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# TOP IN CONTEMP MATH II MATH 166

Texas A&M

GPA 3.6

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This 24 page Study Guide was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Study Guide belongs to MATH 166 at Texas A&M University taught by Kendra Kilmer in Fall. Since its upload, it has received 31 views. For similar materials see /class/225997/math-166-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15

Math 166 Spring 2008 Week in Review 3 Exam 1 Review courtesy Kendra Kilmer covering Sections 2127 AlA2 6162 Please note that this review is not all inclusive 1 Determine whether each of the following is proposition a Five is an even number b There are 150 named streets in College Station c Turn left at the second light d 2x7 7 2 15 2 Let p and q denote the propositions p The number of faculty members at Texas AampM increased over the past year q The average class size at Texas AampM decreased over the past year Express the following compound propositions in words a 7A q 03 N p V q c N pi q Let p and q denote the propositions p John went to the basketball game q Crystal went to the basketball game Express each of the following statements symbolically Equot a Crystal did not go to the basketball game but John did b Crystal went to the basketball game or John did not go to the basketball game c Neither John nor Crystal went to the basketball game 4 Construct a truth table for each compound proposition a MIDYNP b MWWQ 5 Let U 12345678910A 13579 B 2468 10 C 1589 D 8 10 Findthe following a AUC b B D c C d Cm AUDC 6 Using the following sets determine whether each statement is True or False U 123459121518 A 24915B 131518C 5915 a 215 6 A b BUCV A Z c A has 16 subsets d D C B e UC 5915 f A and B are disjoint sets 7 For each part draw a Venn Diagram with 3 subsets A B and C and shade the region that represents each set a BnAchC b C AUB c AUB C 8 A survey was conducted of 100 student s dining preferences They were all asked their preference on eating at Chipotle Fitzwilly s and Potbelly s The following information was determined 0 20 students like only Fitzwilly s o 30 students like exactly two of the restaurants 0 50 students like Potbelly s o 5 students like all three restaurants 0 15 students like Chipotle and Fitzwilly s o 8 students like only Fitzwilly s and Potbelly s o 60 students do not like Chipotle Fill out the Venn Diagram with this information and use it to determine how many students do not like Fitzwilly s 9 Transportation Services is trying to decide on the width of the parking spaces in a newly created parking lot To do so they have been doing a study on the size of cars that are on campus During their study they discovered that there are a total of 12000 cars on campus that can be classi ed into three categories compact cars mediumsized cars and trucks The number of trucks is the same as the number of compact and mediumsized cars combined There are half as many compact cars as mediumsized cars and trucks combined Determine the number of each type of vehicle on campus during their study 10 Solve the following system of linear equations by doing GaussJordan Elimination by hand 2x 7 7y 9 y73x5 11 Each of the following augmented matrices represents a system of linear equations Determine whether or not the augmented matrix is in rowreduced form If it IS in rowreduced form give the solution to the system of linear equations If it IS NOT in rowreduced form perform the next row operation that would need to be performed to get the augmented matrix in rowreduced form 10004 a 012073 00012 108 b 0177 015 10725 01077 00016 0000 12 Given the follwing matrices with the indicated dimensions which of the following are valid matrix operations A4gtlt3yB3gtlt27C4gtlt27D3gtlt4 a 3AD b A SDT 7 B c BCT 7 5D d AB 20 13 Solve the given matrix equation for a I and c 5 a 77 1 9 T 752 5 3 72 5 7117745773 19 78 72 1 3 3 b c 9 Whatis2al7707 14 Consider a simple economy consisting of two sectors goods and services The production of 1 unit of goods requires the consumption of 01 unit of goods and 02 unit of services The production of 1 unit of services requires the consumption of 03 unit of goods and 025 unit of services Find the level of production for each sector in order to satisfy the demand for 100 million worth of goods and 400 million worth of services How much of each sector is consumed intenially while satisfying this demand Spring 2008 Math 166 Exam 1 Topics courtesy Kendra Kilmer covering Sections 21 27 A1 A2 61 62 Section 21 0 When solving a system of linear equations our solution will be in one of the following forms 0 Unique Solution 0 No Solution 0 In nitely Many Solutions 0 When setting up a word problem always make sure to de ne your variables rst Sections 22 23 o The goal of the GaussJordan Elimination Method is to get the augmented matrix in Row Reduced Form A matrix is in Row Reduced Form when a Each row of the coef cient matrix consisting entirely of zeros lies below any other row having nonzero entries b The rst nonzero entry in each row is 1 called a leading 1 c In any two successive nonzero rows the leading 1 in the lower row lies to the right of the leading 1 in the upper row d If a column contains a leading 1 then the other entries in that column are zeros Note We only consider the coef cient side left side of the augmented matrix when determining whether the ma trix is in rowreduced form 0 To put a matrix in Row Reduced Form there are three valid Row Operations a Interchange any two rows Ri lt gt R j b Replace any row by a nonzero constant multiple of itself cRi c Replace any row by the sum of that row and a constant multiple of any other row Ri ch o If you are not required to work the problem by hand you can use the rref function on your calculator to get the augmented matrix in rowreduced form 0 Once the system is in rowreduced form make sure you know what leads you to conclude no solution or in nitely many solutions Section 24 o A matrix is an ordered rectangular array of numbers A matrix with m rows andn columns has dimensions m X n 0 Our basic matrix operations are addition matrices must have the same dimensions subtraction matrices must have the same dimen sions scalar multiplication multiply every entry in the matrix by the scalar and transpose interchange rows and columns Section 25 0 To multiply two matrices the number of columns in the rst matrix must equal the number of rows in the second matrix The resulting matrix will have the same number of rows as the rst matrix and the same number of columns as the second matrix 0 Be able to multiply matrices by hand 0 Matrix Multiplication is NOT Commutative Order matters 0 The identity matrix is a square matrix with 1 s along the main diag onal and 0 s everywhere else Section 26 o If a square matrix A has an inverse then AA 1 A IA In where In is the identity matrix of size n o A square matrix that has an inverse is nonsingular If a square matrix does not have an inverse then it is singu ar 0 You are not responsible for knowing how to compute an inverse by hand You may use your calculator 0 Be able to solve systems of linear equations using inverses Section 27 o In a Leontief inputoutput model the solution to the matrix equation X iAX D whereX is the total output A is the inputoutput matrix and D is the consumer demand is X I 7 A 1D assuming 17A 1 exists Section A1 o A proposition is a declarative smtement that can be class ed as ei ther true or false but not both 0 There are four connectives that we use to combine propositions The conjunction written symbolically p q is of the form p and q It is true when both statements are true it is false otherwise Thquotj quotorquot39 39j quot 39 quot pV q is of the form p or q It is false when both smtements are false it is true otherwise The exclusive disjunction written symbolically p X q is of the form either p or q It is true when either p or q is true but not both Otherwise it is false The negation written symbolically N p is of the form not p It is true when p is false and vice versa 0 Truth Tables are used to determine the truth value of a compound proposition Section 61 0 We can either use roster notation or setbuilder notation to repre sent a set 0 Two sets A and B are equal written A B if and only if they have exactly the same elements 0 If every element of a set A is also an element of a set B then we say thatA is a subset ofB and writeA Q B o IfA andB are sets such thatA Q B butA B then we say A is a proper subset of B written A C B o The set that contains no elements is called the empty set and is de noted by 0 It is a subset of all sets 0 The universal set is the set of all elements of interest in a particular problem 0 We use Venn Diagrams to visually represent sets The universal set U is represented by a rectangle and subsets of U are represented by circles inside of the rectangle o The union ofA and B written A U B is the set of all elements that belong to A or B o The set of elements in common with the sets A and B written A B is called the intersection ofA an B o The complement ofA denoted A is the set of all elements in U that are not in A 0 Two sets A and B are disjoint ifA B 0 0 De Morgan s Laws Let A and B be sets Then A UB A B and A B AEUB Section 62 o nA represents the number of elements in a set 0 nA UB nA nB 7 nA QB 0 We can label each region in a Venn Diagram with the number of elements in it to sort out the given information 1 Q L U1 Math 166 Spring 2008 Week in Review 11 Final Exam Review courtesy Kendra Kilmer covering A1 amp A2 amp Chap 25678amp9 excluding Sec 54amp93 Please note that this review is not all inclusive How many 3 digit numbers can be formed from the numbers 01 25 7 8 9 if no digit can be repeated the rst digit must be nonzero and each number formed must be even A 10000 lyear term life insurance policy costs 150 If the probability that a particular person dies in the next year is 0002 nd the company s expected net gain Assume that a small town has only three industries auto energy and transportation Each of these industries acts as a supplier and a user of their products The inputoutput matrix A for this economy is given by auto energy transportation auto 02 0 4 0 1 ener y 0 1 0 2 0 2 transportation 02 0 2 0 1 Find the value of the goods consumed in the internal process of production in order to satisfy an external demand for automobiles of 474 million external demand for energy of 948 million and an external demand for transportation of 474 million Determine whether each of the following is a proposition a Take me to school b The school is on the righthand side of the road c 3x5 20 Given the payoff matrix below 71 3 72 5 74 2 Find the expected payoff of the game if the row player selects row 1 18 of the time and the column player selects column 1 15 of the time and column 2 35 of the time 6 An experiment consists of observing whether cars are kept in the driveway or in the garage It was found that 75 of cars are kept in the driveway and the remaining cars are kept in the garage Of those cars kept in the driveway 90 were worth less than 15000 Of those kept in the garage 80 were worth 15000 or more If a car worth less than 15000 is randomly selected what is the probability that it is in the driveway Solve the following system of linear equation using inverses 3x 7 7y 7 l5z 9 z 5x 7 9y 7 7 2x 7 5y z 9 8 A girl scout convention is being held in College Station Troop 647 is bringing 7 girls Troop 916 is bringing 5 girls and Troop 525 is bringing 3 girls In how many ways can these girls sit in a row if each troop wants to sit together 9 A research study was done on conusmers behavior regarding their toilet paper preferences It was found that each month of the consumers using Brand A 70 will continue to use Brand A whereas 25 will switch to Brand B and 5 will switch to Brand C Of the consumers using Brand B 80 will continue to use Brand B whereas 5 will switch to Brand A and 15 will switch to Brand C Of the consumers using Brand C 95 will continue to use Brand C whereas 1 will switch to Brand A and 4 will switch to Brand B At the beginning of January it is known that 33 of the consumers are using Brand A 62 of the consumers are using Brand B and 5 of the consumers are using Brand C What percent of the consumers will be using Brand A at the beginning of June What percent of the consumers will be using Brand A in the long run 10 Pivot the given matrix about the element an 1 72 5 9 0 4 78 l6 0 5 3 9 11 Solve the following matrix equation for a b and c 75617 92a9278196108 9704257105971008475 12 Given U 123456 A 15 B 1356 and C 246 determine Whether each statement is true or false a 13 6 B b U has 64 proper subsets c A and B are disjoint sets d A B UC 246 e 2 e A 019V 13 What is the effective rate of interest of an account if the interest is compounded monthly and the investment Will double in 20 years 14 A group of 488 Aggies are going on a road trip to the bowl game They are trying to decide how many cars vans and buses to take for the trip They has determined that each car Will seat 6 people and the expenses involved for each car come out to be 100 Each van holds 14 people and costs 195 per van Each bus holds 40 people and costs 475 If they have decided that they want to take twice as many buses as vans how many vehicles of each type should they take if the budget is 6025 and they want to exactly use all of their budget and have no empty seats 15 Amy purchases a 250000 house She makes a 5 down payment and then nances the remaining balance with a 30 year mortgage that charges a 625 annual interest rate compounded monthly on the unpaid balance a What is her required monthly payment b Assuming she makes exactly the required payment each month create an amortization schedule detailing the 1st 2nd 241st and 242nd payments 16 Consider the economy of a small village consisting of two industries farming and weaving The farmers produce food for themselves and the weavers along with extra food for the remaining people The weavers produce cloth for themselves and the farmers along with extra cloth for the remaining people The villagers have found that to produce 100 of food 040 of food and 010 worth of cloth is needed locally to feed and clothe the farmers To produce 100 of cloth 030 of food and 020 of cloth is needed locally to feed and clothe the weavers The additional food and cloth produced is then available to export to the city If the city demands 77 200 worth of food and 27 700 worth of cloth each month how much food and cloth should be produced by the village to meet its own needs and to supply the city 1 A survey of 550 freshmen revealed that 300 own a car 200 own a computer and 100 own neither a car nor a computer If a student is randomly selected from this group what is the probability that they own a car but not a computer m An experiment consists of randomly selecting one card out of a standard deck of 52 cards Let E be the event that a red card is drawn and let F be the event that a club is drawn Are the events E and F mutually exclusive events Are the events E and F independent events 19 Katie bought a house in 2000 for 300000 She nanced it with a 30 year mortgage at an annual interest rate of 725 compounded monthly on the unpaid balance a What are her current monthly payments b After making six years of payments she decides to re nance her home in 2006 with a 30 year mortgage that has an annual interest rate of 525 compounded monthly on the unpaid balance What are her new monthly payments c How much total interest will she end up paying for this house 20 The independent probabilities that Jenny Karen and Lindsay will attend class are 0 35 095 and 076 respectively What is the probability that exactly two of the girls will attend class 21 For each of the following payoff matrices i Determine the maximin and minimax strategies ii Is the game strictly determined If so what is the saddle pointvalue of the game Who does the game favor lt2 31 34 3 22 A jar contains 10 pennies and 5 dimes Three coins are randomly selected from the jar Let X represent the value of the three coins ie how much money you have Find the probability distribution of X 23 How many days will it take for a sum of 2000 to earn 50 interest if it is deposited in an account earning simple interest at the rate of 8 per year 24 Classify each of the random variables as nite discrete in nite discrete or continuous a Let X represent the weight of a newborn kitten in ounces b Cough drops are randomly selected with replacement from a bag containing 30 cherry 20 menthol and 35 lemon Let Y represent the number of cough drops drawn until a cherry cough drop is selected c Jane serves the volleyball 100 times Let Z represent the number of times she serves an ace 25 Two marbles are randomly selected from a bin containing 8 red and 5 green marbles If this experiment is repeated 10 times and the marbles are replaced each time what is the probability of getting 2 red marbles at least 3 times 26 Sarah is trying to arrange 5 identical blue candle sticks 7 identical red candle sticks and 3 identical green candle sticks on her shelf How many distinguishable arrangements are possible 27 Given the follwing matrices with the indicated dimensions which of the following are valid matrix operations A2gtlt4yB3gtlt17C4gtlt37D3gtlt2 a AC76DT b SDTB c 2C8 d AT 7 CD 28 An experiment consists of rolling a fair sixsided die 1000 times Find the probability of rolling a 5 at most 170 times Use the appropriate normal distribution to approximate the binomial probability 29 Valerie decides to put 30 into an account every week If the account earns 3 45year compounded weekly how much will she have in the account 15 years from now 30 Given the following probability distribution x 715 75 20 30 PXx 0056 0 72 019 0034 Find E X VarX 039 median and mode 31 Given 3 24681012A 2612B 81012 C 2468 10 nd the following a Aan b AUB C c BmAmC 32 In College Station 85 of the population is currently an Aggie fan Each year 30 of the nonAggie fans become Aggie fans and 1 of the Aggie fans become non Aggie fans If this trend continues what percent of the population will be an Aggie fan 5 years from now 33 If nA UB 80 nA 30 and 113 75 What is nA BC 34 Jenny and Lisa have recently acquired some leftover inventory from a small online baby supply store Jenny has acquired 20 paci ers 18 bottles and 9 cloth diapers Lisa has acquired 15 paci ers 7 bottles and 12 cloth diapers a Organize the above information into a matrix and label it A b The paci ers Will sell for 2 each the bottles Will sell for 4 each and the cloth diapers Will sell for 12 each Write a matrix B summarizing this data c Write a matrix product involving the matrices A and B that Will produce a new matrix giving the total amount of money each girl Will make if they sell all of their inventory Compute the desired product and interpret the result 35 Let X be a normal random variable With a mean of O and a standard deviation of 8 Find the value of b if P7b lt X lt b 09826 36 Given the payoff matrix below determine the optimal mixed strategy for the row and column player and then compute the expected payoff of the game Who does the game favor 65 2 37 Let p and q denote the propositions p It will be hot this summer q I will spend the summer on the beach Express the following compound propositions in words a NpAq b qu c N Na 38 A TrueFalse Exam consists of 15 questions In how many ways can the exam be answered so that the student answers at least 10 questions correctly 39 An experiment consists of rolling a fair sixsided die and ipping a fair coin a Find the Sample Space associated with this experiment b Find the event E that an even number is rolled c Find the PE 40 Gabe has 2500 in credit card debt on a card that charges 196 per year compounded monthly on the unpaid balance If he pays the minimum payment of 50 each month and he doesn t use it to make any more purchases how long will it take Gabe to pay off his debt 41 An experiment consists of rolling a pair of fair sixsided dice What is the probability that the sum is at least 9 or exactly one 5 is rolled 42 An experiment consists of randomly selecting 1 marble from a bowl containing 5 red 3 blue and 10 white marbles What is the probability that the marble selected is not white 43 Solve the systems of linear equation using the GaussJordan elimination method a 712x742y748z 7114 2x 7y 8z 19 3x 2y lOz 9 b 2x3y72z 10 x3yiz 73 3x4yiz7 c 3x79y6z12 x73y2z74 2x76y4z8 44 If E andF are mutually exclusive events PE 0 5 and PF 03 nd PE F U E9 45 A crate contains 90 strawberries of Which 10 are rotten If a customer randomly selects 15 strawberries What is the probability that at least 3 of the strawberries are rotten 46 A fruit stand has 30 apples 14 peaches 6 plums and 15 pears If you randomly select 8 pieces of fruit how many ways can you select exactly 4 plums or exactly 3 pears 47 In a group of 10 people what is the probability that at least 2 of them were boni in the same month 48 An experiment consists of rst performing a task in which A B or C can occur Then a second task is performed in which D or E can occur The following probabilities are known PA 0 3 PC 0 5 PDjA 0 25 PEjB 0 7 PDjC 064 Find the following a P E b PBlE c PA U D 49 The scores on an English exam were normally distributed with a mean of 65 and a standard deviation of 5 If 10 of the class received an A on the exam and 15 of the class received a B on the exam what was the cutoff for a B 50 A club consists of 35 nance majors 30 accounting majors and 40 info majors This club must elect a president and vicepresident from the nance majors and then elect 2 representatives from each of the two remaining majors How many ways can this be done 51 The following is the transition matrix for a Markov process that has 3 states A B and C Note the rows and columns are represented in the matrix in the order listed above 0 1 0 5 03 T 0 9 02 0 65 0 03 005 a Is the above matrix stochastic b Is the above matrix regular c What does the entry 32 represent d What does the entry 13 represent 52 A game consists of rolling a pair of fair sixsided dice If the sum of the numbers is less than 5 the player wins 5 If the sum of the numbers is greater than 9 the player wins 4 Otherwise the player loses A Find the value of A that makes this game fair 53 It is known that 62 of Aggies purchased an allsports pass If 50 Aggies are randomly selected what is the probability that at most 29 of them own a sports pass 54 Shade the following regions on a Venn Diagram a A BUC b Bncc c A New Benc 55 It is known that the length of newbonis is normal distributed with a mean of 21 inches and standard deviation of 3 inches What is the probability that a randomly selected newborn has a length between 18 and 24 inches 56 A game consists of two players simultaneously showing one two or three ngers Carl and Rico decide to play this game If the sum of the numbers showing is less than four Carl pays Rico 3 If the sum of the numbers showing is greater than four Rico pays Carl 2 Otherwise Carl pays Rico 1 a Find the payoff matrix for this twoplayer zerosum game b Find the maximin and minimax strategies for Rico and Carl Is this game strictly determined If so what is the value of the game Who does this game favor 57 Danny put a 5500 down payment on a car He nanced the remaining balance with a 60 month loan at a 55 annual interest rate compounded monthly on the unpaid balance If the monthly car payments are 450 what is the cash price of the car 58 Determine the value of b for which the system of linear equations 4x 7 y 6 8x by 10 has no solution 59 A survey was done of 700 students on their preference of fruits The following information was found 275 students like bananas and strawberries 250 students like all three fruits 400 students like grapes 50 students like only bananas and grapes 20 students like only grapes 205 students do not like bananas 60 students like only strawberries How many students like exactly one of these kinds of fruit 60 Construct a truth table for each compound proposition a p pM1 b WIDMNPM Math 166 Spring 2008 Week in Review 6 Exam 2 Review courtesy Kendra Kilmer covering Sections 6364 7176 l A survey is done of student s participation in activities at Parent s Weekend The results are shown below a What is the probability that a student attended the Variety Show b What is the probability that a student attended only Midnight Yell c Given that a student attended Midnight Yell what is the probability that the student did not attend the Variety Show d What is the probability that a student who attended the Variety Show also attended Midnight Yell 2 If two fair sixsided dice are rolled what is the probability that the sum is 8 given that the sum is greater than 7 3 A survey has shown that 95 of the women in a certain community work outside the home Of these women 56 are married while 85 of the women who do not work outside the home are married Find the probabilities that a woman in the community can be categorized as follows a A single woman working outside the home b Married 4 On a typical spring day in College Station the probability of rain is 035 the probability of a traf c jam is 024 and the probability of rain or a traf c jam is 045 Are the events it rains and a traf c jam occurs independent 5 In a home theater system the probability that the video component needs repair within 1 year is 002 the probability that the electronic component needs repair within 1 year is 003 and the probability that the audio component needs repair within 1 year is 001 Assuming the probabilities are independent what is the probability that exactly two components will need repair within 1 year 6 Unfortunately two lights have been added to Mrs Kilmer s commute to work After a few weeks of collecting data she has determined that the probability that she will be stopped by the light at 2818 and Holleman is 02857 whereas the probability that she will be stopped by the light at 2818 and Luther is 00476 Assuming these events are independent what is the probability that she will be stopped by at least one of the two lights on her drive to work tomorrow moniing 7 A crate contains 8 basketballs and 5 footballs A bag contains 6 basketballs and 9 footballs An experiment consists of selecting a ball at random from the crate and placing it in the bag A ball is then randomly selected from the bag If a football is selected from the bag what is the probability that the transferred ball was a football 8 The probability that a person with certain symptoms has a disease is 08 The blood test used to con rm this diagnosis gives positive results for 90 of people with the disease and 5 of those without the disease What is the probability that an individual who has the symptoms and who reacts positively to the test actually has the disease 9 How many three digit numbers can be formed from the digits 1 3 4 5 6 9 if the numbers formed must all be even and no digit can be repeated 10 Susie has ve different pieces of fruit eight different vegetables and seven different cookies If she is going to pack a lunch that contains ve items how many ways can she have exactly two pieces of fruit or exactly two cookies 11 Sarah is trying to arrange 5 identical blue candle sticks 7 identical red candle sticks and 3 identical green candle sticks on her shelf How many distinguishable arrangements are possible 12 An experiment consists of selecting 2 coins without replacement from a bowl containing a penny a nickel a dime and a quarter Note We are only observing which coins we select The order we select them in does not matter a Give the sample space for this experiment b Find the event E where E is the event that the sum of the coins is greater than 33011 c Find the probability of the event E 13 For two events E and F we know the PE F 0 2 PF 0 5 and PE F 04 Find PE F U E9 14 Sue lives in a house divided That is her mom is an Aggie and her dad is a Longhoni She is packing her bag to go to Austin for the football game Her drawer contains twelve Aggie shirts and nine longhorn shirts In packing her bag she randomly selects ve shirts a What is the probability that she packs exactly two Aggie shirts b What is the probability that the second shirt she packs is an Aggie shirt if the rst shirt is a longhorn shirt 15 An accounting rm employs 20 accountants of whom 8 are CPAs If a delegation of 7 accountants is randomly selected from the rm to attend a conference what is the probability that at least 2 CPAs will be selected 16 A box contains ve red marbles and 18 green marbles An experiment consists of randomly selecting a sample of 4 marbles out of the box and observing the number of red marbles in the sample Find the probability distribution for this experiment 1 Determine whether each statement is true or false a Given any two events A and B PA B PA PB b The numbers 1 2 and 3 are written separately on 3 pieces of paper An experiment consists of drawing two slips from the bowl and observing the numbers This experiment has 3 events Note The order in which the numbers are selected is not observed c An experiment consists of casting two fair dice and recording the sum of the numbers appearing uppermost The sample space for this experiment has equally likely outcomes d The sample space associated with an experiment is given by S abcde The events E ab and F cd are mutually exclusive Hence the events E and F C are mutually exclusive

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