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# Class Note for MATH 1313 at UH

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Date Created: 02/06/15

Math 1313 Online Summer 09 Week 6 Lj 1463 1L Section 64 Permutations and Combinations Section 64 Permutations and Combinations Definition nFactorial 1 b I y natura num er n M 7 is an arrangement of a speci c set where the order in which the objects are A 3amp9 nw aer as gt gt Formula 39 1 53 e t Ac quot3 where n is the number of distinct objects and r is the number of distinct objects taken r at a time Formula Permutations of n objects not all distinct Given a set ofn objects in which 111 objects are alike and of one kind 112 objects are alike and of another kind and finally 117 objects are alike and ofyet another kind so that n1n2nT n then the number of permutations ofthese n objects taken I at a time is given by n 111an 117 A is an arrangement ofa speci c set where the order in which the objects are t Formula r S n where n is the number of distinct objects and r is the number of distinct objects taken r at a time Example 1 You are in charge ofseating 5 honored guests at the head table ofa conference How many seating arrangements are possible ifthe 5 chairs are on one side ofthe head table 3331 5539 QtK mtA 6 0quot 1 QR J V 1 0 F A V 1 Section 64 Permutations and Combinations Example 2 Find the number ofways 9 people can arrange themselves in a line for a group picture 6 cs 3 l gtnukc xxlak QLag bbl o Example 3 In how many ways can ards be drawn from a wellshuf ed deck of ing Cw k boa DT39Abr Mbsvkqu J O 13 LLSL Example 4 An organization has Wrs In how many ways can the positions of president vicepresident secretary treasurer and historian be lled if not one person can ll more than one position 39 33 32 1 23 339 2amp953 62 4quot e 1oo 93934 Example 5 An organizations needs to make up a social committee Ifthe organization has 25 members in how many ways can a 1 0 person committee be made CLBJSJWD gun PM Example 6 Ifthere are 40 contestants in a beauty pageant in how many ways can the judges award lst prize and 2nd prize if not one person can be awarded lst and 2nd AWLB Sbo Section 64 Permutations and Combinations Example 7 In a production of West Side Story eight actors are considered for the male roles of Tony Riff and Bernardo In how many ways can the director cast the male roles A CL A c 3 53w VH9 v 1 I 1ID i IA I rL39 1y LYE Ri 39 I 2x I m i y we Com a C lbgt 5K9 COMB C DHi Lio Car3 QLHDJD QD3gt 7 39 5 Bums 9 HS 10 T 17be Section 64 Permutations and Combinations Example 12 A coin is tossed 5 times a In how many outcomes do exactly 3 heads occur Irnww quotTLT1 INTI r w mm rumquot sh Cs3 LN Li or g kcuib n suav CKSMA 3 CL S J LL93 3 733347 5 3 404 gIDK o 1 56 0 55 1 Elm x391 139 5 1 A 3 4 1 L10 0 a L10 1 L b0 us Wankva In 10 w 9 c 101373 if01 O illiJofL 0Lv lt c ns 2 LL 3 5 a 19 3LAUgtL L I 3 aka 0X16 305 lt Section 64 Permutations and Combinations Example 15 A committee of 1 6 peopleW is forming a subcommittee that is to be made up of and In how many ways c he subcommittee be formed chug c 49 saw Example 16 A customer at a fruit stand picks a sample of 6 avocados at random from a crate containing 35 avocados ofwhich 8 are rotten In how many ways can the batch contain an rotten avocados c395 l o39LbJD Pbglg0 95H C13QCi logt1awo0 W W 4 L 1 uak 6 3quot V 311lto Popper 1 Order is important for 7 a Combination c Neither d Both Section 71 Experiments Sample Spaces and Events Section 71 Experiments Sample Spaces and Events Anm is an activity with observable n A mt is mm A sample space is a set consisting of all possible sample points of an experiment A Finite Sample Space is a sample space with nitely many outcomes Anmis a m of Wf an experiment Given two events E and F The union ofE and F is denoted by EU F The intersection ofE and F is denoted by En F IfE F 5 then E and F are called mutually exclusive An event is mutually exclusive also means that two events that cannot happen at the same time such as getting a head and a tail on the same toss ofa coin The complement of an event is EC and is the set of all outcomes in a sample space that is not in E Example 1 Consider the experiment of tossing a die a Describe the sample space S ofthis experiment g 5 gagbaud b Let E be the event that an even number is tossed and F be the event that a prime number is tossed Describe E and F in set notation then find the following Ei1b1l 15 F i stk EuF iilSJHlf91 axis swag EUFsCC Ecl 135 Section 71 Experiments Sample Spaces and Events Example 2 A sample of 3 apples taken from a fruit stand is examined to determine whether they are good or rotten The sample space S GGG GGR GRG GRR RGG RGR RRG RRR Let E be the event that at least 2 apples are good and let F be the event that exactly 2 apples are rotten Find the event 23 M 661 616118HK c g unaltul bg Example 3 An experiment consists of selecting a letter at random from the letters in the word DALLAS and observing the outcomes a What is an appropriate sample space for this experiment 5 EDNL15 b Describe the event l39the letter selected is a vowelquot Etszk Example 4 Describe a sample space associated with the experiment oftossing 2 fair coins u lt gfzw ww s T 1 Describe the event of having the same outcome on each coin E23 HHTTE Section 72 De nition of Probability Section 72 Definition of Probability The ratio is the m of an event E that occurs m times after n repetitions Note The Ms a number that lies m If is a nite sample space with n outcomes then the events IE2 are calledm of the experiment ce irobabilities are assigned to each of these simple events we obtain a m The probabilities Ps1 Psz PSn have the following properties 10 s Psi s 1 i 123 n 2 Psl Psz Psn 1 3Ps u 5 Ps Psj iijand ij 123 n Example 1 A fair die is cast List the simple evens 135113 34 313 363 Example 2 The accompanying data were obtained from a survey of Americans who were asked How safe are Americanmade consumer products Rating Number of Respondents Very Safe 95 gr 00 T D 39 396 Somewhat safe 305 Ear70 392 o 40 Nottoo safe 75 37700 o 53 Notsafe atall 10 IO OO 9 q o39 Don39t know 15 1 5 g u 039 lo 3 5 00 Find the probability distribution associated with this experiment IS 35 Ms rash bk W 39Yr0gt o 00 oi a op39L 05 Section 72 De nition of Probability A sample space in which the outcomes of an experiment are equally likely to occur is called a 01quot quotloss a gran l39 A4 a Let Ss1 52 squot be a uniform sample space Then 1 Wm PSz PSn 5 Finding the probability of an Event E 1 Determine the sample space S 2 Assign probabilities to each ofthe simple events of S 31fEs1 52 5k where 51 52 5k are simple events then PU P51 P52 quot39 PSk Note IfE 0then PE 0 Example 3 A pair of fair clice is cast What is the probability that a the sum ofthe numbers shown is less than 5 E 54quot lt 51 Yke3 quotPM 7 Olp b at least one 6 is cast Cc 2A was 0M 15 VWBT bx Discs c you roll doubles SECOND DIE I 39 39i gownsg LJ 39 z 1 II I ul 4 15 y I 23 24 25 26 l M 32 38 341 35 36 2 911 u FIRST DIE 005 I 41 42 43 44 45 46 L 1 039 3 51 52 53 54 55 55 f 61 62 63 64 6 5 66 Section 72 De nition of Probability Example 4 Ifone card is drawn from a wellshuf ed standard 52card deck what is the probability that the card drawn is aAclub 3 n t M new a a m b A red card Lb O 1x cmb Q WAD r1 39 0 5 CAseven q qu WSLti 1 B OQL cl A face card 1 1 1 QLSALVIB L 39L 01quot D eAblack9 f v 3 9 vgguw b 1 1 oosqlt oo 140 290 200 to 0 in 0 I on o 9 to 02 0 vv 0 03 v5 Evvlov 339 v vv o no Aafa ogon a C C C C C O In D N H 7 6 4 44 O 6 O 44 J my 112 i t i 4 a 41quot 1 l 0 900 m o O O m o 0 v 0 no 0 N Section 72 De nition of Probability Example 5 A survey was taken in a certain community about the number ofthe radios in the house the probability distribution was constructed Number of Radios 0 1 2 3 Probability 001 009 053 037 What is the probability of a house chosen at random from this community having a 1 or 2 radios 3quot 1 mm W 7 00 gt053 o L b more than 1 radio 7 U0 3 gt c not even one radio QED 00 o 53 039 gt 060 Popper 2 Four students are running for class president Liz Sam Sue and Tom The probabilities of Sam Sue and Tom winning are 625 1 150 and 320 respectively What is the probability that Tom loses 015 9 01 031 076 3w 39 1339 39 9 Tom ollt 1 m L1 Popper 3 Suppose iou buy a piece of of ce equipment for m Afterlears you sell it for a scrap value of The equipment is depreciated linearly over 5 years The rate of depreciation for the piece of equipment is boob 00 Alcoa 39 h 39 g 2000 g 3800 lJ 600 14000 Popper 4 Find the inverse ofthe given matrix ifit exists a f7 85 3quot oo 6 7 D 43 1V5 v2 13 13 T d Noinverse exist K s 4 Popper 5 A problem is listed below Identify its type Beginning in 2 years a new Booster Club at a certain high school would like to award an outstanding senior on the varsity soccer team with a check of 4000 towards the student39s college tuition Since there is a boy and girl team they wish to award one senior boy and one senior girl the same amount ofmoney Thus the club will need 8000 at the end of 2 years How much must the booster club invest each semiannual 39 din an account that pays 4 per year compounded semiannually to have the desired amp a Amortization Lo ek39 Ab Smr Q Future Value of an Annuity 9 Present Value of an Annuity m Popper 6 Charles will supplement his diet by adding at least 76 mg ofVitamin B and at least 52 mg of Vitamin C per day He can get the desired amount of vitamins from two brands of vitamins Feel Great and Build Up Each pill of Feel Great contains 10 mg ofVitamin B and 5 mg ofVitamin C Each pill ofBuild Up contains 8 mg ofVitamin B and 9 mg ofVitamin C Each pill of Feel Great costs nd each pill ofBuild Up costs Suppose you want to know how many pills per day of each brand Charles s Let the number of pills of Feel Great and the number of pills ofBuild Up Which ofthe following if any would be an m 7 a MinimizeC 11226 9y MinimizeC 11226 11531 c MinimizeC 11526 By d MinimizeC 11526 11231 Popper 7 Given that the augmented matrix in rowreduced form below is equivalent to the augmented matrix of a system oflinear equations Determine whether the system has a solution and nd the solutions to the system ifthey exist 100l 7 1 39739 010l 10 0006 O v 391 b x 7y 10z6 cx 7y 10 DltW a 167 10 V Qlyl 3 Lo Lu 07 b Popper 8 Give the optimal solution point and value KO 3 Maximize P 5x By xys9 quot39 x2ySlO s x20y20 039 t j a P SOat100 cub b P72at09 drab J 9 r 9V5 d P40at05 1 3 0 quot5 3 8 L xquot Popper 9 Your brother wou d H shoud he make Into an account 0 m s mums Cm ars forthe purchase of a new car What to attain his goa a b 60350 c 61350 003 3 d 62352 35 DOD k 11 D L quot T39 Mg m D L v Popper 1 0 Mark B

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