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Stat 111 Collected by Awatef al mutairi STAT 111 Chapter Zero Question1 Suppose two dice are tossed and the numbers on the upper faces are observed. Let S denote the set of all possible pairs that can be observed. Define the following subsets of S. A: the number on the second die is even. B: the sum of the two numbers is even. C: at least one number in the pair is odd. List the elements in A,CC,A∩B ,A∩BC,AC∪B, and AC∩C. Answer1 S = { (1,1),(1,2),(1,3),(1,4),(1,5), (1,6),(2,1),(2,2)….(6,6). N(S) = 6X6 = 36. A = {(1,2),(2,2),(3,2),(4,2),(5,2),(6,2), (1,4),(2,4),(3,4),(4,4),(5,4),(6,4),(1,6),(2,6), (3,6),(4,6),(5,6),(6,6). B = {(2,2),(4,2),(6,2),(2,4),(4,4),(6,4),(2,6),(4,6),(6,6). C = {(1,1),(1,2),(1,3),(1,4),(1,5),(2,1),(2,3),(2,5),(3,1), (3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,5),(5,1), (5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,3),(6,5). Cont..Answer1 CC =S – C ={(2,2),(2,4),(2,6),(4,2),(4,4),(4,6), (6,2),(6,4),(6,6)} A∩B = B A∩ BC = A – B ={(1,2),(3,2),(5,2),(1,4),(3,4),(5,4),(1,6), (3,6),(5,6)} AC = {(1,3),(2,3),(3,3),(4,3),(5,3),(6,3),(1,5),(2,5),(3,5), (4,5),(5,5),(6,5),(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)} AC ∪B ={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(4,2),(2,2),(6,2), (1,3),(2,3),(3,3),(4,3),(5,3),(6,3),(2,4),(4,4),(6,4),(1,5), (2,5),(3,5),(4,5),(5,5),(6,5),(2,6),(4,6),(6,6)} AC ∩C= AC Question2 Suppose a family contains two children of different ages, and we are interested in the sex of these children. Let F denote that a child is female and M that the child is male, and let a pair such as FM denote that the older child is female and the younger male. There are four elements in the set S of possible observation. S= { FF ,FM,MF,MM} Let A denote the subset of possibilities containing no males, B the subset containing two males, and C the subset containing at least one male. List the elements of A , B ,C ,A ∩ B ,A ∪ B , A ∩ C ,A∪ C ,B ∩ C and C ∩ BC Answer2 A={FF}, B={MM}, C={ FM,MF,MM} A∩B=φ A ∪ B ={FF,MM} A∩C=φ A∪C=S B ∩ C ={MM} C ∩ BC = C- B ={FM,MF} Question3 A total of 36 members of club play tennis, 28 play squash, and 18 play badminton. Furthermore, 22 of the members play tennis and squash, 12 play both tennis and badminton, 9 play both squash and badminton, and 4 play all three sport. How many members of this club play at least one of these sports. Answer3 N(T)=36 N(S)=28 N(b)=18 N(T ∩ S)= 22 N(S ∩B)=9 N(T ∩S ∩B)=4 N(T ∩B)=12 N(T ∩S ∩B)=N(T)+N(S)+N(B)-N(T ∩S)-N(T ∩B)N(S ∩B)+N(T ∩S ∩B)=36+28+18-22-9-12+4=43 Question4 In a survey carried out in a school snack shop. The following results were obtained. Of 100 boys questioned, 78 linked sweets, 74 ice cream, 53cake, 57linked both sweets an ice cream, 46 liked both sweets and cake while only 31 boys liked all three. If all the boys interviewed linked at least one item, draw a Venn diagram to illustrate the results. How many boys both ice cream and cakes? Answer4 N(S)=100, 6 26 N(SW)=78, y 31 N(I)=74, 15 x N( C) =53, N(SW ∩I)=57, z N(SW∩C)=4, N(SW∩I ∩C)=31. N(S)=N(SW ∪I ∪C)=N(SW+N(I)+N( C ) –N(SW ∩C)N(SW ∩C)-N(I ∩C)+N(SW ∩I ∩C)=100. N(I ∩C)=33. STAT 111 Chapter One Question1 Find the number of ways in which 6 teacher can be assigned to 4 section of an introductory psychology course if no teacher is assigned to more than one section? Answer1 6P 4 = 6 x 5 x 4 x 3= 360 Question2 A toy manufacture makes a wooden toy in tow parts; the top part may be colored red , white, or blue and the bottom part brown, orange, yellow or green. How many differently colored toys can be produces? Answer2 3 X 4 = 12 Question3 How many different signals may be formed by displaying 6 flags in row if there are 3 blue flags, 2 red flags, and 1 white flags available if all flags of the same color are identical? Answer3 6! 60 3!2!1! Question4 Proteins in living cells are composed of 20 different kinds of amino acids. Most proteins consist of several hundred amino acids in along chin structure. How many different proteins of length 100 can be constructed? Answer4 2 20 20 20..... 20 (20) 100tim e Question5 Find the number of subsets of a set X containing n elements? Answer5 Either the subsets containing no element n , 0 n n n 1 elements1 2 elements 2 , n elements n 2 n n n ........ (1 1) 0 1 n Question6 In how many can a person gathering data for a market research organization interview 3 of the 20 families living in a certain apartment house? Answer6 20 20 c3 1140 3 Question7 Suppose that someone wants to go by bus, by train, or by place on a week's vacation to one of the five East North states. Find the number of different ways in which this can be done? Answer7 3 X 5 = 15 Question8 a)How many ways can one make a true –false test consisting of 20 questions? In how many ways can they be marked true or false so that b) 7 are right and 13 are wrong? c) at least 17 are right? Answer8 a) 2x2x2x2Xx….x2 = 20 = 1048,576 b) 20 20 20 77,520 7 7,13 c7 20 20 20 c10 184,756 c) 10,10 10 d) c 20 17 c18 c19 c20 1351 20 20 20 Question9 How many license plates may be formed beginning with 2 different letters of the Arabic alphabet following by 4 different digits? How many be formed if repetition of letters and digits is allowed? Answer9 Without repetition : 28x27x10x9x8x7= 3,810,240. With repetition : 28x28x10x10x10x10=7,840,000. Question10 A test has 10 true-false questions and 6 multiple-choice questions with 5 possible choices for each. How many possible sets of answers are there? Answer10 2 5 10 6 16,000 ,000 Question11 How many different 6-digits numbers may be formed using the digits from the 4,4,5,6,6,6. Answer11 6! 60 2!1!3! Question12 A telephone company in a certain area has all telephone numbers prefixed by either 465,475, or 482, followed by 4 digits. How many different telephone numbers are possible in this area? How many if repetition in each number is allowed? Answer12 With repetition 3 104 30,000 Without repetition 3 7 6 5 4 2,520 Question13 A student may select one of 3 English classes, one of 2 mathematics classes ,and one of 2 history classes for his program. In how many ways may he build his program? Answer13 c c c 3 2 3 1 1 1 3 2 2 12 Question14 A person has 8 friends, of whom 5 will be invited to a party. a)How many choices are there if 2 of the friends are feuding and will not attend together? b) How many choice if 2 of the friend will not attend together? Answer14 2 6 2 6 a) 6 2 15 36 0 5 1 4 2 6 2 6 b) 26 2 3 0 5 Question15 How many ways can we arrange the letters in the word cold? Answer15 4! = 24 Question16 How many ways may 5 people be seated in a 5-passenger vehicle if one of two people must drive? Answer16 2 4! 1 The drive seat must be filed in 2 ways, after that, the remaining 4 can be arranged in 4! Question17 A student is to answer 7 out of 10 questions in an examination. How many choices have she? How many if she must answer at least 3 of the first questions? Answer17 a) c7 120 10 b) c3 c4 c4 c3 c5 c2 110 5 5 5 5 5 5 Question18 In how many ways can 2 oaks, 3 pines, and 2 maples be arranged in a straight line if one does not distinguish between trees of the same kind? Answer18 7! 210 2!3!2! Question19 Ten persons have organized a club. How many different committees consisting of 3 persons may be formed from these 10 people? In how many ways may a president a secretary and treasure be selected? Answer19 a) c7 120 10 10 b) p3 720 Question20 If eight persons are having dinner together, in how many different ways can three order chicken, four order steak, and one order lobster? Answer 20 8! 280 3!4! Question21 a)How many distinct permutations are there of the letters in the word statistics? b) How many of these begin and end with letters? Answer 21 1) 10! 50,400 3!3!2! 8! 2) 3,360 3!2! Question22 In how many ways can 7 books be arranged on a shelf if a) Any arrangement is possible, b) 3 particular books must always stand together, c) 2 particular books must occupy the ends? Answer 22 1) 7! 5,040 2) 3!5! 720 3) 2 5! 240 Question23 How many numbers consisting of five digits each can be made from the digits 1,2,3,…,9 if a) the numbers must be odd. b) the first two of each number are even? Answer 23 a) 5 6 7 8 5 75,600 b) 6 7 3 4 504 Question24 In how many ways can 3 women and 3 children be seated at a round table if a) No restriction is imposed. b) Two particular children must not sit together. c) Each child is to be seated between two women? Answer 24 a) 5! 120 b) 5!2 4! 72 c) (3 1)!3! 12 Question25 From a group of 8 women and 6 children a committee consisting of 3 women and 3 children is to be formed. How many different committees are possible if a)2 of the children refuse to serve together? b) 2 of the women refuse to serve together? c) 1 child and 1 woman reuse to serve together? Answer 25 a) c c c c c c 2 4 8 2 4 8 0 3 3 1 2 3 896 6 2 6 2 6 b) 1000 3 0 3 1 2 2 7 5 7 5 7 5 c) 1 1 910 0 3 3 3 2 2 3 Question26 There are 3 paths leading from A to B and 2 paths leading from B to C. In how many ways may one make the round trip from A to C and back without retracing any paths? Answer 26 ABC 3x2x1x2=12 Question27 A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue (blocks of the same color are identical). If the child puts the blocks in a line, how many arrangements are possible? Answer 27 12! 6!4!1!1! Question28 A robot must pick up ten items from the floor. In how many ways can the task be performed? If the items are divided into two sub collections, the first containing six items and the second containing four items, and if, once an item from a sub collection is selected, the robot is programmed to pick up the remaining items in that sub collection before proceeding to the order sub collection, in how many ways can the task be performed? Answer 28 a) b) 10! = 3628,800 6! 4! 2 = 34,560 Question29 In a class of 10 students, how many ways can the students be seated so that there are 1 student in each of the five rows and 5 students in the last row? Answer 29 10 30,240 .....5 Question30 a small community consists of 10 women, each of whom has 3 children. If one woman and one of her children are to be chosen as mother and child of the year, how many different choices are possible? Answer 30 c c 10 3 1 1 30 Question31 How many functions defined on n points are possible if each functional value is either 0 or 1 ? Answer 31 2 2 2 ... 2 2n n _ tim e Question32 How many even three digit number can be formed from the digits 1,2,5,6 and 9 if each digit can be used only once. Answer 32 3 4 2 24 Question33 How many sample points are in the sample space when a pair of dice is thrown once? Answer 33 6X6=36 S={(1,1),(1,2),…..(1,6)} Question34 Calculate the number of permutations of the letters a, b, c, d had taken two at a time? Answer 34 p 4 2 12 Question35 four names are drawn from the 24 members of a club, for the offices of president, vice - president , treasurer, and secretary. In how many differently ways can this be done? Answer 35 24 p 4 255,024 Question36 In how many ways may 3 books be placed next to each other on a shelf? Answer 36 3! 6 Question37 Four different mathematics books, 6 different physics books, and 2 different arrangements are possible if a)The books in each in each particular subject must all stand together, b) Only the mathematics books must stand together? Answer 37 no rest = 12! a) 3! (4! 6! 2! ) = 207360 b) 4! 9! = 8,709,120 Question38 Four women and four children to be seated in a row of chairs numbered 1 thought 8; a) How many total arrangements are possible? b) How many arrangements are possible if the women are required to sit in alternate chairs? c) How many arrangements are possible if the four women are considered indistinguishable and the four children are considered indistinguishable? d) How many arrangements are possible if the four women are considered indistinguishable but the four children are considered indistinguishable Answer 38 a) 8! = 40, 320 b) 2! (4! 4!) + 1,152 c) 8! 70 4!4! 8! 1680 4! d) Question39 From 4 chemists and 3 physicists find the number of committees that be formed consisting of 2 chemists and 1 physicist? Answer 39 4 3 18 2 1 Question40 Form a group of 5 teachers and 7 students, how different committees consisting of 2 teachers and 3 students can be formed? What if 2 of the students refuse to serve on the committee together? Answer 40 a) c2 c3 350 2 5 5 2 5 5 b) c0 c3 c2 c1 c2 c2 300 5 7 or c c c c c 300 5 2 5 2 5 2 0 3 1 2 Question41 From 5 statisticians and 6 economists a committee consisting of 3 statisticians and 2 economists is to be formed. How many different committees can be formed if a) no restrictions are imposed, b) Two particular statisticians must be on the committee. c) One particular economist cannot be on the committee. Answer 41 a) b) c) 150 5 cc 6 3 2 2 3 6 2 1 2 5 5 3 2 ccc cc 45 100 Question42 A shipment of 10 television sets includes three that are defective. In how many ways can a hotel purchase four of these and receive at least two at least two of the defective sets? Answer 42 c c c c 3 7 3 7 2 2 3 1 70 Question43 In how many ways can a set of four objects be partitioned into three subsets containing, respectively, 2,1 and 1 of the objects? Answer 43 4 12 2,1,1 Question44 In how many ways can seven scientists be assigned to one triple and two double hotel rooms? Answer 44 7 210 3,2,2 Question45 a)How many license plates are there if the first three place are form the Arabic letters and the last three are numbers? b)If each number can be used any one time? Answer 45 a)28 28 28101010 21,952,000 b)28 27 2610 9 8 14,152,320 Question46 If a travel agency offers special weekend trip to 12 different cities, by air, rail or bus. In how many different ways can such a trip be arranged? Answer 46 12 3 36 Question47 In an experiment consists of throwing a die and then drawing a letter at random from the English alphabet, how many points are possible? Answer 47 6 26 156 Question48 In a medical study are patients are classified in 8 way according to whether they have blood type AB +, AB - , A+ , A - ,B +,B –, O +, O - , also according to whether their blood pressure is low, normal, or high. Find the number of ways in which a patient can be classified? Answer 48 8 3 24 Question49 If 4 teachers, 3 engineers, and 3 doctors are to be seated in a row, how many seating arrangement are possible when people of the same jobs must sit next to each other? Answer 49 3!(4!3!3!) 5,184 Question50 Find the number of ways in which one a, three B's, two C's and one F can be distributed among seven students taking a course in statistics? Answer 50 7 420 1,3,2,1 Question51 10 Math student, 5 chemistry students and 5 geo;ogy students. In how many different ways we can select 6 such that a)Any 6; b)2 from chemistry; c)Number of Math students range from 2 to 4? Answer 51 a) c6 38,760 20 b) c2 c4 13,650 5 10 c ) c2 15 c c c c c 10 10 10 10 10 4 3 3 4 2 33,300 STAT 111 Chapter Two Question1 The following data were given in a study of a group of 1000 subscribers to a certain magazine. In reference to sex, marital status, and education, there were 312 males, 470 married persons , 525 college graduates, 42 male college graduates, 147 married college graduates, 86 married males, and 25 married male college graduates. Show that the numbers reported in the study must be incorrect. Answer 1 M Male Mr Married G College P (M) 312 P (MR) .470 P (G) .525 P (M G) .042 P (Mr G) .147 P (M Mr G) .25 P (M Mr G) 1.057 1 Question2 The mathematics department consists of full professors, 15 associate professors, and 35 assistant professors; a committee of 6 is selected at random from the faculty of the department. Find the probability that all the members of the committee are assistant professors. Find also the probability that the committee of 6 is composed of 2 full professors, 3 associate professors, and 1 assistant professor. Answer 2 75 n( s ) 6 35 6 p ( A) .00806 75 6 2515 35 2 3 1 p ( A) .029 75 6 Question3 Let P be a probability measure such that P(A)=1/3 , p(b)= 1/2 , P(A∪B)= 2/3 . Find P(A∩B), P(A∩BC), P(AC∪ BC), P(AC∪ B). Answer 3 P (A B) 1/6 P (A Bc ) 1/6 P (Ac Bc ) 5/6 P (Ac B) 5/6 A AC B 1/6 2/6 1/2 BC 1/6 2/6 1/2 1/3 2/3 1 Question4 If 3 books are picked at random from a shelf containing 5 mathematics, 3 books of statistics, and a chemistry, what is the probability that a) the chemistry is selected b)2 mathematics and 1 book of static are selected Answer 4 1 8 1 2 a) .33 9 3 5 3 2 1 b) .3571 9 3 Question5 A system containing two components A and B is wired in such a way that it will work if either component works. If it is known from previous experimentation that the probability of A working is 0.9, that of B working is 0.8, and the probability that both work is 0.72, determine the probability that the system will work? Answer 5 p( A) 0.9 p( B) 0.8 p( A B) 0.72 p (The system will work)= p( A B) P( A) P( B) P( A B) .9 .8 .72 .98 Question6 Let A and B be events with P(A)=1/2 , P(A∪B)= 3/4,P(Bc)=5/8 . Find P(A∩B), P(Ac ∪ Bc) c c c ,P(A ∩ B ),and P(B ∩A ). A B Bc Ac Answer 6 P (A B) 1/8 P (Ac B c ) 1/4 P (Ac Bc ) 7/8 P (B A c ) 2/8 Question7 A pair of fair dice is tossed. Find the probability that the maximum of the two numbers is greater than 4? Answer 7 20 p .5556 30 Question8 Of 120 students, 60 are studying French, 50 are studying Spanish, and 20 are studying French and Spanish. If a student is chosen at random, Find the probability that the student? a) is studying French or Spanish. b)is studying neither French nor Spanish. c)is studying exactly of them. Answer8 n(S) 120,n(F) 60,n(SP ) 50,n(F SP ) 20 60 50 P (F) , P (SP ) 120 120 20 1 40 C P (F SP ) , P( F SP ) 120 6 120 30 C P( F SP) 120 a) p( F SP) .75 b) P( F C SPC ) 1 P( F SP) .. c) P( F SPC ) P( F C SP) .5833 Question9 A committee of 5 is to be selected from a group of 6 teachers and 9 students. If the selection is made randomly, what is the probability that the committee consists of teachers and 2 students? Answer9 6 9 3 2 .2398 15 5 Question10 A jar contains 3 red, 2 green,4blue, and 2 white marbles. Four marbles are selected at random without replacement from this jar. What is the probability of drawing 2 red, a blue, and a white marble? Answer10 3 2 4 2 2 0 1 1 .0727 11 4 Question11 If 2 balls are randomly drawn from a bowl containing 6 white and 5 black balls, what is the probability that one of the drawn balls is white and the other black? Answer11 6 5 1 1 .5455 11 2 Question12 If the probability that a student A will fail a certain statistics examination is 0.5, the probability that student B will fail the examination is 0.2, and the probability that both student A and student B will fail the examination 0.1, what is the probability that at least one of these two student will fail the examination? Answer12 P(A) .5 P(B) .2 P(A B) .1 P(A B) P(A) P(B)- P(A B) .6 Question13 An experiment consists of tossing a die and then flipping a coin once if the number on the die is even. If the number on the die is odd, the coin is flipped twice. List the element of the sample space S? Answer13 S {(2,H),(2,T ),(4,H),(6,H)} {(1,H, H), (1,H, T ),(1,T ,H),(1,T ,T ),.} Question14 If the probability are, respectively, 0.09, 0.15, 0.21, and 0.23, that a person purchasing a new automobile will choose the color green, white, red, or blue, what is the probability that a given buyer will purchase a new automobile that comes in one of those colors? Answer14 P(g w r b ) p(g) p(w) p( r ) p(b) .68 (mutuallyexclusive) [ or at least one of thosecolures] Question15 A die is loaded in such a way that the probability of any particular face's showing is directly proportional to the number on that face. What is the probability that an even number appears? Answer15 P (S) w 1 2w 3w 4w 5w 6w 1 21w 1 1 w 21 12 P (E) P (2) P (4) P (6) 21 Question16 A jar contains 12 marbles, 2 of which are red, 2 green, 4 blue, and 4 white. A marble is selected at random from the jar. What is the probability that it is blue? Answer16 4 1 4 12 12 1 Question17 It is known that a patient will respond to a treatment of a particular disease with probability equal to 0.9. If there patients are treated in an independent manner, find the probability that at least one will respond? Answer17 P(at least one )= p( A1 A2 A3 ) 1 P( A1 A3 A3 ) 1 P( A A2 A3 ) 0.9 C 1 C C Question18 If A and B are independent events with P(A)=0.5, and P(B)=0.2, find the following a) P(A∪ B) b)P(Ac ∩ Bc ) c)P(Ac ∪ Bc ) Answer18 a) p( A) p( B) p( A) p( B) .6 b) p( AC ) p( BC ) .4( AC , BC are indep also) Question19 A mixture of candies contains 6 mint, 4 toffees, and 3 chocolates. If a person makes a random selection of one candies, Find the probability of getting a)a mint. b) a toffee or a chocolate. Answer19 6 1 6 a) .4615 or 13 13 1 Question20 A die is tossed 50 times. The following table gives the six numbers and their frequency of occurrence Number 1 2 3 4 5 6 Frequency 7 9 8 7 9 10 Find the relative frequency of the event a) a 4 appears. b) and odd numbers appears. c)a prime number appears. Answer20 07 a) .14 50 789 b) .48 50 989 c) .52 50 Question21 Three women and three children sit in a row. Find the probability that a) the 3 children sit together. b)the woman and children sit in alternate seats. Answer21 (3!4!) a) .2 6! 3!3!2! b) .1 6! Question22 Let A and B be events with P(A∪ B)=7/8 , P(A∩ B)= 1/4 ,P(AC)=5/8 . Find P(A),P(B), and P(A∩ BC). Answer22 3 P (A) 8 6 P (B) 8 A B BC 1 P (A B ) 8 c AC 2/8 4/8 6/8 1/8 1/8 1/8 3/8 5/8 1 Question23 A balanced die is tossed twice. If A is the event that an even number comes up on the first toss, B is the event that an even number comes up on the second toss and C is the event that both toss result in the same number, are the events A, B and C independents? Answer23 A {(1,2).(2,6),(4,1),....(4,6),(6,1),.....(6,6)} B {(1,2),...(6,2),(1,4)..(6,4),(1,6)....(6,6)} C {(1,1),(2,2),..(6,6)} n(A) 18,n(B) 18,n(S) 36,n( C ) 6 A B C {(2,2),(4,4),(6,6)} 3 P (A B C) .083 36 1818 6 P( A) P( B) P(C ) .041 36 P( A B C ) P( A) P( B) P(C ) .:not indep Question24 Three names to be selected from a list of seven names for use in a particular public opinion survey. Find the probability that the first on the list is selected for the survey? Answer24 1 6 1 2 .4286 7 3 Question25 A hat contains twenty white slips of paper numbered from 1 through 20, ten red slips of paper numbered from 1 through 10, forty yellow slips of paper numbered from 1 through 40, and ten blue slips of paper numbered from 1 through 10. If these 80 slips of paper are thoroughly shuffled so that each slip has the same probability of being drawn, find the probabilities of drawing a slip of paper which is a)blue or white . b) numbered 1,2,3,4 or 5. c) red or yellow and numbered 1,2,3,or4. d) numbered 5,15,25,or 35. e)white and numbered higher than 12 or yellow and numbered higher than 26. Answer25 a)(10 20)/80 .375 b)20/80 .25 c)8/80 .1 d)8/80 .1 e).275 Question26 Three students A,B and C are in a swimming race. A and B have the same probability of winning and each is twice as likely to win as C. Find the probability that B or C wins. Answer26 P(a) 2/5 P(b) 2/5 P(c) 1/5 P(c b) p(c ) p(b) - p(c b) 3/6 .6 Question27 Of 10 girls in a class, 3 have blue eyes. If two of the girls are chosen at random, what is the probability that a)both have blue eyes. b) Neither have blue eyes. c)at least one has blue eyes. Answer27 3 7 2 0 a) .06 10 2 3 7 0 2 b) .4667 10 2 3 7 3 7 1 1 2 0 c) .533 10 2 Question28 Consider families with two children. Let E be the event that a randomly chosen family has at most one girl, and F ,the event that the family has children of both sexes. Show that E and F are not independent. Answer 28 S {BB, GG, GB, BG} E {GB, BG, BB} F {GB, BG} P (E F) 2/4 P (E) P (E) 3/4 2/4 6 / 16 3/8 2/4 Question29 Find the probability of getting three heads in three (independent) tosses of a balanced coin. Answer 29 p(HHH,HHT ,HT H,T HH,T T H,T HT ,HT T ,T T T } 1 P (HHH) 8 Question30 Relays used in the construction of electric circuits function properly with probability 0.9. A assuming that the circuits operate independently, which of the designs in Figure 0-2 yields the higher probability that current will flow when the relays are activated? Answer 30 For design(1): =.9 For design(2): =.96639 Question31 Find the probability of getting at least one head in 5 tossed of a balanced coin? Answer 31 P(at least one head)=1-p(HHH)=.969 Question32 Among a shipment of 4 electrical components of types A,B,C and D, there are 3 of type A,4 of type B, 5 of type C, and 6 of type D. From this shipment 3 components are randomly selected. Find the probability that 1. all are of type C. 2. one of each of the type B,C,D 3. at least 2 of type B and nothing of types A,D? Answer 32 5 3 10 a) .0123 18 816 3 4 5 6 1 1 1 b) .1471 18 13 4 5 4 5 2 1 3 0 c) .041 18 3 Question33 An urn contains M white and N black balls. If a random sample of size r is chosen, what is the probability that it will contain exactly K white balls? What if M=K=1? Answer 33 M N N K r K r 1 or if M K 1 M N N 1 r r Question34 A single die is tossed. Find the probability of a 3 or 6 turning up? Answer 34 P(3or 6) p(3) p(6) 1/6 1/6 2/6 1/3 .33 Question35 If A and B are mutually exclusive events and P(A)=0.3,and P(B)=0.5, find 1. p ( AC ) 2. p ( A B) 3. p ( A C B ) Answer 35 1. p ( AC ) 0.7 2. p ( A B) p ( A) p( B) .8 3. p ( AC B) p ( B) .5 Question36 If three events A ,B , and C are independent, show that 1. A and B∩C are independent. 2. A and B ∪C are independent . 3. AC and B ∩CC are independent. 1 3 B A 2 4 1 3 A B 2 4 Answer 36 1)p(A ∩(B ∩C))=P(A ∩B ∩C)= P(A)x P(B)x P(C)=P(A)x P(B ∩C) 2)P(A ∩(B ∩C))=P((A ∩B) ∪ (A ∩C))=P(A).P(B ∪C) 3)P(AC ∩(B ∩CC )=P(AC )x P(B)x P(CC ) P(AC )P(B ∩CC ) If T,F independent => TC and FC independent STAT 111 Chapter Three Question1 Find P(A|B)if a) B is a student of A, b)A and B are mutually exclusive. Answer 1 P A B a) p( A B) 1 P( B) 0 b) p ( A B ) 0 P( B) Question2 A box contains three coins, one coin is fair, one coin is twoheaded, and one coin is weighted so that the probability of head appearing is 1/3 . A coin is selected at random and tossed. Find the probability that heads appears. Answer 2 P( H ) P( H I ) P( I ) P( H II ) P( II ) P( H III) P( III) 0.61111 Question3 An urn contains 3 red marbles and 7 white marbles. A marble is drawn from the urn and a marble of the other color is then put into the urn. A second marble is drawn from the urn. A) Find the probability that the second marble b)If both marbles were of the same color, what is the probability that they were both white? Answer 3 p(both white both same color) = p (both white same color) = p(same color) p (both white) p( (both red)+p(both white) = 7 6 10 10 0.875 p( R1 ) p( R2 R2 ) p( w1 ) p( w2 w2 ) Question4 The probability that three men hit a target are respectively , and each shoots once at the target (independent) . a) Find the probability that exactly one of them hits the target. b) If only one hit the target, what is the probability that it was the first men? Answer 4 a) p(exactly one man)= p(E) P( M 1 M 2 M 3 ) P( M 1 M 2M 3 ) P( M 1 M 2 M 3 ) C C C C 31 0.43 72 P( M 1 E ) C C p(M1 E) P(M1 M 2 M 3 ) 0.1935 P( E ) C C Question5 Only one in 1000 adults is affected with a rare disease for which a diagnostic test has been developed. The test is such that, when and individual actually has the disease, a positive result will occur 99% of the time, while an individual without the disease will show a positive test result only 2% of the time. If a randomly selected individual is tested and the result is positive, what is the probability that the individual has the disease? Answer 5 P(With ve) p(ve with) p( with) p(ve) p(ve) p(ve with) p( with) p(ve without) p( without) 0.99 0.001 0.02 0.999 0.02097 0.99 0.001 p( with ve) 0.0472 0.02097 Question6 The members of a consulting firm rent cars from three agencies; 60 percent from agency 1,30 percent from agency 2, and 10 percent from agency 3. If 9 percent of the cars 1 need a tune-up, 20 percent of the cars from agency 2 need a tune-up, and 6 percent of the cars from agency 3 need a tune-up , what is the probability that a rental car delivered to the firm will need a tuneup. Answer 6 P(I)=0.6 P(T | I)=0.09 P(II)=0.3 P(T| II)=O.2 P(III)=0.1 P(T| II)=0.06 P(T)=0.09x0.6+0.2x0.3+0.06x0.1=0.12 Question7 Each of 2 cabinets identical in appearance has 2 drawers. Cabinet A contains a silver coin in each drawer, and cabinet B contains a silver coin in one of its drawers and a gold coin in the other. A cabinet is randomly selected, one of its drawers is opened, and a silver coin is found. What is the probability that there is silver coin in the other drawer? Answer7 1 1 1 P( S ) 1 0.75 2 2 2 p( S A) P( A) P (A | S ) 0.6667 P( S ) Question8 Suppose we have 10 coin such that if the ith coin is flipped, head will appear with probability , i=1,…,10. When one of the coin is randomly selected and flipped, it shows a head. What is the conditional probability that it was the fifth coin? Answer8 P(5th| H) 0.09 P(H|) 0.55 Question9 The probability that a regularly scheduled flight departs on time is 0.83, the probability that it arrives on time is 0.82; and the probability that it departs and arrives on time is 0.78. Find the probability that a plane a)arrives on time given that it departed on time, and b) departed on time given that it has arrived on time. Answer9 P (D) 0.38 P (A) 0.83 P (DA) 0.78 a)P (A| D) 0.94 b)P (D| A) 0.95 Question10 An urn contains 10 white, 5 yellow, and 10 black marbles. A marble is chosen at random from the urn, and it is noted that it is not one of the black marbles. What is the probability that it is yellow? Answer10 P(Y| 5) 0.33 Question11 Suppose that a fair coin is tossed until a head appears for the first time. Determine the probability that exactly n tosses will be required. Answer11 1 1 1 1 P(T , T ,....T , H ) P(T ) XP(T ) X ...P( H ) X X 2 2 2 2 n Question12 A die is tossed. If the number is odd, what is the probability that it is prime? Answer12 P( prim eodd) p( pr odd) p(odd) 0.667 Question13 A box of fuses contains 20 fuses, of which 5 are defective. If 3 of the fuses are selected at random and removed from the box in succession without replacement, what is the probability that all there fuses are defective? Answer13 p( D1 D2 D3 ) 0.00877 Question14 A number is picked at random from { 1,2,3,…,100}. Given that the chosen number is divisible by 2 what is the probability it is divisible by 3 or 5? Answer14 Let A1 {the numbers which divisible by 2} A3 {the numbers which divisible by 3} A5 {the numbers which divisible by 5} P (A3? A5 | A2) 0.46 P (A3 A2) 0.16 P (A5 A2) 0.1 P (A2 A3 A5) 0.03 Question15 Three members of a private country club have been nominated for the office of president. The probability that the first will be elected is 0.3, the probability that the second member will be elected is 0.5, the probability that the third member will be elected is 0.2. If the first member is elected, the probability for an increase in membership fees is 0.8, If the second or third member be elected, the corresponding probabilities for an increase in fees are 0.1, and 0.4. What is the probability that there will be an increase in membership fees? If someone is considering joining the club but delays his or her decision for several weeks only to find out that the fees have been increased, what is the probability that the third member was elected president of the club? Answer15 a)0.37 b)0.2162 Question16 Suppose that 5 percent of men and 0.25 percent of women are color blind. A color blind person is chosen at random. What is the probability of this person's being male? Assume that there are an equal number of males and females? Answer16 P (m) 1/2 P (c| m) 0.05 P (c| f) 0.0025 P (c) 0.02625 P (m| c) 0.9524 Question17 In recent years much has been written about the possible link between cigarette smoking and lung cancer. Suppose that in a large medical centre ,,of all the smokers who were suspected of having lung cancer,90 percent of them did, while only 5 percent of the nonsmokers who were suspected of having lung cancer actually did. If the proportion of smokers is 0.45, what is the probability that a lung cancer patient who is selected by chance is a smoker? Answer17 P(c ) 0.4325 P(s| c) 0.9364 Question18 A laboratory blood test is 95 percent effective in detecting a certain disease . when it is, in fact, percent. However, the test also yields a 'false positive" result for 1 percent of the healthy persons tested. ( That is, if a healthy person is tested, then, with probability 0.01, the test result will imply he or she the disease.) If 0.5 person of the population actually has the disease, what is the probability a person has the disease given that the test result is positive? Answer18 P( ve) 0.0147 P(D| ve) 0.3231 Question19 In a certain town, 40% of the people have brown hair,25% have brown eyes, and 15% have both brown hair an brown eyes. A person is selected at random from the town If he has brown hair, what is the probability that he also has brown eyes; b) if he has brown eyes, what is the probability that he does not have brown hair c) what is the probability that he has neither brown hair nor brown eyes. Answer19 a)0.375 b)0.4 c)0.5 Question20 twenty percent of the employees of a company are college graduates. Of these, 75% are in supervisory position. Of those who did not attend college, 20% are in supervisory position. What is the probability that a randomly selected supervisor is a college graduate? Answer20 P (c1 ) 0.2 P (s1 | c1 ) 0.75 P (c2 ) 0.8 P (s1 | c 2 ) 0.2 P (s2 | c 2 ) 0.8 P (c1 | s) 0.4839 STAT 111 Chapter Four Question1 Two fair dice tossed, and let X denote the sum of the spots that appears on the top face. a)Obtain the probability distribution for X. b) Construct a graph for this probability distribution. Answer1 x 2 F(x) 1/36 3 4 5 6 2/36 3/36 4/36 5/36 7 8 9 10 6/36 5/36 4/36 3/36 11 12 2/36 1/36 Question2 A machine has been producing ball-point pens with a defective rate of 0.02. A sample of size 5 is taken from a carton of pens produced by the machine. Let Y represent the number of defective pens in the sample. What is the value of f(0),f(5),f(3.5)? Answer2 F(0) p ( y 0) p ( NNNNN ) (.98) 5 .904 F(5) p ( y 4) p ( DDDDD) (.02) 5 3.2 10-9 F(3.5) 0 Question3 A urn holds 5 white and 3 black marbles. If two marbles are drawn at random without replacement and Z denote the number of white marbles 1.Find the probability distribution for Z Graph the distribution. Answer3 z 0 1 2 sum 3/28 15/28 10/28 28/28 Question4 For each of the following functions determine if it is a probability function or not and sketch the distribution ( )= ( )= ( )= Answer4 a) Yes b) Yes c) No Question5 Two dice are rolled. Let X be the difference of the face numbers showing, the higher minus the lower, and 0 for ties. Find the probability mass function of X. Answer5 x 0 1 2 3 4 5 F(x) 6/36 10/36 8/36 6/36 4/36 2/36 Question6 6. The probability mass function of X is given by F(x)=k|x-2| x=-1,1,3,5 Find a)K b)The cumulative function of X and plot its graph c)P(X d)P(-0.4<X<4) e)P(X>1) Answer6 x2 1 a) k , f ( x ) 8 8 b) 5 c) 8 1 d) 4 1 e) 2 x -1 1 3 5 F(x) 3/8 1/8 1/8 3/8 Question7 Let S={(I,j):I,j be the set of all subsets of S. Let ((I,j))=for all 62 pairs(I,j) in S. Define X(I, j) i j 1 i j 6 Find Find the distribution F(x)for the random variable X Graph this distribution function. Answer7 F(x)= Question8 A box contains good defective items. If an item drawn is good. We assign the number 1 to the drawing; otherwise, the number 0. Find the p.m.f and the c.d.f? Answer8 F(0) p(x 0) 1 - p F(1) p(x 1) p Question9 Consider the toss of balanced dice. Let X denote the random variable representing the sum of the two faces,find a)The probability distribution of X. b) Draw the graph of p.m.f c)find P(X>7),P(5 X 9). Answer9 Question10 Suppose that a random variable X has a discrete distribution with the following p.m.f (x)= Find the value of the constant c. Answer10 c 1 1 1 1 1 x 1 f ( x) x 1 x cx 1 x c 2 3 ..... 2 2 2 2 2 c 1 c c 1 2 1 1 2 Question11 A fair coin is tossed until a head or five tails occurs. Find the p.m.f and the c.d.f Answer11 F(1) 1/4 1 F(2) f(T H) 4 1 F(3) F(T T H) 8 1 F(4) 16 1 1 2 1 F(5) F(T T T T H) F(T T T T T T) 32 32 32 16 Question12 Independent trials, consisting of the flipping of a coin having probability p of coming up heads, are continually performed until either a head occurs or a total of n flips in made. Let Y denote the number of times the coin is flipped. a)Find the probability distribution of Y. b)Graph the distribution. Answer12 F(1) P F(2) (1- P )P F(3) (1- P ) P . . . F(N - 1) (1- P ) P F(N) (1- P ) Question13 The distribution function of the random variable X is given by F(x)= 1.Graph F(X) 2.Determine the p.m.f 3.Compute P(X<3),P(X 2)and P(1 X 3),P(X>1.4) Answer13 x 1 2 3 4 F(x) 1/8 2/8 3/8 2/8 3 P(X 3) , 8 7 P(X 2) , 8 6 P(1 X 3) , 8 7 P(X 1.4) 8 Question14 a certain school gives only three letter grades in its course: p(pass),F(fail),and W (withdrew) . In computing grade points p =1 point, F=-1 point, and W=0 points . A student has enrolled in a mathematics course and a history course. Let x represent the total of the grade points that the student may earn in the two classes , Using the letter grade to represent the outcome in the course ,describe a sample space for the possible grades of the student Question14 1)List the outcomes in the events a){ x=0} b){x>-1} c){x > 1 } 2 d){x <-1} e){x1} f){-1 x <2} The student estimates the probability of passing any course is 0.7, of failing is 0.1,and of withdrawing is 0.2. Question14 2) Using the above figures, express in tabular form the probability function f(x) induced by X. A assume independence of grades between classes. 3)Graph the probability of X. Answer14 X-{-2,-1,0,1,2} S={FF.FP,FW,PP,PF,PW,WW,WF,WP} 1) a){ FP, PF, WW} b){X=0,1,2}={PW,WP,FP,PF,PP,WW} c){X=1,2}={PW,WP,PP} d){X=-2}={FF} e){X=0,-1,-2}={FP,PF,FW,WF,FF} f){X=-1,1}={FP,PF,FW,WF} P(P)=.7,P(F)=.1,P(W)=.2 2) X -2 -1 0 F(X) .01 .04 .18 1 2 .28 .49 Question15 Starting at a fixed time, we observe the sex of each newborn child at a certain hospital until a boy(B) is born. Let p=P(B), assume that successive births are independent, and define the random variable X by X=number of births observed. Find the probability mass function of X. Answer15 X=1,2,3,…. F(1)=P(B)=P F(2)=P(GB)=(1-P)P F(3)=P(GGB)=(1-P) P . . . F(X)= Question16 c Suppose that f(x)= 3c x=1.2…. Is the probability function for a random variable X. Determine c Find P(2 X <5). Find P(X 3). Answer16 c 1 c c 1 1 c 1 a)1 x 1 x c x 1 c 2 3 .... 1 2 3 3 3 3 3 1 2 c 2 3 2 f ( x) x 3 2 2 2 26 b) p(2 x 5) 2 3 4 4 0.32099 3 3 3 3 c 2 2 2 c) p( x 3) 3 x 3 4 5 .... 3 3 3 3 2 1 1 3 1 2 ..... 3 3 3 1 9 x Question17 The probability mass function of X is given as F(X)= Find 1.k 2.the cumulative function of X. 3. P(X>2) 4.P(-0.4<X<2) 5.P(X>1) Answer17 2 2 4 1) 1 f ( x) 0 k k k 2) F(X)== 4 1 3) 8 2 2 1 4) 8 2 6 3 5) 8 4 8 k 8 k Question18 Suppose X is random variable having density f given by X -3 -1 0 1 2 3 5 8 F(X) 0.1 0.2 0.15 0.2 0.1 0.15 0.05 0.05 Compute the following probabilities a)X is negative b) X takes a value between 1 and 8 inclusive Answer18 a) f (3) f (1) 0.3 b) p(1 x 8) 0.55 Question19 A fair die is tossed. Let X denote twice the number appearing ,and let Y denote 1 or 3 according as an odd or an even number appears. Find the probability distribution of X and Y. Answer19 x 2 4 6 8 10 12 F(x) 1/6 1/6 1/6 1/6 1/6 1/6 1 y 1 3 F(y) 1/2 1/2 Question20 Suppose a box has 12 balls labeled 1,2,…,12 Two independent repetitions are made of the experiment of selecting a ball at random from the box. Let X denote the larger of the two numbers on the balls selected. Compute the density of X. Question20 Suppose a box has 12 balls labeled 1,2,…,12 Two independent repetitions are made of the experiment of selecting a ball at random from the box. Let X denote the larger of the two numbers on the balls selected. Compute the density of X. STAT 111 Chapter Five Question1 Among the 16 application for a job, ten have college degree. If three of the application are randomly chosen for interviews, what are the probability that a)non has a college degree. b) one has a college degree. c)all three have college degree. Answer1 10 6 a) c0 16c3 0.0357 c b) c c c c) c c c 3 10 6 1 2 16 0.268 3 10 6 3 0 16 3 0.214 Question2 Find the probability that 7 of 10 persons will recover from a tropical disease, where the probability is 0.8 that any one of them will recover from the disease. Answer2 P(x 7) c7 (0.8) 7 (0.2) 3 0.201 10 Question3 The average number of days school is closed due to snow during the winter in a certain city is 4. What is the probability that the schools in this city will close for 6 days during a winter. Answer3 e 1 46 P(x 6) 0.1042 6! Question4 A manufacturer of automobile tires reports that among a shipment of 500 sent to a local distributor, 1000 are slightly blemished. If one purchases 10 of these tires at random from the distribution, what is the probability that exactly 3 will be blemished? Answer4 P(x 3) c3 (0.8) 3 (0.2) 7 0.201 10 Question5 The probability that a certain kind of component will survive a given shock test is 3/4. Find the probability that exactly 2 of the next 4 components tested survive? Answer5 1 4 3 P(x 2) c2 ( ) 2 ( ) 2 0.2109 4 4 Question6 Suppose X has a geometric distribution with p=0.8. Compute the probability of the following events. a) X 3 b) 4 x 7 c)3 X 5 or 7 X 10 Answer6 a) p( x 3) (1 p)3 0.008 b) p(4 X 7) p( x 4) p( x 7) 0.008 c) p(3 x 5( p(7 x 10) 0.04 Question7 In a manufacturing process in which glass items are being produced, defects or bubbles occur, occasionally rendering the piece undesirable for marketing. It is known that on the average 1 in every 1000 of these items produced has one or more bubbles. What is the probability that a random sample of 8000 will yield fewer than 7 items possessing bubbles? Answer7 p( x 7) p( X 6) 0.3134 Question8 Let X be uniformly distributed on 0,1,…,99. Calculate a) P( X 25) b) P(2.6 x 12.2) c) P(8 X 10 or 2 X 32 d ) P(25 X 30) Answer8 1 1 a ) p ( x 25) p ( x 26)..... p ( x 99) ... 0.75 100 100 10 b) p (3 x 12) p ( x 3) .. p ( x 12) 0. 1 100 c) p (3 x 32) p (4 x 32) p ( x 4) .. p ( x 32) 0.29 d ) p ( x 25) p ( x 26) .. p ( x 30) 0.06 Question9 As part of air pollution survey , an inspector decides to examine the exhaust of 6 of a company's 24 trucks. If 4 of the company's trucks emit excessive amounts of pollutants, what is the probability that none of them will be included in the inspector's sample? Answer9 4 20 6 p( x 0) c0 c 0.288 24 c 6 Question10 A fair die is rolled 4 times. Find a) The probability of obtaining exactly one 6. b)The probability of obtaining no 6. c)The probability of obtaining at least one 6. Answer10 3 1 5 a) p( x 1) c1 0.386 8 6 4 0 4 4 1 5 b) p( x 0) c0 0.482 6 6 c) p( x 1orx 2orx 3orx 4) 1 p( x 0) 0.518 Question11 Of a population of consumers, 60% is reputed to prefer a particular brand A of toothpaste. If a group of consumers is interviewed, what is the probability that exactly five people have to be interviewed to encounter the first consumer who prefers brand A. Answer11 p( x 5) 0.60.4 0.01536 4 Question11 Of a population of consumers, 60% is reputed to prefer a particular brand A of toothpaste. If a group of consumers is interviewed, what is the probability that exactly five people have to be interviewed to encounter the first consumer who prefers brand A. Answer11 p( x 5) 0.60.4 0.01536 4 Question12 Team A has probability 2/5 of winning whenever it plays. If A plays 4 games, find the probability that A ins a)2 games. b)at least 1 game c)more than half of the games. Answer12 2 2 4 2 3 a) p( x 2) c2 0.3456 5 5 4 3 b) p( x 1) 1 p( x 1) 1 p( x 0) 1 0.8704 5 c) p( x 2) 1 p( x 2) 1 0.8208 0.1792 Question13 The telephone company reports that among 5000 telephones installed in a new subdivision 4000 have push-buttons. If 10 people are called at random, what is the probability that exactly 3 will be talking on dial telephones? Answer13 3 7 10 1 4 p( x 3) c3 0.201 5 5 Question14 Suppose that 30% of the application for a certain industrial job have advanced training in computer programming. Application are interviewed sequentially and are selected at random from the pool. Find the probability that the first application having advanced in programming is found on the fifth interview. Answer14 p( x 5) (0.3) (0.7)4 0.07203 Question15 Suppose 2% of the items made by a factory are defective. Find the probability that there are 3 defective items in a sample of 100 items. Answer15 e 2 23 p( x 3) 0.180 3! Question16 From a group of twenty PhD engineers, ten are selected for employment . What is the probability that the ten selected include all the five best engineers in the group of twenty. Answer16 x~ hyp(n, N , K ) hyp(10,20,5) 5 15 5 5 f (5) 0.0163 20 10 Question17 If the probability is 0.40 that a child exposed to a certain contagious disease will catch it, what is the probability that the tenth child exposed to the disease will be the third to catch it. Answer17 x~ Nb(3,0.4) 9 3 7 f (10) 0.4 0.6 0.0645 2 Question18 Past experience has shown that the occurrence of defects in a telephone line being produced by a certain machine generated a Poisson process with 5 defects per kilometer occurring on the average. a) what is the probability that there will be 5 or less defects in 2 kilometers of cable? b)what is the probability that there will be exactly 3 defects in ¼ kilometers of cable? Answer18 10y e 10 a) f ( x) y! p( y 5) f (5) 0.0671 5 4 5 e 4 0.093 b) f (3) p( z 3) 3! Question19 An inspector in a television manufacturing plant has observed that defective tuners occur at a rate of 3 per 100 sets inspected. What is the probability that in 30 sets inspected, 2 or few will have defective tuners? Answer19 p( x 2) 0.9371 Question20 The painted light bulbs produced. By a company are 50% red, 30% blue and 20% green. In a sample of 5 bulbs, find the probability that 2 are red, 1 is green and 2 are blue. Answer20 5 0.52 0.31 0.135 p( A) 212 Question21 Find the probability of getting 5 heads and 7 tails in 12 of flips a balanced coin? Answer21 1 x ~ bin(12, ) 2 5 7 12 1 1 p( x 5) c5 0.193 2 2 Question22 If the probability is 0.75 that an application for a driver's license will pass the road test on any given try, what is the probability that an application will finally pass the test on the fourth try. Answer22 x ~ G(0.75) p( x 4) (0.75)(0.25)3 0.0117 Question23 The manufacturer of parts that are needed in an electronic device guarantees that a box of its parts will contain at most two defective parts. If the box holds 20 parts and experience has shown that the manufacturer process produces 2 percent defective items, what is the probability that a box of the parts will satisfy the guarantee? Answer23 x ~ b(20,0.02) x ~ position(0.4) f (2)) p( x 2) 0.9921 Question24 In an assembly process, the finished items are inspected by a vision sensor, the image data is processed , and a determination is made by computer as to whether or not a unt is satisfactory. If it is assumed that 2% of the units will be rejected, then what is the probability that the thirtieth unit observed will be second rejected unit? Answer24 x ~ Nb(2,0.02) P( X 30) c1 (0.02) 2 (0.98) 28 0.0066 29 Question25 In an interactive time-sharing environment it is found that, on average, a job arrives for CPU service every 6 seconds. What is the probability that there will be less than or equal to 4 arrivals in a given minute? What is the probability that there will be inclusively between 8 and 12 jobs arriving in a given minute? Answer25 p( x 4) 0.0293 p(8 x 12) f (12) f (7) 0.5714 Question26 Lots of 40 components each are called acceptable if they contain no more than 3 defective. The produce for sampling the lot is to select 5 components at random and to reject the lot if a defective is found. What is the probability that exactly 1 defective will be found in the sample if there are 3 defective in the entire lot? Answer26 x ~ hyp(5,40,3) 3 37 4 p( x 1) c1 c 0.30111 40 c 5 Question27 Find the probability of obtaining exactly three 2's if an ordinary die is tossed 5 times. Answer27 1 x ~ b(5, ) 6 5 1 p( x 3) 3 6 3 2 5 0.03215 6 Question28 Find the probability that a person tossing three coins will get either all heads or all tails for the second time on the fifth. Answer28 2 x ~ Nb(2, ) 8 p p( HHH TTT ) P( HHH ) P(TTT ) 4 2 6 p( x 5) 0.1055 1 8 8 2 3 2 8 Question29 A geological study indicates that an exploratory oil well drilled in a particular region should strike oil with probability 0.2. Find the probability that the third oil strike comes on the fifth well drilled. Answer29 x ~ Nb(3,0.2) p( x 5) c2 (0.2)3 (0.8) 2 0.03072 4 STAT 111 Chapter six Question1 Two refills for a ballpoint pen are selected at random from a box that contains 3 blue refills, 2 red refills, and 3 green refills. If X is the number of blue refills and Y is the number of red refills selected, find a) the joint probability function. b) P{(X,Y)} where A is the region { (x,y):x+y1}. Answer1 Y 0 3/28 9/28 3/28 15/28 1 6/28 6/28 0 12/28 2 1/28 0 0 1/28 F(X) 10/28 15/28 3/28 1 18 A f (0,0) f (1,0) f (0,1) 26 Question2 From a sack of fruit containing 3 oranges, 2 apples, and 3 bananas a random sample of 4 pieces of fruit is selected. If X is the number of oranges and Y is the number of apples in the sample, find a)the joint probability distribution of X and Y; b)P[(X,Y), where A is the region {(x,y) x+y ≤ 2} Answer2 x/y 0 1 2 3 f(y) 0 0 3/70 3/70 3/70 15/70 1 2/70 18/70 18/70 2/70 40/70 2 3/70 9/70 3/70 0 15/70 F(X) 5/70 30/70 30/70 5/70 1 A f (0,0) f (0,1) f (0.2) f (1,0) f (1,1) f (2,0) 0.5 Question3 Suppose an experiment consists of three flips of a fair coin, with each outcome being equally likely. Let X denote the number of heads on the last flip. Y, the total number of heads for the three tosses. Find the joint probability mass function. Answer3 x/y 0 1 2 0 1/8 0 1/8 1 2/8 1/8 3/8 2 1/8 2/8 3/8 3 0 1/8 1/8 F(X) 4/8 4/8 1 S {HHH , HTH , THH , HHT , TTH , THT , HTT , TTT } X 1 1 1 0 1 0 0 0 Y 3 2 2 2 1 1 1 0 Question4 Two tablets are selected at random from a bottle containing 3 aspirin, 2 sedative, and 4 laxative tablets, If X and Y are , respective, the number of aspirin tablets and the number of sedative tablets included among the two tablets drawn from the bottle, find a)the probabilities associated with all possible pairs of values(x,y). b)the marginal distribution of X and Y. c)the conditional distribution of X given Y=1 Answer4 x/y 0 1 2 f(y) 0 6/36 12/36 3/36 21/36 1 8/36 6/36 0 14/36 2 1/36 0 0 1/36 F(X) 15/36 18/36 3/36 1 Y 0 1 2 F(Y) 21/36 14/36 1/36 X 0 1 2 F(X) 15/36 18/36 3/36 x 0 1 2 F(x) 8/14 6/14 0 X:num of as , y:num of se 1 1 1 Question5 Lets X and Y denote the number of black and white balls, respectively, that will be obtained in drawing two balls from a bag that contains two black and two white balls. Find the joint probability .mass function of X and Y. Answer5 x/y 0 1 2 f(y) 0 0 0 1/6 1/6 1 0 4/6 0 4/6 2 1/6 0 0 1/6 F(X) 1/6 4/6 1/6 1 X:black balls Y:white balls Question6 Suppose that X and Y have following joint probability function y/x 1 2 3 1 0 1/6 1/12 2 1/5 1/9 0 3 2/15 1/4 1/18 Find 1.The marginal distribution of the random variable X. 2.The marginal distribution of the random variable Y. 3. P(Y=3\X=2) Answer6 x/y 0 1 2 f(y) 1 0 30/180 15/180 45/180 2 36/180 20/180 0 56/180 3 24/180 45/180 10/180 79/180 F(x) 60/180 95/180 25/180 1 x 1 2 3 F(x) 60/180 95/180 25/180 y 1 2 3 F(y) 45/180 56/180 79/180 45 f ( x 2, y 3) 180 45 3. 0.474 95 180 f ( x 2) 180 1 1 Question7 Consider an experiment that consists of 2 rolls of a balanced die. If X is the number of 4's and Y is the number of 5's obtained in the 2 rolls of the die, find a)the joint probability distribution of X and Y; b)P[(X,Y) A] where A is the region given by {(x,y): 2x+y<3} Answer7 x/y 0 1 2 f(y) 0 16/36 8/36 1/36 25/36 1 8/36 2/36 0 10/36 2 1/36 0 0 1/36 F(X) 25/36 10/36 1/36 1 16 f (0.0) , f (0,1) f {(1,5), (2,5), (3,5), (6,5) 36 8 (5,1), (5,2), (5,3), (6,5)} 36 33 p{( x, y ) : 2 x y 3} f (0,0) f (0,1) f (0,2), f (1,0) 36 Question8 A fair coin is tossed three times. Let X denote 0 or 1 according as a head or a tail occurs on the first toss, and let Y denote the number of heads which occur. Determine a)the distribution of X and the distribution of Y. b) the joint probability mass function Answer8 x/y 0 1 f(y) 0 0 1/8 1/8 1 1/8 2/8 3/8 2 2/8 1/8 3/8 3 1/8 0 1/8 F(X) 4/8 4/8 1 x 0 1 F(x) 1/2 1/2 1 y 0 1 2 3 F(y) 1/8 3/8 3/8 1/8 1 Question9 From a group of three Republicans, two Democrats, and one one Independent, a committee of two people is to be randomly selected, Let X denote the number of Republicans and Y the number of Democrats on the committee. Find A) the joint probability distribution of X and Y, and then find the marginal distribution of X. b)the conditional distribution of X given that Y=1. Answer9 x/y 0 1 2 f(y) 0 0 3/15 3/15 6/15 1 2/15 6/15 0 8/15 2 1/15 0 0 1/15 F(X) 3/15 9/15 3/15 1 f ( x,1) f ( x,1) f ( x / y 1) 8 f ( y,1) 15 x 0 1 2 F(x) 2/8 6/8 0 1 Question10 Consider the joint probability distribution defined by the formula 1 f ( x, y ) ( x 2 y) 27 x=0,1,2 y=0,1,2 Find the marginal distribution of X and Y, and f(x/y ) Answer10 x/y 0 1 2 f(y) 0 0 1/27 2/27 3/27 1 2/27 3/27 4/27 9/27 2 4/27 5/27 6/27 15/27 F(X) 6/27 9/27 12/27 1 If(y/x)=f(x,y)/f(x) If x=0 y 0 1 2 Fy(x=0) 0 1/3 21/3 if x=1 if x=2 y 0 1 2 Fy(x =1) 1/9 3/9 5/9 1 y 0 1 2 Fx/x =2 1/6 2/6 3/6 1 Question11 The joint probability function of two random variables X and Y is given as F(x ,y )=c(2x+y) x=0,1,2 y=0,1,2,3 a)Find the value of the constant c b)Find P(X=2,Y=1) C)Find P(X 1, Y 2) d)Find the marginal distribution of X and Y. e)Find f() , and P(Y=1\X=2) f)Determine whether the random variables X and Y are independent . Answer11 x/y 0 1 2 3 0 0 2c 4c 6c 1 c 3c 5c 9c 2 2c 4c 6c 12c 3 3c 5c 7c 15c F(X) 6c 14c 22c 42c Answer11 a ) x y f ( x, y ) 1 c b) f (2,1) 5c 1 42 5 42 c) f (1,2) f (1,1) f (1,0) f (2,2) f (2,1) f (2,0) 24c f ( y,2) fx(2) f (2,1) 5 e) fx(2) 22 f ) no, for example: f(0,0)=0 but fx(0)fy(0)#0 d ) f ( y,2) 24 42 Question12 Let the joint probability mass function of X and Y is given in the following table Find P (X 2, Y 2) P (X 1) P (X Y 4) P (X 2, Y 1) F(-1,2) F(1.5,2) F(5,7) X/Y 1 2 3 4 1 0.1 0 0.1 0 2 0.3 0 0.1 .20 3 0 0.2 0 0 Answer12 X/Y 1 2 3 4 F(x) 1 0.1 0 0.1 0 0.2 2 0.3 0 0.1 .20 0.6 3 0 0.2 0 0 0.2 F(Y) 0.4 0.2 0.2 0.2 1 1) 0 .5 2 ) 0 .6 3) 0 .1 4)0 5) 0 6 ) 0 .1 7 )1 Question13 Let X and Y be independent random variables with the following distribution ; X 1 2 Y 5 10 15 F(X) 0.6 0.4 F(Y) 0,2 0.5 0.3 Find the joint distribution of X and Y. Answer13 x/y 0 1 f(y) 5 0.12 0.08 0.2 10 0.3 0.2 0.5 15 0.18 0.12 0.2 F(X) 0.6 0.4 1 f (1,5) 0.12 f (1,10) 0.30 f (1,15) 0.18 f (2,5( 0.08 f (2,10) 0.20 f (2,15) 0.12 Question14 Suppose that the joint probability mass function of X and Y, is given by F(0,0)=0.4 f(0,1)=0.2 f(1,0)=0.1 f(1,1)=0.3 Calculate the conditional probability mass function of X, given that Y=1 Answer14 x/y 0 1 f(y) 0 0.4 0.1 0.5 1 0.2 0.3 0.5 F(X) 0.6 0.4 1 f y (1) x f ( x,1) 0.2 0.3 0.5 0.2 0 . 4 , x 0 f ( x,1) x f ( x,1) 0.5 f x ( x / y) 0 . 5 f y (1) 0.5 0.6, x 1 0.5 Question15 Suppose that X and Y have following joint probability function X/Y 1 2 3 4 1 0.1 0 0.1 0 2 0.3 0 0.1 0.2 3 0 0.2 0 0 Find The marginal distribution of the random variable X; The marginal distribution of the random variable y; Determine whether the random variables X and Y are independent. Answer15 X 1 2 3 F(x) 0.2 0.6 0.2 1 y 1 2 3 4 F(y) 0.4 0.2 0.2 0.2 No,for example: f(1,2)=0, but fx(1)=0x0.2#1 1 STAT 111 Chapter seven Question1 Let (x,y) have the following joint distribution function x/y 1 2 3 sum 1 0 1/6 1/6 1/3 2 1/6 0 1/6 1/3 3 1/6 1/6 0 1/3 sum 1/3 1/3 1/3 1 Find a)the probability mass function of X+Y. b)the probability mass function of XY. c)the probability mass function of X2. d)the probability mass function of Y2. Answer1 a) b) X+y 2 3 4 5 6 sum F(x+y) 0 2/6 2/6 2/6 0 1 xy 1 2 3 4 6 9 sum F(xy) 0 2/6 2/6 0 2/6 0 1 x2 1 4 9 sum F(x2) 1/3 1/3 1/3 1 y2 1 4 9 sum F(y2) 1/3 1/3 1/3 1 c) d) Question2 Lets X be a random variable with probability distribution F(x)= Find the probability distribution of the random variable Y=2X-1 Answer2 x 1 2 3 sum y 1 3 5 sum F(y) 1/3 1/3 1/3 1 Y=2x-1, x= -1 = g (y)=>fy (y)=fx (g (y)) Question3 Let X1 and X2 be discrete random variable with joint probability distribution F(x)= Find the probability distribution of the random variable Y=X1X2 Answer3 y 1 2 3 4 6 sum Fy(y) 1/18 4/18 3/18 4/18 6/18 1 X1/x2 1 2 sum 1 1/18 2/18 3/18 2 2/18 4/18 6/18 3 3/18 6/18 9/18 sum 8/18 12/8 1 Question4 Let X be a random variable with probability distribution F(x)= Find the probability distribution of the random variable Y=X2 Answer4 x 1 2 3 F(x) 1/n 1/n 1/n …… n 1/n y 1 4 9 F(x) 1/n 1/n 1/n ……25 N2 1/n Question5 Let X be a random variable with probability distribution F(x)= Find the probability distribution of the random variable Y=X2 Answer5 x 1 ….. 2 n y 1 4 9 n Fy(Y) 2/2n 2/2n 2/2n 2/2n STAT 111 Chapter eight Question1 A lot twelve television sets includes two that are defective. If three of the sets are chosen at random, how many defective sets can they expect? Answer1 x 0 1 2 sum F(x) 6/11 9/11 1/22 1 Xf(x) 0 9/22 2/22 11/22 2 10 x s x f ( x) 12 x x ~ hyp(3,12,2) nk 2 1 E ( x) 3 N 12 2 Question2 If the random variable X is the two faces when rolling a pair of balanced dice, find the expected value of X. Answer2 X 2 3 4 5 6 7 8 9 10 F(X) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 11 12 SUM 2/3 1/36 6 1 Question3 Let X have the following probability mass function F(x)= Find E(X),E(X2), and e(x+2)2 Answer3 X uniform n 5 E(X) 3 E(X2 ) 11 E(X 2) 2 27 Question4 The probability mass function of the random variable X is given by x 1 2 3 4 F(x) 4/10 1/10 3/10 2/10 Find the expected value of X. sum Answer3 X 1 2 3 4 SUM F(X) 4/10 1/10 3/10 2/10 1 XF(X) 4/10 2/10 9/10 8/10 2.3>E(x) Question5 The probability mass function of the random variable X is given by x 4 5 6 7 8 9 F(x) 1/12 1/12 1/4 1/4 1/6 1/6 Find the expected value of g(x)=2x-1. sum Answer5 x 4 5 6 7 8 9 sum F(x) 1/12 1/12 1/4 1/4 1/6 1/6 1 E(x)=Xf(x) 4/12 5/12 6/4 7/4 8/6 9/6 82/1 2 E(X)=82/12=41/6 E(2X-1)=2E(X)-1=2(41/6)-1=12.7 Question6 Let the random variable X represent the number of automobiles that are used for official business purposes on any give workday. The probability distribution for company A is given by x 1 2 3 F(x) 0.3 0.4 0.3 And for company B is given by x 0 1 2 3 4 F(x) 0.2 0.1 0.3 0.3 0.1 Show that the variance of the probability distribution for company B is greater than that of company A Answer6 x 1 2 3 sum F(x) 0.3 0.4 0.3 1 E(X)=X f(x) 0.3 0.8 0.9 2 E(X2)=X2 f(x) 0.3 1.6 2.7 4.6 Va=4.6-22 =0.6 X 0 1 2 3 4 F(X) 0.2 0.1 0.3 0.3 0.1 XF(X) 0 0.1 0.6 0.9 0.4 X2 f(x) 0 0.1 1.2 2.7 1.6 5.6->E(X 2) Vb=5.6-22 =1.6 Vb(x)> Va(x) 1.6 0.6 SUM 2->E(X) Question7 Find the moment generating function of the discrete random variable X has the probability distribution F(x)=2x x=1,2,… And use it to find µ1 µ2 Answer7 Question8 An urn contains nine chips, five red and four white. Three are drawn out at random without replacement . Let X denote the number of red chips in the sample. Find E(X). Answer8 x ~ hyp(3,9,5) 5 4 x 3 x f ( x) 9 3 n 5 E ( x) n 3 1.667 N 9 Question9 Suppose that a sequence of independent tosses are made with a coin for which the probability of obtaining a head on any given toss is 1/30. Find the expected number of tosses that will be required in order to five heads? Answer9 X:number of heads Y: number of tosses 1 X bin(n, ) 30 E ( x ) np 5 n 5 n 150 30 Question10 Let X be binomial distribution random variable with E(X)=2 , and V(X)=4/3 . Find the distribution on X. Answer10 X 4 bin (n,p) n? p? E(x) np 2...... equation(1) 3 V(x) npq ....... equation(2) 4 4 4 2 q 3 6 3 2 1 p 1 q 1 3 3 2q 1 2n6 3 1 x ~ bin(6, 3) n Question11 Let x1,x2 ,and x3 be independent random variables having finite positive variances 21 , 22, 23, respectively. Find the correlation between x1-x2 and x2+x3. Answer11 cor(x1 - x2 , x2 x3 ) cov(x1 x2 , x2 x3 ) v( ( x1 x2 )v( x2 x3 ) Question12 A box has 3 red balls and 2 black balls. A random sample of size 2 is drawn without replacement. Let U be the number of red balls selected let V be the number of black balls selected. Compute (U,V). Answer12 UR/VB 0 1 2 Fv(V) 0 0 0 3/10 3/10 1 0 6/10 0 6/10 2 1/10 0 0 1/10 Fu(U) 1/10 6/10 3/10 1 E(U)=6/5 E(U.U)=18/10 V(U)=.36 E(UV)=-.36 (U,V)=-1 , , , E(V)=4/5 E(V.V)=1 V(V)=.36