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# 518 Class Note for STAT 22500 with Professor Martin at Purdue

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Date Created: 02/06/15

Stat 225 Lecture Notes Set Theory 8 Fundamentals of Probability Ryan Martin Spring 2009 1 Set Theory Set theory is concerned with mathematical properties of very abstract collections of ob jects ln Stat 2257 we will need only the very basics7 which we discuss below This material is taken primarily from Weiss7 Section 12 11 Set Notation A set is just a collection of objects7 called elements Some common notations follow 0 Universal and Empty Sets It is assumed that there is a universal set7 denoted by U or 97 that contains everything There is also an empty set7 denoted by 07 that contains nothing 0 Containment If x is an element of a set A7 we write x E A If x is not an element of A7 we write z A o Subsets If for two sets A and B7 all elements in A are also in B7 then we A a subset of B and write A C B If A is not a subset of B7 we write A B To verify that A C B7 one must show that if z E A then x E B Sets are often de ned by some property that its elements satisfy lf Pz is a statement about 7 then x E Q is the set of all z E 9 such that PW is true For example7 Pz could be z is less than or equal to 2777 in which case7 x E Q oo 2 De nition 11 Let Q be a set with subsets A and B i The complement of A is A0 x E Q z A ii The unionoanndB isAUB Qz AoerB iii The intersection of A and B is A B x E Q z E A and z E B Exercise 12 Let Q 1234567 A 1234 and B 246 Write out the elements in each of A U B7 A B and BC Exercise 13 lfA x x S r and B x z gt s for s S 7 describe AC and A B Those of you familiar with formal logic might see some similarities between the nota tion there and the set notation introduced above In particular U 0177 m andn LGotn To illustrate this relationship let Pz and Qz be statements concerning an element x and let A x and B x Then AUBx QP gQ A Bx QP QQ ACEQLmPx 96 96 De nition 14 Two sets A and B are said to be disjoint if A B 0 In other words the sets A and B have no elements in common Exercise 15 For an arbitrary set A give an example of a set B such that A B 0 12 Venn Diagrams Venn diagrams are an easy way to visualize sets and set operations Let Q be the universal set with subsets A B C etc Figure 1 shows an example of a Venn diagram for three sets A B and 0 Exercise 16 Draw a Venn Diagram of A A B and A BC Exercise 17 Draw a Venn Diagram of disjoint A and B Exercise 18 Draw a Venn diagram of sets A and B such that A C B Shade in the regions A U B and A B Exercise 19 In a class with 100 students 50 like math 60 like statistics and 16 like math and statistics Draw a Venn Diagram to nd the following a How many students like only math b How many student like neither math nor statistics Be careful It is not always possible to ll in the areas of a Venn diagram directly A convenient way to handle this is to let one area be a variable say x ll in everything else as if z where known and then solve for x The next two exercises illustrate this idea Exercise 110 There are fty ve athletes training for a triathalonia race that consists of running swimming and biking On a given day you nd that o Twenty six of the athletes will swim o Twenty seven will bike 0 Thirty will run Aquot Fig 1 A Venn diagram for A7 B and 0 Can you shade in the region 0 A U BY 0 Fourteen will run and bike 0 Three will bike and swim7 but not run 0 Eight will run7 bike and swim 0 Four will take the day off and not train Draw a three way Venn diagram and ll in the regions with the appropriate counts Hint Let x be the number in R S BC and work out everything else assuming z is known then solve for Exercise 111 Data collected over the last calendar year showed that 68 of the days where sunny7 24 of the days had min7 and 10 of the days had snow 15 of the days had sun and min7 4 had min and snow7 and surprisingly 1 of the days had min7 sun and snow Finally7 there were 22 of the days that did not t in any of these three categories Create a three way Venn diagram with all the numbers lled in 13 Order of Set Operations ln algebra7 the operations 777 7777 gtlt and 77 can be applied in many di erent orders and the order in which they are carried out matters eg ab c 31 ab ac There are similar rules for ordering the set operations 0 Commutative Laws DAUBBUA 00A BB A o Associative Laws i AUBUO AUBUO ii A B O A B O o Distributive Laws 0A wuo 4A B A m mAUw O 4AUBMMAUm 0 De Morgan7s Laws 1 A U B0 AC BC ii A BC A0 U B0 Exercise 112 Draw Venn diagrams to verify that DeMorgan7s laws are true To prove these results formally7 what one must do is show that the event on the left hand side of the equal sign is a subset of the event on the right hand side7 and vice versa in other words7 events A and B are the same if and only if A C B and B C A We illustrate this logic below Proof of DeMorgtm s Law Suppose z E AUB Then z Z AUB or7 in other words7 z Z A AND z Z B Therefore7 z E A0 AND z 6 B07 or z E A0 B07 which proves A U B0 C A0 BC If we start with z E A0 BC7 the argument is the same in reverse7 showing Ac B0 C A U B Therefore7 x AUBC ltgt x AUB ltgt x Aandx B ltgt AcandzEBc ltgt xEAc Bc which proves DeMorgan7s law D Exercise 113 Prove the following a AAUB ifandonlyifB CA b AA B ifandonlyifACB c AA BUA BC d A C B implies B0 C AC 2 From Set Theory to Probability ln probability7 there is an underlying random experimentirandom in the sense that the outcome cannot be determined ahead of time Set theory allows us to be able to describe the outcomes of the random experiment 21 Sample Space and Events The sample space 9 is the collection of all possible outcomes and an event is a subset of the possible outcomes For example consider the experiment where you draw a random card from a standard deck The sample space 9 consists of the 52 cards and one event is E card drawn is red De nition 21 The set of all possible outcomes of a random experiment is called the sample space and is denoted 9 Elements in in Q are called simple outcomes De nition 22 An cycrit is a subset of the sample space We say that an event occurs if and only if the outcome of the random experiment is an element of the event Exercise 23 Suppose the experiment consists of tossing a coin twice What is 9 Write out the event E rst toss is Heads Exercise 24 Consider the experiment where a pair of 6 sided die are rolled What is the sample space Also write out the outcomes in E1 the second roll is 3 and E2 the sum of the two rolls is 10 Exercise 25 Suppose we roll a 6 sided die arid toss a coin For the events A the die is even and B the coin is Heads write out the following 9 A B AC AUBVA Bm A Bq De nition 26 Two events A and B are said to be mutually cmclusiyc if they cannot occur at the same time In terms of sets this means that A and B are disjoint ie A B0 Exercise 27 Draw a card at random from a standard deck of cards For the events A Q9 B K Q J and C 7 which of the three pairs are mutually exclusive Exercise 28 An urn contains 10 balls numbered 0 through 9 Three balls are removed one at a time without replacement a What is the sample space 9 b Write out the event E an even number of odd numbered balls are chosen 22 Axioms of Probability An axiom is a primary assumption about something it cannot be proved or disproved eg in arithmetic one assumes that a b b a this cannot be proved A theory can be built up upon a set of axioms using deductive reasoning Kolmogorov proposed a set of three axioms upon which all of probability theory is built In calculus you encountered functions which take a real number x as input and produce y fx as output A probability or probability measure is just a real valued function whose input is a set Other examples of set functions are area volume etc De nition 29 Kolmogorov7s Axioms Let Q be the sample space for a random ex periment A function l de ned on the events of Q is called a probability or probability measure if it satis es the following three conditions 5 K1 llDA 2 0 for each event A K2 rm 1 K3 lf A1 A2 are mutually exclusive events7 then r U A EMA n1 n1 From these three axioms7 we can derive many results about a probability ll Below are four results which follow immediately from the axioms Can you prove them Domination Principle If A C B7 then MA 3 MB Complementation Rule MAC 1 7 llDA Proof By de nition7 A and A0 are mutually exclusive and A U A0 9 By Axiom K27 lP A U AC 1 and by Axiom K37 llDA lP AC 1 Now just solve for MAC D Addition Rule lP A U B MA MB 7 lP A B InclusionExclusion Principle For three events A7 B7 and C7 lP A U B U C lP A MB MO ilP A B ilP A O ilP B O MA B 0 Our next result concerns a partition of the sample space7 which we now de ne De nition 210 Events A1A2 for a partition of the sample space 9 if i they are mutually exclusive ie Am An Q for m 31 717 and ii together they form 9 ie Ufa An 9 Exercise 211 Find a partition for the experiment where a 6 sided die is rolled and a coin is tossed Law of Partitions lf A1 A2 forms a partition of Q then for any B7 MB EMA m B n1 Proof For any event B7 we can write B Uf1An B prove thisl Since the An7s are mutually exclusive7 so are An B prove thisl Now use Axiom K3 D Example 212 Two friends Adam and Brian are visiting from out of town They are scheduled to arrive at the same time but on different ights Let A be the event that Adams ight is late and B the event that Brian7s ight is late Past experience with these two airlines suggests that EMA 055 lP B 060 and lP A B 025 If you are to pick your friends up at the airport what is the probability you will have to wait The event that you have to wait is the event that either Adam or Brian arrive late ie A U B By the addition rule MA 0 B MA iPgtB 7 MA 0 B 055 06 i 025 090 Example 213 Suppose that in a community 23 are male democrats and 25 are female democrats If everyone in the community votes what percentage will vote for democrat Let M be the event that a randomly chosen person is male and D the event that a randomly chosen person votes Democrat We can partition the event D by gender ie D D M U D Me By the Law of Partitions we have MD MD 0 M Mo 0 M0 023 025 048 Since any probability measure l satis es Kolmogorov7s axioms the results above are completely general Note that how l assigns probability is completely arbitrary as long as the three axioms are satis ed We will spend the remainder of the course studying the properties of speci c probability measures lP beginning with the classical model in the next section 23 Classical Probability A probability model is a mathematical description of a random experiment based on certain assumptions The classical or equal likelihood model is one in which all outcomes 0 E 9 have the same probability De nition 214 Let Q be a nite sample space with equally likely outcomes Then for each event E where NE is the number of outcomes in E Exercise 215 Suppose I toss a fair coin three times Find the following probabilities a What is the probability you get HTT b What is the probability of getting two H7s c What is the probability of getting at least two H7s Exercise 216 Suppose I roll a pair of 6 sided die one red and one green Find the following probabilities a What is the probability that the red die is 4 and the green die is 3 b What is the probability that the red and gree die will be the same c What is the probability the sum of the two rolls is 5 Exercise 217 Show that the classical model satis es Kolmogorov7s three axioms More Practice Problems Let Q 12 107 let A be the even numbers in 9 let B be the prime numbers in Q and let G be the numbers larger than 3 in 9 Find A07 B07 A B7 A U B and AC 0 Prove the following lP A BC llDA 7 lP A B A 5th grade teacher is looking over the grade distribution for her 32 students She notices that 3 students aced77 Math7 Science and English while 13 did not ace any of the three subjects There were 7 who aced Science and English7 4 who aced Science and Math7 and 3 who aced Math and English 11 total aced Science and 9 total aced Math Draw a Venn Diagram with the numbers lled in Suppose a city has three newspapers7 the Times7 Herald and Emamineiquot Circulation information indicates that 47 of households get the Times7 334 get the Hemld7 346 get the Examiner7 119 get the Times and Hemld7 151 get the Times and Emamineiquot7 104 get the Herald and Emamiiieiquot and 48 get all three If a household is selected at random7 what is the probability that the household receives a either the Times or the Hemld7 but not both b c d e exactly two of the newspapers AA exactly one of the three papers A none of the three newspapers the Times and Hemld7 but not the Esamiiieiquot A 35 of Purdue students wear either sneakers or a hat 20 wear a hat and 30 wear sneakers Find the probability that a randomly chosen student a is wearing a hat or sneakers b is wearing a hat and sneakers c is wearing a hat but not wearing sneakers d is not wearing sneakers A standard roulette wheel has 38 numbers7 18 red7 18 black and 2 green When the wheel is spun7 the ball is equally likely to land on any of the 38 numbers Find the probability that a ball lands on red black green a b c d e f AAAA black or green not black A A not green

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