Homework 2(1).pdf

Problem Set 2Due on 2018/3/161.Each time a shopper purchases a tube of toothpaste,he choose either brand A or brandB.Suppose that for each purchase after the first,the probability is13that he will choosethe same brand that he chose on his preceding purchase and the probability is23thathe will switch brands.If he is equally likely to choose either brand A or brand B onhis first purchase,what is the probability that both his first and second purchases willbe brand A and both his third and fourth purchases will be brand B?2.In the World Series of baseball,two teams A and B play a sequence of games againsteach other,and the first team that wins a total of four games becomes the winner ofthe World series.If the probability that team A will win any particular game againstteam B is 1/3,what is the probability that team A will win the World series?3.Suppose that A1,A2,.,Akare a sequence of k independent events.Let B1,B2,.,Bkbe another sequence of k events such that for each value of j(j=1,2,.,k),eitherBj=Ajor Bj=ACj.Prove that B1,B2,.,Bkare also independent events.Hint:Use an induction argument based on the number of events Bjfor which Bj=ACj.4.Suppose that when a machine is adjusted properly,50 percent of the items producedby 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 andthat,under these conditions 25 percent of the items produced by it are of high quality1and 75 percent are of medium quality.1)suppose that five items produced by the machine at a certain time are selected atrandom and inspected.If four of these items are of high quality and one item is ofmedium quality.What is the probability that the machine was adjusted properly atthat time?2)Suppose that one additional item,which was produced by the machine at the sametime as the other five items,is selected and found to be of medium quality.What isthe new posterior probability that the machine was adjusted properly?2