpaste, he chooses either brand A or brand B. Suppose that for each purchase after the first, the probability is 1/3 that he will choose the same brand that he chose on his preceding purchase and the probability is 2/3 that he will switch brands. If he is equally likely to choose either brand A or brand B on his first purchase, what is the probability that both his first and second purchases will be brand A and both his third and fourth purchases will be brand B?

Suppose a person rolls two balanced dice twice in succession. Determine the probability that the sum of the two numbers that appear on each of the three rolls will be 7.

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, 65percent 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?

A box contains r red balls and b blue balls. One ball is selected at random and its color is observed. The ball is then returned to the box and k additional balls of the same color are also put into the box. A second ball is then selected at random, its color is observed, and it is returned to the box together with k additional balls of the same color. Each time another ball is selected, the process is repeated. If four balls are selected, what is the probability that the first three balls will be red and the fourth ball will be blue?

Suppose that in 10 rolls of a balanced die, the number 6 appeared exactly three times. What is the probability that the first three rolls each yielded the number 6?