MidtermI.310.problembank

Midterm I-Problem Bank1.b black balls and r red balls are in a hat.(a)Draw5of them with replacement.What is the probability of getting exactly3blacks?(b)Draw5of them without replacement.What is the probability of getting exactly3blacks?(c)Draw5of them without replacement.What is the probability of getting exactly3blacks,giventhat the second one is red?2.(a)What does it mean for two events to be independent?(b)If E,E are independent events,show that P(E|E )=P(E).(c)LetΩ={(x,y)0≤x,y≤1,the points inside the unit square.Let E be the event x<1,andE the event x+y<1.Are these events independent?Why or why not?3.Suppose a fair die is rolled4times.(a)What is the probability that a6will be rolled at least tiwce?(b)What is the probability that a6will be rolled at least twice,and afive will be rolled at leasttwice?(c)What is the probability that a6will be rolled at least twice,or afive will be rolled at leasttwice,or both?4.Let X be a discrete random variable on a probability space,Ω.Are the events E x={i∈Ω|X(i)=x}disjoint?5.What is the probability that a random poker hand will be aflush(five cards of the same suit)?6.An elevator takes on8people and stops at15floors.We assume that people in the elevator areequally likely to get offat anyfloor.(a)Compute the probability that they all get offat differentfloors.(b)Compute the probability that two of them get offat the samefloor,and everybody else getsoffat a differentfloor.7.A child hasfive identical red blocks,four identical green blocks,and one yellow block.Supposethe child stacks these ten blocks randomly on top of one another(a)In how many distinguishable ways can this be done?(b)Compute the probability that blocks of like color end up stacked together(i.e.,the5red blocksare stacked consecutively and the4green blocks are stacked consecutively)?8.A balanced die is rolled12times.Given that a3appears at least once,what is the probabilitythat it occurs exactly4times?9.In a twenty-question true-false exam,find the probability that a students gets70percent correctby guessing.10.A box contains b black balls and r red balls.One of the balls is drawn at random.It is thenreturned to the box together with c additional balls of the same colour.A second ball is then drawn at random.(a)What is the probability that the second ball chosen is red?(b)What is the probability that thefirst ball drawn was black if the second ball drawn is red?11.Your sock drawer contains8grey socks,10brown socks,and12blue socks.(a)If you grab4socks(at once)from the drawer randomly without looking,what is the proba-bility you grabbed exactly2blue socks?(b)You randomly grab4socks from the drawer and note that one of the socks is blue.What isthe probability that all4socks are blue?12.Someone has the following habit on shopping an entertaining item.The item is bought withprobability1/2if it is not on sale,but it is bought with probability2/3if the item is on sale.Assume that the probability that the item is on sale is1/4.If the item was bought,what is the probability that it was on sale?13.The members of a consultingfirm rent cars from3rental companies:60percent from CompanyA,30percent from Company B,and10percent from Company C.The past statistics show that9 percent of the cars from Company A need a tune-up,20percent of the cars from Company B needa tune-up,and6percent of the cars from Company C need a tune-up.If a rental car delivered tothefirm needs a tune-up,what is the probability that it came from Company B?14.A group of5people are interviewed in order to determine the month in which they were born.Assume for simplicity that all months are equally probable.Compute the probabilities of the following events:(a)All5people have different birth-months.(b)One pair of people has the same birth-month,a second pair of people has the same birthmonth(but a different month from thefirst pair),and thefifth person has a different birth month from the rest of the group.15.Consider a succession of rolls of a fair die.(a)Define the random variable X to be the number of rolls at which6appears for thefirst time.pute P(X=4).pute P(X=even).pute P(X>5|X=even).(b)Define the random variable Y to be the number of rolls at which6appears for the4th time.Compute P(Y=20).。