Finite Math with Applications
Finite Math with Applications MATH 1313
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Math 1313 Section 73 Rules of Probability In this section you will learn five rules that you can use to compute probabilities Here they are Let S be the sample space of an experiment and suppose E and F are events of the experiment Rule 1 PE Z Ofor all E Rule 2 PS 1 Rule 3 HE and F are mutually exclusive events then PE U F PE PF Rule 4 If E and F are any two events of an experiment then PEUF PEPF PEnF Rule 5 If E is an event of an experiment and E 5 denotes the complement of E then PEPE 1 and therefore PE 1 PE and PE1 PE We will use Rules 3 4 and 5 extensively in solving the problems in this lesson Example 1 The SATMath scores of the senior class at ahigh school are shown in the table 600lt xS 700 500 lt xS 600 400 lt xS 500 300lt xS 400 xlt300 a Construct the probability distribution for this data Math 1313 Class Notes 7 Section 73 Page 1 of5 b If a student is selected at random what is the probability that his score i more than 500 ii less than or equal to 400 iii greater than 400 but less than or equal to 700 Example 2 Suppose a card is drawn from a wellshuf ed deck of 52 playing cards LetE be the event that the card is a face card LetF be the event that the card is a heart Find PE UF Find the probability that the card is either a 4 or a 7 Find the probability that the card is a heart or a diamond Math 1313 Class Notes 7 Section 73 Page 2 of5 Example 3 Among 500 freshmen enrolled in the business school 320 are taking an economics class 225 are taking a math class and 140 are enrolled in both What is the probability that a student selected at random from this group is enrolled in a economics andor math b exactly one ofthese courses c neither ofthese courses Example 4 Let E and F be two events of an experiment with sample space S Suppose PE 45 PF 7 and PEmF 31 Compute a PE UF b PE nF Math 1313 Class Notes 7 Section 73 Page 3 of5 c PEuF d PEuF Example 5 Let E and F be two events of an experiment with sample space S Suppose PE 38 PF 36 and PE OF 41 Compute a PE U F b PE UF 0 PEuF d PE UF Math 1313 Class Notes 7 Section 73 Page 4 of5 Example 6 You are the chief for an electric utility company The employees in your section cut down trees climb poles and splice wire You report that of the 96 employees in your section 6 cannot do any of the three management trainees 25 can cut trees and climb poles only 20 can cut trees and splice wire but cannot climb poles 10 can do all three 15 can only cut trees 12 only climb poles and a total of 30 can only do one task Find the probability that a randomly selected employee in your section can to exactly 2 tasks Find the probability that a randomly selected employee in your section cannot cut down trees Math 1313 Class Notes 7 Section 73 Page 5 of5 Math 1313 Section 81 Distributions of Random Variables In Chapter 8 you will learn about random variables their probabilities distributions and three quantities that you can compute from a distribution of a random variable expected value variance and standard deviation You will also work some problems concerning two very common distributions of random variables the binomial distribution and the normal distribution The entire chapter relies on the concept of a random variable which we now de ne De nition A random variable is a function which assigns a number to each outcome of a chance experiment The domain of this function is the set of outcomes of the experiment and the range is the set of real numbers Example 1 An experiment consists of rolling a pair of fair dice and observing the numbers that fall uppermost on each die We can define a random variable X by assigning to each outcome of the experiment the number that is the absolute value of the difference between the two numbers rolled Notation We use capital letters printed or block letters hand written to denote random variables We usually use X first Then if the problem requires more than one random variable we use Y and then Z Example 2 Suppose a coin is tossed three times and we observe the whether heads or tails lands up on each toss and record the sequence of heads and tails Let the random variable X denote the number of tails observed in each outcome of the experiment a Find the value assigned to each outcome of the experiment by the random variable X Math 1313 Class Notes 7 Section 81 Page 1 of5 b Find the event comprising the outcomes to which a value of 2 has been assigned by the random variable X c Find the probability distribution of the random variable X Notation The question in part b above can be summarized by this notation X 2 Example 3 LetX denote the random variable that gives the sum of the faces that fall uppermost when two dice are cast a What are the possible values assigned to the random variable b Find the event comprising the outcomes to which a value of 7 has been assigned by the random variable Math 1313 Class Notes 7 Section 81 Page 2 of5 c List the outcomes of the experiment in the event for which the random variable is assigned the value of 10 X10 Example 4 A survey was conducted by a technology firm among 1000 families to determine the distribution of televisions in average households Here are the results IIN00fTVs 01 2 3 4 I5I6II FREQUENCY OF OUTCOME 10 230 295 325 116 20 4 Construct a probability distribution of the random variable X where X denotes the number of TVs in a randomly selected household Find the probability that a randomly selected family has more than 2 TVs This can be written as nd PXgt2 Math 1313 Class Notes 7 Section 81 Page 3 of5 De nition A histogram is a bar graph of a probability distribution Example 5 Draw a histogram from the probability distribution in Example 2 Example 6 Draw the histogram for the following probability distribution Math 1313 Class Notes 7 Section 81 Page 4 of5 You ll be able to solve some problems using a probability distribution of a random variable Example 7 Consider the probability distribution of the random variable X x 02468 PXx12214715 5 Find a PXZ4 b PXlt8 c P2ltXS6 d PXZO e PXS10 Math 1313 Class Notes 7 Section 81 Page 5 of5 Math 1313 Section 86 Applications of the Normal Distribution We have reached the last section of the text that we will cover in the course This section covers some word problems involving normally distributed random variables You may notice that there is some similarity between the wording of these problems and the wording of the Chebychev s Inequality problems You will know to use the normal distribution to solve a problem when you are told that the random variable is normally distributed If those words aren t there then you are probably looking at a Chebychev s Inequality problem Example 1 According to the data released by the Chamber of Commerce the weekly wages of factory workers in a small city are normally distributed with a mean of 1000 and a standard deviation of 100 What is the probability that a randomly selected factory worker who lives in the city makes a weekly wage of a less than 900 b more than 1200 c between 850 and 1150 Math 1313 Class Notes 7 Section 86 Page 1 of4 Example 2 The heights of a certain species of plant are normally distributed with a mean of 30 inches and a standard deviation of 3 inches What is the probability that a plant chosen at random will be between 24 and 32 inches tall Sometimes we can use a normal distribution to approximate a binomial distribution We will want to do this when the number of trials of the binomial distribution gets larger Suppose we are given a binomial distribution associated with a binomial experiment involving 71 independent trials each with a probability of success p and a probability of failure q Then ifn is large andp is not close to 0 or 1 the binomial distribution can be approximated by a normal distribution with unpand 0quot1 q Using this mean and standard deviation these problems work just like the last few except for ONE LITTLE TWIST When we are approximating the binomial distribution by the normal distribution we are actually estimating the area of the rectangles in the histogram by the area under a normal curve But for X5 PX5 is represented by a rectangle that runs from 45 to 55 with height PX5 So the area for less than ve successes actually starts at 45 not 5 In picture form Math 1313 Class Notes 7 Section 86 Page 2 of4 My particular technique for keeping this straight I Start with the binomial random variable X and write the probability in question in the form PX Z a or PX lt b or Pa lt X S b as determined by the question 2 Then create a normal random variable Y that gives the appropriate edges for the rectangles in the histogram we wish to add the areas of So for example if my question asks for P4 S X lt 9 I want to include the rectangle centered at 4 but not the one centered at 9 ie I want to estimate the areas of the rectangles at 45678 My new normal variable becomes P35 ltY lt 85 3 Then follow the procedure above to turn the normal variable into a standard normal variable so we can read the probability off the chart 35 ultZlt85 u 039 039 P35 lt Y lt 85 P 4 Calculate the numbers above and use the looking up the probability on the standard normal distribution table techniques for the last section Example 3 Use the normal distribution to approximate the binomial distribution A coin is weighted so that the probability of obtaining a head on a single toss is 03 If the coin is tossed 25 times what is the probability of obtaining a fewer than 10 heads b Between 9 and 12 heads inclusive c More than 8 heads Math 1313 Class Notes 7 Section 86 Page 3 of4 Example 4 A company claims that 35 of the households in a certain community use their Squeaky Clean cleanser In a neighborhood of 10 households what is the probability that between 40 and 45 households use the cleanser This is a binomial experiment Use the normal distribution to approximate the probability Math 1313 Class Notes 7 Section 86 Page 4 of4 Mann 1313 Sectinn 71 Experimenls Sample Spaces and Events T ermlnnlngy 1n thls chapter you wlll learn the basles ofprobablllty We needlo startwlth several defmmons De niu39n An experimenl ls an activle Wlth observable results Examples ofexpenments are 0 o ll g a dle and observlng be number on the uppermostface o Fllppmg a eoln and observlng wnellner ll lands on leads ortalls o Tesung a battery from a batch at a manufaelurer39 s and observlng ll ll works orls defeeuve De nidnn A Samplepnintls one outcome ofan expenmenl De nidnn A sample spacers the set of all outcomes of an expenmenl If a sample spaee nas aflmte number of elements ll ls ealleol aflmte sample spaee De nitin An evenl ls a subset othe sample spaee of an expenmenl whatwe eovereolln chapter 6 ln our study of expenmenls and events for an expenmentls are sample spaee so when we deal Wlth he s t probablllty we eall the unwersal spaee S We glve events names sueln as E F G etc When we referto a due we mean Math1313 ClassNotesSecuon71agelof4 Example 1 Suppose E is the event of rolling a die and observing an even number on the uppermost face Express E using set notation E 1 1 1 I Definition The impossible event is an event that cannot occur De nition The certain event is an event that is guaranteed to occur You will need to be able to do state the sample space associated with an experiment You will also need to be able to use set notation to describe stated events Example 2 Suppose you roll a die and observe the number on the uppermost face a Describe the sample space for this experiment b Suppose an experiment consists of rolling a die an observing the number that appears on the uppermost face and E is the event of rolling a number greater than 10 Describe E using set notation 021 Ezigr wssc c Suppose an experiment consists of rolling a die an observing the number that appears on the uppermost face and F is the event of rolling a number 1essthan10 DescribeFusing setnotation F1 IQ33L EgQ 33 MW Set Operations Since each event is a set we can use the set operations that we learned about in Chapter 6 Notation and Venn diagrams are the same as previously discussed Definition Two events E and F are said to be mutually exclusive if E H F p That is the events are mutually exclusive if the sets E andF are disjoint Math 1313 Class Notes 7 Section 71 Page 2 of4 Example 3 Suppose an experiment consists of rolling a die and observing the number on the uppermost face Then S 1 2 3 4 5 6 Suppose E andF are events of the experiment Where E is rolling an odd number and F is rolling a prime number Find aEnF E g H 3 5 F QgEB Ear 4315 bEuF EIXCQIBISE 0E ECquot iQl fl g PC l i dEnF EMEC 3E ch EcIFC t gt Example 4 An experiment consists of ipping a coin three times and observing the resulting sequence of heads and tails a Describe the sample space 5 HHHHHT HTH H77 T4474ij TTTH T rf b Determine the event E that exactly one tails appears E Mar HTMTHH c Determine the eventF that the rst and third tosses are different p 511117 HTT TH Riff3 Math 1313 Class Notes 7 Section 71 Page 3 of4 Example 5 Suppose an experiment consists of rolling a pair of dice and observing the number that falls uppermost on each die a Describe an appropriate sample space SECOND DIE 11 12 13 14 15 16 3 1 3 2 33 84 36 E1 1 211 22 23 24 25 E1 1 FIRST DE 411 42 4 3 45 46 0 51 52 54 55 5 6 63 6 4 65 66 b Deterrninethe eventEthatthe sum ofthe pperrnostnumbe is 8 11 gain C9 33 93 3 23 than 9 c Determine the eventF that the sum of the uppermost numbers is greater z 1 5 a 4345 g 16 616 Math 1313 Class Notes 7 Section 71 Page 4 of4 1313isect7601 Page 1 of8 Math 1313 ection 61 Sets and Set Operations In Chapter 6 we will learn to work with sets This chapter sets the stage for everything else we will do in this course so this section is important Definitions and Notation The word set is undefined but we will use the common understanding that a set is a collection of objects From the description of a set you should be able to decide if an object is in the set or not Definition An element is an item or object in a set Notation We name sets with capital letters like A B C and usually eleme are listed rdescribedb some ro ert fat Awa yefm x 3 Roster otation for a set names the set and lists the elements in braces SetBuilder Notation for a set names a set and describes the set as the set ofall elements that have a given property Example 1 Our set is the first five counting numbers and we will call the set A Roster Notation A 7 l A 3 F g LQQk m l 39 39 U Om SetBuilder Notation A k 5 3 I Q 1 L l 3 7c 15 oohl39fn V 73 A 65 im at W Example 2 Describe the set Bred white blue in setbuilder notation 0 96 ryi g 4Co 6r an U Example 3 Describe the set gtltgtlt is a natural number less than 13 in roster notation 39 21 3 45 477 2 2 Chm1 1313isect7601 Page 2 of8 Example 4 Write in set builder notation the set C with elements the people in this class rc walhg m VllocH 39 59 cl oil 7 9 36 Definition Two sets A and B are equal written AB if every element of A is also an element of B and every element of B is also an element of A In other words two sets are equal if they have exactly the same elements Example 5 Suppose F 246810 and G gtltgtlt is an even natural number less than 11 Does F G lt 19 x F kw Flh an eveVt Wo t UVMm ll M F m 2 6 L as rl Definition The set A is a subset of the set B if every element ofA is also an element of B This is written A g B Example 6 Suppose G 12346810 H 2468 and J 23510 Which th gigowingig ue 6g H 3 k can 4 Q I f bJgG NO W 5 1 in J W 39T l 6 Leg ouoz A SA CGQH Na 3 l rtce ls l 0 56 m H W No 3 25 but NOT m H Definition The set A is a proper subset of B if A is a subset of B and there is at least one element of B that is not in A The statement A is a proper subset of B is written A C B N 39 07 5 llCl l7 1313isect7601 Page 3 of8 Example 7 Suppose A13579 B135 C13567 and D13579 A Is BCA Ll as B IsCCD N Problem l9 C ISACD I00 bgca ug are Howtgtell if a set A is NOT a subset of the set B rim ONE elemba cg fl lg NOT M elm Definition The set with no elements is called the empty set We write the empty set as Q The empty set is a subset of every set Why Try to find an element in the empty set that is not in the other set You CAN T because there are no elements in the empty set So the empty set qualifies by default as a subset of every set p f i Example 8 List all the subsets of the set C 12 I a z 1 3 z J 2 3 K7 at PFoPef 1b 5 S Venn Diagrams A Venn diagram is a rectangle which represents the set of all of the items we are interested in with circles which may overlap inside the rectangle representing subsets of the larger set We use Venn diagrams to picture how sets are related U Zilll3l f5 a U 5 A Ll3lf l 151 Q3 5 1313isect7601 Page 4 of8 Definition The universal set is the set of all items of interest In this chapter we will denote the universal set by U Example 9 Suppose A and B are subsets of U Use Venn diagrams to illustrate each of the following statements The sets A and B are equal V A E B The set A is a proper subset of the set B U 6 6 C The sets A and B have no elements in common l b s N 97 Pom Fl J M 7 L l39l V 5 D The sets A and B are not subsets of each other Set Operations I A B A O39T w39r 4 Suppose U is a universal set and A and B are subsets of U We have three operations that can be performed on these sets Operation Notation Meaning Venn Union A B U ix 7c lg n A Q in 3 E 1313isect7601 Page 5 of8 Intersection A m B Ifl MM Complement A NW rf A 7 Example 10 Suppose U 12345678910 A 13579 B 246810 and C 23467 Find DBoC iiAUB lg i ll l g75719gizlc9M iii B iv A B BCZI735 7I 7Q AGB E it Km 8 f Definition Two sets A and B are called disjoint if AnB Q that is A and B have no elements in common Properties of Set Operations Let U be the universal set and let A B and C be subset of U Properties of Set Complements v 39 111543 Ucsltk o a l lll l 5 In U ZVSQ V 2 U M 0m 4 m 12 58 U 3 A A 4 AUA U 7 m 5 AHA 1313isect7601 Page 6 of8 Properties of Intersection and Union J W AUBBUA 7 6 W N 5 mi D oc AnBBnA 6K dam AUBUCAUBUC 3 W m M A BnCAnB C AUB CAUB AUC arrw ti 991 P N AnBUCA BUA C DeMorgan s Laws 1 AUEEAEHBE OWLakde UM 3 Q o quot in ify sec 1cm 0 2 A B A UB 1 com m Of in rs airc II I U IWiCO 0 QQMJ Example 11 Suppose U12345678910 A12357 B23569 and C348910 Find i AnBY UC firiB fag53 An3 Q lo AMT uc t 5 545759 9 d a iiA BUC AC E4IGI7 O No8 i lc C 5r254gt73 CGBU UCC f 3919 g 7 MBUCYDAC jall 3 Lf g Q g 71 0 BUCWQ 7 6UCC 7 1313isect7601 Page 7 of8 Example 12 Let Ugtltgtlt is an enrolled student at the University of Houston Agtltgtlt is a student at UH enrolled in Math 1313 Bgtltgtlt is a student at UH enrolled in a history class Cgtltgtlt is a student at UH enrolled in a business class Describe in words IA B 39 d9 l bat01 at Aisllayjclass ii A UC H C sfuMS Nof A ACUC Slum axle Ads MSS 01 U OCEAVBYGC llgcl by C In 1 l3 ll 59 U D A S C MGR w hogi f UL ClW NW iv A UB Pl VD C N07quot 3 07 i 5 My fic mat in 33 AKRD UfAKms gusM 96 W fatCl nf AIS 1467 l Actec Nc el l33 01 40 6 A 51579 Describe in set notation v The set of all enrolled UH students who are taking a history classa rli are not takin Math 1313 C a C lb n R c viThe set of all enrolled UH students who are not taking Math 13L13 nor are they taking a busin ass N 39 L C lt ham 5 vii The set of all enrolled UH students who are taking Math 1313 and a business class but not a history class VH W A0530 6 1313isect7601 Page 8 of8 Example 13 Let U be a universal set and let A B and C be subsets of U W Describe using Venn diagrams C U5 e aquot L r 5 7 Vic9 a 5 9 c H 4 GA AG ii AmBY iii AmB iv AnB C V AUB C Math 1313 Section 84 Binomial Distribution We will nish the course by looking at two common distributions of random variables the binomial distribution and the normal distribution In this section we ll take a look at the binomial distribution De nition A binomial trial or a Bernoulli trial is an experiment which has exactly two outcomes which we can label as success and failure De nition A binomial experiment is a sequence of binomial trials Binomial experiments have these properties in common the number of trials is xed there are 2 outcomes success and failure the probability of success is the same in each trial the trials are independent ie one doesn t depend on the outcome of another bP N Example 1 A fair die is cast four times Find the probability of obtaining exactly one three in the four trials Is this a binomial experiment We use some standard notation in working with binomial experiments p probability of success q probability of failure pqlsoqlipandpliq n number of trials In a binomial experiment we can define a random variable X to be the number of successes in n independent trials Computing the Probability of x Success in n Trials of a Binomial Experiment We can compute the probability of x successes in n independent trials of a binomial experiment with p probability of success and q probability of failure by PX x Cn xp qquotquot Math 1313 Class Notes 7 Section 84 Page 1 of5 Example 2 A fair die is cast four times Find the probability of obtaining exactly one three in the four trials Example 3 Consider the following binomial experiment The probability that a randomly selected student at a certain college will graduate with a bachelor s degree after four years of study is 78 From among a group of 15 students at this college what is the probability that a all of them will graduate after four years of study b exactly 10 of them will graduate after four years c At least one graduate after four years We compute the mean variance and standard deViation of binomial probability distributions using different formulas from the ones we have already learned Math 1313 Class Notes 7 Section 84 Page 2 of5 Mean Variance and Standard Deviation of a Binomial Random Variable If X is a binomial random variable associated with a binomial experiment consisting of 71 trials with probability of success p and probability of failure q then 1 E X quotp VarX npq 039 VarX M Example 4 Find the mean variance and standard deviation of the experiment described in example 3 Example 5 Determine the probability of each event a The probability of at least four successes in 5 trials of a binomial experiment in which p 65 State the mean variance and standard deviation b LetX be the number of successes in 7 independent trials of a binomial experiment in which the probability of failure is 78 Find P2 S X S 5 Math 1313 Class Notes 7 Section 84 Page 3 of5 Example 6 In a certain congressional district it is know that 40 of the registered voters classify themselves as conservatives If ten registered voters are selected at random from the district what is the probability that four of them will say they are conservatives Example 7 Six newly married couples agree to be part of a 20 year survey Studies show that the probability that a marriage will end in divorce is 6 within 20 years of its start What is the probability that out of the 6 surveyed couples at the end of 20 years a none will be divorced b all will be divorced c at least two couples will be divorced Example 8 State the mean variance and standard deviation of the binomial experiment state in Example 7 Math 1313 Class Notes 7 Section 84 Page 4 of5 Example 9 The Krazy Toy Company makes toy phones The quality control department estimates that 5 of the phones made are defective A random sample of 20 phones is made from a large shipment of toy phones What is the probability that the sample contains at most 2 defective phones What is the probability that at least one phone is defective Math 1313 Class Notes 7 Section 84 Page 5 of5 13137sect7705 Page 1 of8 Math 1313 Section 75 Conditional Probability We ll start this section with an example Example 1 Two cards are drawn at random and without replacement from a wellshuffled deck of 52 playing cards a What is the probability that the first card drawn is an ace b What is the probability that the second card drawn is an ace given that the first card drawn was not an ace c What is the probability that the second card drawn is an ace given that the rst card drawn was an ace In this experiment we start with a sample space which contains 52 elements We have two events the first draw and the second draw Since there is no replacement the number in the sample space becomes smaller In parts b and c of Example 1 we learn more about the experiment which changes the sample space which changes the probabilities These are examples of conditional probability Math 1313 Class Notes 7 Section 75 Page 1 of8 13137sect7705 Page 2 of 8 We can demonstrate conditional probability using Venn diagrams Suppose we have an experiment with sample space Sand suppose Eand Fare events of the experiment We can draw a Venn diagram of this situation Now suppose we know that event Ehas occurred This gives us this picture So the probability that Foccurs given that Ehas already occurred can be expressed as nEnF nEnF nS PEnF HF m M PE 39 nS Conditional Probability of an Event If Eand Fare events of an experiment withPE 2 0 then the conditional probability that the event Fwill occur given that the event Ehas already occurred is PE nF PFlE HE Math 1313 Class Notes 7 Section 75 Page 2 of8 13137sect7705 Page 3 of 8 Example 2 Let A and Bbe events of an experiment with sample space 5 Suppose PA6 PB7 and PAnB45 Find a PAiB b PBiA Example 3 Let A and Bbe events of an experiment with sample space 5 Suppose PA48 PB 63 and PAnB 35 Find a PAiB b PBiA c PB iA d PAiB Math 1313 Class Notes 7 Section 75 Page 3 of8 13137sect7705 Page 4 of 8 Example 4 A pair of fair dice is tossed and the number on the uppermost face is observed What is the probability that the sum of the numbers falling uppermost is 6 if is it known that one of the numbers was 2 Example 5 A survey showed that 40 of all convenience store shoppers buy milk 30 buy bread and 25 buy both milk and bread a If a randomly selected shopper buys milk what is the probability that she will also buy bread b What is the probability that a randomly selected shopper buys ony bread Math 1313 Class Notes 7 Section 75 Page 4 of8 13137sect7705 Page 5 of 8 Product Rule Sometimes we know the conditional probability and are interested in finding PEnF We can solve the conditional probability formula for PEnF to get the product rule HF E PEnF PE PE FPEPFlE We will use tree diagrams for these kinds of problems to help organize the information we know The first branch of the tree is the first trial and the second branch is the second For the above formula we could illustrate the product rule by Example 6 An urn contains 5 blue marbles and 7 green marbles a Two marbles are drawn in succession and without replacement from the urn What is the probability that both marbles are green b Two marbles are drawn in succession without replacement from the urn What is the probability that the second marble is green Math 1313 Class Notes 7 Section 75 Page 5 of8 13137sect7705 Page 6 of 8 Example 7 Urn X contains 9 white marbles and 3 blue marbles Urn Y contains 5 white marbles and 6 blue marbles One of the two urns is chosen at random with equal likelihood of being chosen and then a marble is drawn from that urn What is the probability that the marble drawn is blue Example 8 A new lie detector test has been devised and needs to be tested before it can be used by the police One hundred people are selected at random and each person draws and keeps a card from a box of 100 cards Half the cards instruct the person to lie and the other half instruct the person to tell the truth The test indicated lying in 85 of those who lied and in 7 of those who did not lie What is the probability that for a randomly chosen person the person was instructed not to lie and the test did not indicate lying Math 1313 Class Notes 7 Section 75 Page 6 of8 13137sect7705 Page 7 of 8 Example 9 Mary K Cosmetics estimates that 29 of the country has seen its commercial and if a person sees its commercial there is a 13 chance that the person will not buy its products The company also claims that if a person does not see its commercial they still have a 24 chance of buying the company s products What is the probability that a randomly selected person in the country will not buy its products Independent Events Two events A and Bare independent if the outcome of one event does not depend on the outcome of the other event Test for Independence of Two Events Two events A and Bare independent if and only if PAnBPAPB This can be extended to any finite number of events Math 1313 Class Notes 7 Section 75 Page 7 of8 13137sect7705 Page 8 of 8 Example 10 Determine if the two stated events are independent The experiment is drawing a card from a wellshuffled deck of 52 playing cards a A the event of drawing a face card B the event of drawing a heart b C the event of drawing a heart D the event of drawing a club c E the event of drawing a king F the event of drawing a red card Note It is possible for the events to overlap and still be independent Many students think that mutually exclusive events and independent events mean the same thing They do not Example 11 Two events A and Bare independent PA43 and PB31 Find PAUB Math 1313 Class Notes 7 Section 75 Page 8 of8 Math 1313 Bayes Theorem In section 75 we were interested in nding the probability of the second event on the tree diagram if we knew that the rst event had occurred In this section we reverse the roles If we know that the second event occurred what is the probability of a given outcome of the rst event We can use a principle called Bayes Theorem to solve problems of this type This theorem is an application of conditional probability We ll use the conditional probability formula and the product rule to solve these problems 9 mm 670M133 Mm Example 1 Three factories A B and C produce engine components Factory A produces 40 of the total components Factory B produces 35 and Factory C produces the remaining 25 A quality control analysis shows that 6 of the components from Factory A do not meet speci cations 5 from Factory B do not meet speci cations and 4 from Factory C do not meet speci cations A component is selected at random from total output and is found to be defective What is the probability that it was produced at Factory A PPM PfM39WQ Wm m cw b P H 7th m c Valbg fl KOL t 3 0 42 km W 8 Math 1313 Class Notes 7 Section 76 Page 1 of4 X o LbQ flaw P A D119 fl K40LD3SX03 KW Example 2 In a recent senatorial election 50 of the voters in a certain district were registered Democrats 35 were registered Republicans and 15 were registered Independents The incumbent Democratic senator was reelected over her Republican and Independent challengers Exit polls showed that she gained 75 of the Democratic vote 25 of the Republican vote and 30 of the Independent vote Assuming that the exit poll was accurate what is the probability that a vote for the incumbent was cast by a registgred Republican m w k 9025mm 939219027 W IS M67 Pfegongrgt MRI9i FOP Wm faker agxng 25gtlt35152lt3 1 33 22 WWW st x3 Math 1313 Class Notes 7 Section 76 Page 2 of4 I 7 9 LF 0Q ON Example 3 A screening test has been used to determine if a person has a certain vision impairment Among people Who have the vision problem the screening test has been shown to be accurate 92 of the time The test gives a false positive reading 8 of the time ie detects the condition among people Who do not actually have it The vision impairment affects 7 of the population a Suppose a person test positive for the condition What is the probability that the person actually has the condition b Find the probability that a person tests negative for the condition but actually It TM 7 a 907 3N 73 a Wes M M 007x72 323 gt20 LtG tO 90 n timm 00 X 0 f 0031 6amp3 at PH Ne F09 W 7 6 Math 1313 Class Notes 7 Section 76 Page 3 of4 Example 4 Anurn contains 10 green balls and 6 blue balls Two balls are drawn in succession and without replacement What is the probability that a the second ball drawn was blue b the rst ball drawn was green given that the second ball drawn was blue c the second ball drawn was green d the rst ball drawn was gree given that the second ball drawn was green I 2 32 2 W 435 W 633 0 0 we 749 6437k 6 03 0593 T ni higbfgg ENE 01 GU15 375 Q New 9 7 EM p8623 WW 0WO7 c S OW W652 3 73 g 1an amp mlt3 clkassNogie2ion76flfQe4of4 42 iquot 0 g D Q r 0 w t ghc Math 1313 Section 74 Use of Counting Techniques in Probability Some of the problems we will work will have very large sample spaces or involve multiple events In these cases we will need to use the counting techniques from chapter 6 to help solve the probability problems In particular we ll work with the multiplication principle and combinations In this section we will look only at problems in which each outcome is equally likely The sample spaces for these types of problems are called uniform sample spaces Example 1 Suppose an experiment consists of tossing a coin three times and observing which side faces up after each toss a Describe the sample space for this event b What is the probability that HHH occurs c What is the probability that exactly two tails occur Suppose we tossed the coin ve times instead of three times The tree diagram and sample space would be too cumbersome to work with We ll use a different method Computing the Probability of an Event in a Uniform Sample Space nE nS 39 Let S be a uniform sample space and let E be any event Then PE Math 1313 Class Notes 7 Section 74 Page 1 of5 Example 2 Suppose we toss a coin ve times and observe which side faces up after each toss What is the probability that exactly two tails occur Example 3 Suppose we toss a coin ve times Find the probability that it lands heads more than once Method 1 Method 2 Math 1313 Class Notes 7 Section 74 Page 2 of5 Example 4 Five cards are selected at random without replacement from a well shu led deck of 52 playing cards a Find the probability that at least one of them is a heart b Find the probability that 2 aces and 2 kings are chosen Note that the phrase at least one is a clue that you can use PE l PE Example 5 A department store receives a shipment of 100 computer games of which 6 are defective A customer buys 5 of these games What is the probability that a exactly 2 of them are defective b at least one of them is defective Math 1313 Class Notes 7 Section 74 Page 3 of5 c at most one of them is defective Example 6 From a group of 5 freshmen 6 sophomores 4 juniors and 3 seniors what is the probability that a staff of 3 freshmen 3 sophomores 2 juniors and 2 seniors will be selected for the yearbook staff Assume that each student is equally likely to be chosen Example 7 A fivemember committee is to be formed at random from a group of 7 Democrats and 9 Republicans Find the probability that the committee will consist of a two Democrats and three Republicans Math 1313 Class Notes 7 Section 74 Page 4 of5 b at most 4 Republicans c at least one Democrat Example 8 A class contains 30 students 18 girls and 12 boys A group of 5 students is chosen at random from the class to make a presentation to the school board What is the probability that the group making the presentation is made up of a more girls than boys b at least 1 boy c 2 or 3 boys Math 1313 Class Notes 7 Section 74 Page 5 of5 Math 1313 Section 85 Normal Distribution In addition to these notes you will need to bring a copy of pages 1177 7 1178 the standard normal distribution table to class This table is located in an appendix in the online text In this section we will study normally distributed random variables These are continuous random variables We can see the difference between nite discrete random variables and continuous random variables by looking at graphs of them side by side 2 0311 025 02 5 4 3 2 r r 2 3 a 015 01 005 With a nite discrete random variable we can create a table of values and list each number to which the random variable assigns a value With a continuous random variable we can t do that There are an in nite number of values We will look at a probability density function instead Here are properties of probability density functions 1 fx gt 0 for all values ofx 2 The area between the curve and the x aXis is 1 Some probability density functions have additional properties These are called normal distributions 3 there is a peak at the mean 4 the curve is symmetric about the mean 5 687 of the data is within one standard deviation of the mean 9545 of the area is within two standard deviations of the mean 9973 of the area is within three standard deviations of the mean Math 1313 Class Notes 7 Section 85 Page 1 of7 6 the curve approaches the x axis as x extends indefinitely in either direction Some normal distributions have additional properties These are called standard normal distributions 7 mean is 0 8 standard deviation is l Notation Pa SX SbPaltX bPaSX ltbPaltXltb sowe will use Pa lt X lt b as our standard notation This is true because the area under a point is 0 Notation We will denote the random variable which gives us the standard normal distribution by Z Math 1313 Class Notes 7 Section 85 Page 2 of 7 Using the Standard Normal Distribution Table to Find Probabilities The standard normal distribution mble gives the area between the curve and the x axis to the le of the line x z This area corresponds to the probability that Z is less than 2 or PZ lt z 3 2 1 J 1 2 3 1 We ll use the table together with the properties listed above to answer many types of questions How to read the mble For a particular value 2 of our random variable 2 will be rounded to the nearest hundredth the probability in the table is PZltz This number corresponds to the area under the curve left ofthe vertical line x z Ifz 135 to find the probability PZlt135 from the table find the row with 13 in the rst column and then find the column headed by 005 The number that is in the 13 row and the 005 column is the probability that the random variable will have a value LESS than 135 We will 39 ofprobability J L 39 There are several ditferent techniques to using the table and we will walk through them one at a time 1 Probability Less than a number Example 1 Find PZlt136 Math 1313 Class Notes 7 Section 85 Page 3 of7 Example 2 Find PZ lt 047 11 Probability Greater than a number This is the area under the curve from the vertical line X z to the right The total area under the curve is 1 so 1 7 area to the left ofX 2 equals area to the right ofz OR since the graph of the normal distribution is symmetric with respect to the y aXis the area to the RIGHT of X z is the SAME as the area to the LEFT of X 2 So to nd PZ gt 2 look up z in the chart and that is the probability EXample 3 Find PZ gt 178 EXample 4 Find PZ gt 105 111 Probability Between two numbers The probability that the random variable takes on a value between two numbers is the area under the curve between the vertical lines at these values This area is equivalent to the area to the left of the larger number minus the area to the left of the smaller number So to find the probability between two numbers look up both numbers on the chart and take the difference PaltZltbPZltb7PZlta Example 5 Find P l25 lt Z lt 203 Math 1313 Class Notes 7 Section 85 Page 4 of7 Example 6 Find P 68 lt Z lt 141 Sometimes we want the nd 2 the number inside the parentheses given a probability IV Find z if PZ lt z a palticular number Look up the number in the chart The row heading plus the column heading is 2 Example 7 Find 2 if PZ lt z 8944 Example 8 Find 2 if PZ lt z 0401 V Find z if PZ gt z a palticular number This says that the area to the right of the number 2 is the number given So the area to the LEFT of 72 is this probability Look up the probability on the chart nd the row and column The answer is the NEGATIVE of this number Example 9 Find 2 if PZ gt z 9463 Example 10 Find 2 if PZ gt z 0132 There is also another version of this Example 105 Find 2 if PZlt z 3228 Math 1313 Class Notes 7 Section 85 Page 5 of7 VI Find z if the probability between z and z is given We are given a probability which is the area under the curve from 72 to z for some number 2 How to nd 2 Take the number given add 1 then divide that by 2 Look this number up on the chart the column plus row is 2 Example 11 Findz if P z lt Z lt z 9812 Example 12 Find 2 if P z lt Z lt z 5408 Example 13 Find 2 if P z lt Z lt z 1820 Next we need to look at what to do with a distribution that is normal but not standard normal that is a normally distributed random variable with mean other than 0 and standard deviation other than 1 Math 1313 Class Notes 7 Section 85 Page 6 of7 We can convert any problem involving probability of a normally distributed random variable to one with a standard normal random variable thus allowing us to use the table Here s how Suppose X is a normal random variable with E X u and standard deviation 039 ThenX can be converted to the standard normal random variable using the formula Then we can evaluate the new problem using the techniques presented earlier in the lesson Example 14 Suppose X is a normally distributed random variable with u 50 and 039 30 Find PX lt 95 Example 15 Suppose X is a normally distributed random variable with u 85 and 039 16 Find PX gt 54 Example 16 Suppose X is a normally distributed random variable with u 100 and 039 20 Find P85 lt X lt 110 Math 1313 Class Notes 7 Section 85 Page 7 of7 Math 1313 Section 82 Expected Value You already understand the concept of the average of a set of numbers Sometimes we want to know the average of a set of data but we only have the probability distribution available not the raw data In this case we can sill nd the average but we give it a new name expected value and nd it using a different approach De nition The mean or average of a set of numbers n1 n2 n3 n is n1 n2 n3n r The mean is one measure of central tendency or the center of a group of data When we have de ned a random variable and are working with its probability distribution we talk about its expected value rather than its mean or average De nition LetX denote a random variable that assumes values x1 x2 x with associated probabilities p1 p2 p respectively Then the expected value of X denoted EX is givenby EX xlp1 x2p2 xnpn Example 1 Find the expected value of X given its probability distribution x 012345 PXx 125 25 1875 25 0625 125 Math 1313 Class Notes 7 Section 82 Page 1 of4 Example 2 The owner of a newsstand estimates that the weekly demand for a certain magazine are as follows Demand 101112131415 Probabilit 05 15 25 30 20 05 Find the number of magazines he should expect to be demanded per week Example 3 A group of private investors intends to purchase a hotel currently being offered for sale in a resort city Records obtained from the hotel indicate the occupancy rates with corresponding probabilities during the May 7 September tourist season and are shown in the following table Magic Carpet Hotel IOccupanc Rate 75 80 85 90 95 100l Probability 3521 18 1509 02 Find the expected occupancy rate for the Magic Carpet Hotel b If the Magic Carpet Hotel has 175 rooms how many rooms can the investors expect to be occupied on a randomly selected night during the tourist season c The Magic Carpet Hotel charges 125 per night per room What is the expected income on a randomly selected night in tourist season E Math 1313 Class Notes 7 Section 82 Page 2 of4 Example 4 A bag contains 30 quarters 25 dimes 45 nickels and 50 pennies A coin is drawn at random from the bag What is the expected value of the coin Odds Odds and probability are not the same thing although they are de nitely related We speak of odds in two ways either the odds in favor of an event occurring or the odds against an event occurring Odds are often expressed as a ratio of two numbers such as a b or a to b Here are some formulas which are helpful in solving odds problems Odds in favor of an event If PE is the probability of an event E occurring PE PE PE l then the odds in favor of E are 1 PE PE Odds against an event If PE is the probability of an eventE occurring then 1 PE PE P E P E the odds againstE are PE 0 Probability of an event given the odds If the odds in favor of an eventE occurring are a to b the then probability of event E occurring is PE a b a Math 1313 Class Notes 7 Section 82 Page 3 of4 We ll use the third formula extensively Example 5 The probability that the Texans will win the Super Bowl is 002 What are the odds in favor of the Texans winning the Super Bowl Example 6 The odds that it will rain on Thursday are 3 to 5 What is the subjective probability that it will rain What is the probability that it will not rain Example 7 The odds against Shelly winning a drawing are 991 What is the probability that Shelly will win the drawing Math 1313 Class Notes 7 Section 82 Page 4 of4 Math 1313 Section 64 Permutations and Combinations In this section we will consider the number of ways in which the elements in a set can be arranged Before we can determine the number of possible arrangements we have to know whether or not order matters In other words if our set is 1 2 3 and we are looking for all possible arrangements of the elements of the set and order matters then 123 and 231 are different arrangements However if order does not matter then 123 and 231 would be considered to be the same When does order matter Suppose you and five of your friends are lining up to take a group photo How many ways can this happen In this case order matters You ll get a different photo if you are first in line than you will if you are last in line Or suppose you are in line with five other people In this case order matters Now suppose you are checking your Cash 5 lottery ticket and you see that you have the correct five numbers You are still a winner even if the numbers on your ticket are in a different order than the order in which they were drawn In this game order does not matter The group photo example is an example of a permutation order matters and the lottery example is an example of a combination order does not matter De nition n is read 11 factorial and is defined to be nniln72n7332l Then6654321 395 7amp0 Math 1313 Class Notes 7 Section 64 Page 1 of9 ag NPK A C 3 W You can work this out on your calculator with just a couple of key strokes Be sure you nd the factorial key on your calculator and that you know how to use it Permutations We will consider several types of permutation problems First we ll look at permutations of all of the elements of the set Example 1 How many possible photos are there if siX people line up in single le to take a picture Q39i39 39 i39i 4 c 7510 The number of permutations of n distinct objects taken n at a time written Pn n is n You may also see this notation for permutations P which also means nl NW 074 awn ExampleZ FindP8 8 row 3973 ozcc raw 8 l 40 3 a O P1amp3f Z While you can compute this using factorials it is sometimes more convenient to use the permutation feature on your calculator You need to nd this key and be sure you know how to compute permutations using your calculator pct Math 1313 Class Notes 7 Section 64 Page 2 of9 Often we want to consider arrangements of less than all of the elements from a set For example suppose there are 27 people in a club and the club is electing new officers They will elect a 5 epresident Vicepresident secretary and treasurer We want to o know the number of possible ways that this can happen Order matters since being elected president is not the same as being elected treasurer In this case the number of permutations of n objects taken r at a time where r g n is given by Pn r L nir You may also see this notation for permutations P which also anr means N Pr F 5N PQAOW Q39i 3 459 MW 6 1 ck 7 0 4 Example 3 Find P10 3 13 W 9f l 390 O orgW 10 7033 MW r 0lg 7310 You can also work problems like these on your calculator Example 4 A club with 27 members will elect new officers They will chose a president Vicepresident secretary and treasurer In how many ways can this happen A Q 7 v I F i 5 it oh lizard of cef Pe M39Ea E oq P74 49300 Math 1313 Class Notes 7 Section 64 Page 3 of9 Example 5 Eight people apply for three different jobs Each applicant is quali ed to ll each of the three jobs If the jobs are lled from these 8 people in how many ways can the jobs be 39 7 lled N g r 3 1L kegWoe JO 5 aiOI l3j 33 The nal type of permutation problem involves a set where the objects in the set are not all distinct objects ie there are repeated or identical items in the set Given a set of n objects not all distinct with nl objects alike nZ objects alike n39 objects alike and n1 nZ n n then the number of permutations of o i Example 6 How many permutations of the letters in the word PEPPER are there I 57mm 69 3 M E b a F 3 I l a C L C R r 1 j Jam 0 39 1 quot l P Q 0 Example 7 The Fall Flagg Co would like to create a company password by arranging the letters in the company name FALLFLAGG How many arrangements are possible F p ZL 9030 wa r W t 75w Q39 39Q39o39Z M Math 1313 Cla 5 Notes 7 Section 64 Page 4 of9 these It objects taken n at a time is n lrillurn QZDF 3 1 v9 Combinations We will use combinations throughout the rest of the course You should become very pro cient at working with combinations on your calculator The number of combinations of n distinct objects taken r at a time r gn is given by Cn r L rn 7 r You may also see this notationior combinations C which also 5nr5 means N CQ IN Fgt Example 8 Find C104 A Ofe 9 CASH caJlE a w N Co w QJO of 7 a l 3 4 Camel mu a 9 4 Example 9 How many 5card hands can be dealt from a standard deck of 52 playing cards Csa1 s 62157f 7Q 0 Example 10 A sixmember governing council will be chosen from a 40member group How many possible committees are h 7 tere Comm eef cam lomalvon C40 Q 3338 3 W9 Math 1313 Class Notes 7 Section 64 Page 5 of9 Sometimes problems will involve the use of the multiplication principle In these problems we have more than one task We ll use permutations or combinations to determine the number of ways in which each task can be completed then multiply these answers together Example 11 A company car has a seating capacity of 6 and will be used by a 6person carpool If only 4 of the employees will drive how many possible arrangements are there Perh Eek l 50 Df iver T v 4 P 36 my W Example 12 A committee is made up of 10 men and 8 women A subcommittee will be formed and it must have 3nlen and 3 men as members In how many ways can this happen Combined Id Order Taskl Task Pick Wdrtvm Prcc Mm C I o C 3 36 quot 610 Math 1313 Class Notes 7 Section 64 Page 6 of9 Example 13 A student belongs to a media club This month he must purchase 3 CDs and 4 DVDs from the club s featured list There are 12 featured CDs and 12 featured DVDs ln how many ways can he select his purchase co mb 2 1 as k I 725 k 2 39 39DV D 5 CT gt 0937 COR4 aao L695 00 Note that these questions ask how many ways we can do two or more specific tasks in succession like picking 3 CD s out of 12 or 3 men out of 10 to pick from Sometimes the questions we want to ask involve more than one specific tas I Medfa 7 forms CD on fCD s I 3 CD S 61 Lf C D s 3c b39 4DVD 71 L5 CD 41 30W Cra 3l Ca f c 9 43 39 c 2157 Example 14 A coin is tossed 5 times In how many ways can at least 4 heads appear 4 HQW 0M 5 W HHHHT HHHHH HHTH Lag l Him I THHHHcygLQfDx 775w Math1313ClassNotesiSeiction64Page7of9 S akauQ ginMURR Atmostmeansi a M Q Q 0391 W 0 I 21 Example 15 A hardware store receives a shipment of 100 battery packs A A sample of 4 batteries is taken for testing How many ways can a sample of 4 batteries be chosen C0 M l L 0100 L6 17621 3345 B Suppose 9 of the battery packs are defective How many ways can the selection of 4 batteries oor testing contain 2 k 9 defective battery packs I VWL f a t 7 20 Cms ac C f gl At least means 7 C How many selections contain at most one defective battery tT M057 1 Dev evFAequot O 61 Q235 46009 I 3 6000 chm 39 dam C 71 n C er3 l 2679670 4 1218 3 D How many selections Stain at least one defective 3 7 a 0 3 To D0947 I 91 3 0 pmm 6550 OK augm o 2 Gc e O 3Mmsgttemo ove szgw o go m All GoaQ 39 32 E 736352 Pck 4 quotr 030 W 399 9amp5 Cf ivl C 70 9 mam o Math 1313 Class Notes 7 Section 64 Page 8 of9 L Iamp 565 Example 16 cards are drawn from a wellshuf ed deck of ace35 52 playing cards How many ways can the draw contain at least 4 exactly 2 face cards 5 01 9h f A 11 A 4 4 HM are 5 19M oz 62 or k CQg c as CQg PM cgc ac w 9 W30 50393amp a578039amp r23 in u an 9amp0 Piano aFw X39 i 07 Ca aB C t o 5 660113O 00539140 Example 17 A bag contains 13 balls 7 4 red 5 black and 4 green Five balls are chosen at random How many selections contain at least 3 b1 ck balls My 35m am 0K HILaa 1M 0 SM C6 l3gtC D CSH e d2 MUS5 0 Q8 4 5 lt3 t 352 2 Math 1313 Class Notes 7 Section 64 Page 9 of9