Stat 225 Week 1, 8/20/12-8/24/12, Notes: Set Theory

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1 Stat 225 Week 1, 8/20/12-8/24/12, Notes: Set Theory The Fall 2012 Stat 225 T.A.s September 7, 2012 The material in this handout is intended to cover general set theory topics. Information includes (but is not limited to) introductory probabilities, outcome spaces, sample spaces, laws of probability, and Venn Diagrams. This covers 4.1 through 4.4 in Statistics for Business and Economics. For an additional reference, one might consult A Course in Probability by Neil Weiss, specifically 1.2 and all of chapter 2. 1 Monday, August 20th Material An element is a single item (outcome), typically denoted by ω. A set is a collection of elements. A subset is a set itself, in which every element is contained in a larger set. Suppose the set A is contained in the set B. This is denoted by A B or A B depending on whether or not B has elements which are not in A. If B contains elements that are not in A, then A is called a proper subset of B. A Population is the collection of all individuals or items under consideration. An individual could refer to a person, a playing card, or whatever object we are interested in. A population is used in reference to sampling. However, when we talk about experiments, we use the phrase sample space. Sample space is the set of all possible outcomes for a random experiment and is denoted by Ω. For example, suppose we are interested in whether the price of the S & P 500 decreases, stays the same, or increases. In this scenario Ω = {decreases, stays the same, increases}. The opposite of Ω is the empty (null) set. It is the set with 0 elements in it and is written as. Ω and are complements. A complement is a set that contains all of the elements in the population that are not in the original set. We denote a complement with a superscript c (or C). For example, the complement of A would be denoted as A c or A C. Sometimes the symbol \ is useful when writing complements. The symbol \ means except or everything but. 1

2 Suppose we look at the outcome of 2 rolls of a die. Let A be the event that both rolls are a 5. Then A C = Ω \ {5, 5}. We use the symbol to denote belongs to. Here is the symbol for does not belong to :. What would Ω be if we were to examine the change in the S & P 500 over the past two days? Here are some important sets that pertain to numbers: the real numbers R, the integers Z, the rational numbers Q, the natural (whole) numbers N, and the positive integers Z +. What sets are contained in (or are subsets of) the other sets? Here are some brief exercises. Let us examine what happens in the flip of 3 fair coins. Fair means that the coin has the same probability of landing as a head as it does as landing as a tail. First, define Ω. Let A be the event of exactly 2 tails. Let B be the event that the first 2 tosses are tails. Let C be the event that all 3 tosses are tails. Write out the possible outcomes for each of these 3 events. We will revisit these events later on. 1.1 Example Problems Let Ω, the universal set, be all 26 lower-case letters. Define the sets V, N, E, and G (all of which are subsets of Ω) as follows: V = vowels (here, assume y is a vowel) N = letters next to a vowel (in the natural sequence a - z ) E = every other letter, starting with b G = letters a - g 2

3 List the letters in each of the following sets: 1. V, N, E, and G individually 2. N C 3. G C Start with a standard deck of 52 cards and remove all the hearts and all the spades, leaving 13 red and 13 black cards. List the cards in each of the following sets: 1. N = not a face card 2. R = neither red nor an ace 3. E = either black, even, or a Jack Suppose a fair six-sided die is rolled twice. possible outcomes... Determine the number of 1. for this experiement. 2. in which the sum of the two rolls is in which the two rolls are the same 4. in which the sum of the two rolls is an even number Random Experiment is an action whose outcome cannot be predicted with certainty beforehand. This does not mean that we know nothing about what can happen. An example of a random experiment could be one roll of a die (or multiple rolls), a hand in Texas Hold em, or a grade in a course. Ω represents all possible outcomes from the random experiment or the model under consideration. An event is defined to be any subset of the sample space. It can be one or more outcomes. Typically, when we refer to an event that is a single outcome, it is called a simple event, and subsequently, a simple probability. For an example, you could think of an event as not losing money on the S & P 500 on a given day. This event has 2 outcomes based on our prior example where Ω = {decreases, stays the same, increases}. Instead, suppose you looked at 2 consecutive days for this index. Let A be the event that you made money on the first day. Let B be the event that you had at least one day where you made money. How many outcomes does each event represent? 3

4 The Frequentist Interpretation of Probability states that the probability of an event is the long-run proportion of times that the event occurs in independent repetitions of the random experiment. This is referred to as an empirical probability and can be written as P (E) = N(E) n where n represents the sample size. (For definitions of P(E) and N(E) see the symbols reference.) Long-run means that n is large. There are differing viewpoints on large (typical examples are > 100, > 1,000, > 1,000,000, etc.) We will not use this exact formula for now, but it is essential to the Central Limit Theorem (CLT), which will be covered in MGMT 305. However, the concept is applicable for our purposes. Regardless of the sample size, if we are in an EQUALLY LIKELY FRAMEWORK, then P (E) = N(E) N(Ω). What is meant by an equally likely framework? Well, let us create a scenario that has such a property. Suppose we roll a fair, 6-sided die. Because the die is fair, each side of the die has the same probability of occurring as any other side of the die. Therefore, any individual outcome of the sample space is equally likely as any other outcome in the sample space. Often, the equal-likelihood model is referred to as classical probability. So, in an equally likely framework, the probability of any event is the number of ways the event occurs divided by the number of total events possible. Find the probabilities associated with parts 2-4 of the fair die example above. 1.2 Probability Rules Regardless of whether sample outcomes have the same probabilities, there are rules that probabilities must satisfy. Any probability must be between 0 and 1 inclusive. Additionally, the sum of the probabilities for all the experimental outcomes must equal 1. If a probability model satisfies these 2 rules, it is said to be legitimate. Suppose the event E is composed of several outcomes. Then the probability of E is just the sum of the probabilities of those outcomes. Refer to event B in the coin flip example above for this calculation. What is the probability of Ω,? If A B, what (if anything) can you say about their probabilities? 4

5 1.3 More Example Problems (ASW Chapter 4.1, Problem 6) An experiment with three outcomes has been repeated 50 times, and it was learned that E 1 occurred 20 times, E 2 occurred 13 times, and E 3 occurred 17 times. Assign probabilities to the outcomes. What method did you use? Start with a standard deck of 52 cards and remove all the hearts and all the spades, leaving 13 red and 13 black cards. Suppose a card is randomly drawn from the remaining cards. What are the probabilities of the following events? 1. N = not a face card 2. R = neither red nor an ace 3. E = either black, even, or a Jack (ASW Chapter 4.1, Problem 7) A decision maker subjectively assigned the following probabilities to the four outcomes of an experiment: P (E 1 ) =.10, P (E 2 ) =.15, P (E 3 ) =.40, and P (E 4 ) =.20. Are these probability assignments valid? Explain. 2 Wednesday, August 22nd Material The intersection of the events A and B is written as A B. For an outcome to belong to the intersection, that outcome has to be in both A and B. If we were talking about the intersection of 3 or more events, the outcome would need to be in all of them. The intersection is what is in common. The union of the events A and B is written as A B and it means whatever is in at least one of A or B. Please note that we do not double count. If an outcome was in both A and B, then it is in their union, but it is not in there twice. Refer to example where we flipped 3 fair coins: What are A B, A C, and A B C? Two other useful terms are mutually exclusive and exhaustive. Mutually exclusive refers to two (or more) events that cannot both occur when the random experiment is formed. Can you think of an event that is mutually 5

6 exclusive with event C above? Note that the term disjoint is the same as mutually exclusive except that it refers to sets and not events. One can symbolically denote mutually exclusive events by the following equation: A B =. Exhaustive refers to event(s) that comprise the sample space. In other words, events that are exhaustive have a union that equals the sample space; if A and B are exhaustive, then A B = Ω. What would you call events that are both mutually exclusive and exhaustive? The answer is a partition. What is the simplest partition? Venn Diagrams are useful tools for examining the relationships between events. Tree diagrams are also helpful (more on this when we come to conditional probability, general multiplication rule, etc.) Draw generic diagrams for events that are: mutually exclusive, exhaustive, complements, subsets, and have an intersection but are not subsets. The complement rule is a way to calculate a probability based on the probability of its complement. It is P(A) = 1 - P(A C ). This law is extremely useful. It is often handy in situations where the desired event has many outcomes, but its complement has only a few. For example, suppose we rolled a fair, six-sided die 10 times. Let T be the event that we roll at least 1 three. If one were to calculate T you would need to find the probability of 1 three, 2 threes,..., and 10 threes and add them all up. However, you can use the complement rule. What is P(T)? The general addition rule is a way of finding the probability of a union of 2 events. P(A B) = P(A) + P(B) - P(A B). What does this become if A and B are mutually exclusive? Can you provide a mathematical proof of this? The inclusion-exclusion principle is a way to extend the general addition rule to 3 or more events. Here we will limit it to 3 events. P(A B C) = P(A) + P(B) + P(C) - P(A B) - P(A C) - P(B C) + P(A B C). 6

7 2.1 Venn Diagram Problems Three of the major commercial computer operating systems are Windows, Mac OS, and Red Hat Linux Enterprise. A Computer Science professor selects 50 of her students and asks which of these three operating systems they use. The results for the 50 students are summarized below. 30 students use Windows 16 students use at least two of the operating systems 9 students use all three operating systems 18 students use Mac OS 46 students use at least one of the operating systems 11 students use both Windows and Linux 11 students use both Windows and Mac OS Use the above information to complete a three-way Venn diagram. Windows Mac OS Red Hat Linux Enterprise 7

8 Using the Venn diagram summarizing the distribution of operating system use previously described, calculate the following: Let Windows = A, Mac OS = B, and Red Hat Linux Enterprise = C N(A c B c ) P (A c B c ) N(A B C) Let Ω, the universal set, be all 26 lower-case letters. Define the sets V, N, E, and G (all of which are subsets of Ω) as follows: V = vowels (here, assume y is a vowel) N = letters next to a vowel (in the natural sequence a - z ) E = every other letter, starting with b G = letters a - g Solve for the following quantities: P (consonant) P (G C ) P (E) and P (E C ) In a certain population, 10 % of the population are rich, 5 % are famous, and 3 % are both. Draw a Venn Diagram for the situation described above and label all probabilities. What is the probability a randomly chosen person is not rich? What is the probability a randomly chosen person is rich but not famous? What is the probability a randomly chosen person is either rich or famous? What is the probability a randomly chosen person is either rich or famous but not both? 8

9 What is the probability a randomly chosen person has neither wealth nor fame? The law of partitions is a way to calculate the probability of an event. Let A 1, A 2,..., A k form a partition of Ω. Then, for all events B, P(B) = k i=1 P(A i B). Then, there are DeMorgan s Laws. Let A and B be subsets of Ω. Then (A B) C = A C B C. Furthermore, (A B) C = A C B C. Drew is a risk taker. On any given weekend, Drew takes risks with or without monetary compensation. He gets paid 20 % of the time he takes risks. The risks involved are to either drink something weird (like garlic butter) or do something silly (like shave his head into a mohawk). Drew gets paid and drinks something weird 16 % of the time. Drew does not get paid and drinks something weird 72 % of the time. What is the probability Drew drinks something weird? What is the probability he does something silly? Here are a few of the other laws. Each pair of equations refers to the distributive, associative, and commutative laws respectively. For all of these, let A, B, and C be subsets of Ω. A (B C) = (A B) (A C) A (B C) = (A B) (A C). A B = B A A B = B A A (B C) = (A B) C. A (B C) = (A B) C. Please be aware that the formulas just written can be extended to more than 3 events (even an infinite number of events). 9

10 3 Friday, August 24th Material Let A and B be events. The probability that event B occurs given (knowing) that event A occurs is called a conditional probability. It is denoted as P(B A). Whichever event is considered given or known goes after the in the notation. P (B A) P (B A) =. P (A) The above formula works so long as P(A) > 0. There is an equivalent to the above formula. It is N(A B) P (B A) =. N(A) The idea behind conditional probability is that you have an idea of what occurred, but do not know exactly what happened. Meaning, you can limit the original sample space (Ω) to something smaller. In our above example, we know that the event A occurred, so what we are doing is making A our new Ω. General multiplication rule is defined as P (A B) = P (A) P (B A). This formula is equivalent to the 2 above, just our goal is different now. Before we wanted to figure out a conditional probability, now we want to know a joint probability, or a probability of an intersection of 2 events. This rule can easily be extended to more than 2 events. n P ( A i ) = P (A 1 ) P (A 2 A 1 ) P (A 3 A 2 A 1 )... P (A n A n 1... A 1 ). i=1 Important note: A lot of the formulas in this section (4.4 of ASW) are rearrangements of previous formulas. You use one over another depending on what you are given in the problem and what the goal is. 3.1 Conditional Probability Examples Refer to the example with Drew. Find the following probabilities: What is the probability that Drew drinks something weird, if we know he was paid? What is the probability that Drew does something silly, if we know he was paid? 10

11 What is the probability that Drew drinks something weird, if we know he was not paid? (ASW Chapter 4.4, Problem 38) A Morgan Stanley Consumer Research Survey sampled men and women and asked each whether they preferred to drink plain bottled water or a sports drink such as Gatorade or Propel Fitness water (The Atlanta Journal-Constitution, December 28, 2005). Suppose 200 men and 200 women participated in the study, and 280 reported they preferred plain bottled water. Of the group preferring a sports drink, 80 were men and 40 were women. Let M = the event the consumer is a man W = the event the consumer is a woman B = the event the consumer preferred plain bottled water S = the event the consumer preferred a sports drink Answer the following: 1. What is the probability a person in the study preferred plain bottled water? 2. What is the probability a person in the study preferred a sports drink? 3. What are the conditional probabilities P (M S) and P (W S)? 4. What are the joint probabilities P (M S) and P (W S) 5. Given a consumer is a man, what is the probability he will prefer a sports drink? Using the Venn Diagram summarizing the distribution of operating systems previously described, calculate the following: The probability that a randomly chosen student uses all three operating systems, given the student uses Windows The probability that a randomly chosen student uses all three operating systems, given the student does not use Windows The probability that a randomly chosen student uses Windows, given the student uses Mac OS The probability that a randomly chosen student does not use any of the operating systems, given the student does not use Windows 11

12 Case Problem (Adapted from ASW Chapter 9, Case Problem 2, page 397) Cheating has been a concern of the dean of the College of Business at Bayview University for several years. Some faculty members in the college believe that cheating is more widespread at Bayview than at other universities, while other faculty members think that cheating is not a major problem in the college. To resolve some of these issues, the dean commissioned a study to assess the current ethical behavior of the business students at Bayview. As a part of this study, an anonymous exit survey was administered to this year s graduating class. Responses to the following questions were used to obtain data regarding three types of cheating. Any student who answered Yes to one or more of these questions was considered to have been involved in some type of cheating. 1. During your time at Bayview, did you ever present work copied off the Internet as your own? 2. During your time at Bayview, did you ever copy answers off another student s exam? 3. During your time at Bayview, did you ever collaborate with other students on projects that were supposed to be completed individually? The data are represented in the following Venn diagrams below: MALES Copied off the Internet Copied off an exam Collaborated on Individual projects FEMALES Copied off the Internet Copied off an exam Collaborated on Individual projects 12

13 OVERALL Copied off the Internet Copied off an exam Collaborated on Individual projects Using the law of partitions, fill in the Overall Venn diagram. What is the probability that a randomly chosen student was involved in some type of cheating? Use the inclusion-exclusion principle, then the idea of complements. Which is simpler? Given that a randomly chosen student cheated, what is the probability that student was male? Given that a randomly chosen student is female, what is the probability that student cheated? What is the probability that a randomly chosen student neither presented work from the Internet nor copied answers off another student s exam? What is the probability that a randomly chosen student cheated in all three ways, given that the student copied answers off another student s exam? 3.2 Additional Exercises Extra 1.1 Let A be an event that happens 40% of the time. Let B be an event that happens 75% of the time. Answer the following 4 questions. What 13

14 is the smallest probability the intersection of A and B can have? What is the largest probability the intersection of A and B can have? What is the smallest probability the union of A and B can have? What is the largest probability the union of A and B can have?.15,.4,.75, and 1 Extra 1.2 Suppose we are rolling 2 independent, fair 10-sided die. Let A be the event that the sum of the rolls is a prime number. Let B be the event that the sum of the rolls is odd. Let C be the event that the sum of the rolls is even. Find the following sets: B C, A B, A C, A C, (A C) C, and A (B C) C? set C, set A except the number 2, Ω except 9 and 15, just 2, Ω except 2, and Ω Extra 1.3 Suppose a lottery has balls numbered balls are picked at random and without replacement. Let A be the event that all 4 balls are even. Let B be the event that all 4 balls are less than 10. Let C be the event that all 4 balls are primes. (Allow 1 to be a prime.) Find P(A), P(B), and P(C). (Hint an extended general multiplication rule could be helpful.).0433,.0260, and.0260 Extra 1.4 A school has 100 students. The school offers only 3 language classes, namely Spanish, Italian and Russian. 50 students do not take a language. The Spanish, Italian, and Russian classes have 28, 26, and 16 students respectively. However, 12 students take both Spanish and Italian, 4 students take both Spanish and Russian, and 6 students take both Italian and Russian. How many students take all 3 language classes? 2 What is the probability a randomly chosen student takes exactly 1 language class?.32 You randomly draw 2 students. What is the probability that they are taking at least 1 language class between them?.7525 Extra 1.5 You have eight rooks in your possession. A rook can either move along a row or a column, but it can make no other movement. It can move as many as spots as it wants along the row or column of its choosing. You randomly place these eight rooks on the board, which is an 8 by 8 square. Find the probability that no rook can capture (run into with a legal move) any other rook? * 10 6 Extra 1.6 You roll 2 fair, 6-sided die. The first one is a 3. What is the probability that the second die has a higher value? What is the probability that the first die is higher?.5 and

15 Extra ,000 people are polled. They are asked whether they use Facebook, Twitter, and Myspace. 10,000 use Twitter; 30,000 use Facebook; 5,000 use Myspace; 8,000 use Twitter and Facebook; 2,000 use Twitter and Myspace; 4,000 use Facebook and Myspace; and 1,000 use all 3. What percent of the people polled do not use any of the 3 networking tools? Knowing that a person uses at least one of the networking tools, what is the probability that they use Twitter? Knowing that they use Twitter and Facebook, what is the probability that they do not also use Myspace? 68%,.3125,.875 Extra 1.8 Suppose on a Friday night Jake has the options of going to the bar or going to Karen s apartment. He goes to the bar with a probability of.7. If he goes to the bar he can end up in jail, a friend s place, or back at home with probabilities of.1,.2, and.7 respectively. If he goes to Karen s, he will either stay or go back home. He stays with a probability of.2. Use the law of partitions (combined with the general multiplication rule) to find the probability that Jake is at home on Saturday morning. Find the probability he went to Karen s given that he is not home..73 and

Venn Diagrams; Probability Laws. Notes. Set Operations and Relations. Venn Diagram 2.1. Venn Diagrams; Probability Laws. Notes

Venn Diagrams; Probability Laws. Notes. Set Operations and Relations. Venn Diagram 2.1. Venn Diagrams; Probability Laws. Notes Lecture 2 s; Text: A Course in Probability by Weiss 2.4 STAT 225 Introduction to Probability Models January 8, 2014 s; Whitney Huang Purdue University 2.1 Agenda s; 1 2 2.2 Intersection: the intersection