DERTMENT OF QUNTITTIVE METHODS & INFORMTION SYSTEMS Introduction to usiness Statistics QM 120 Chapter 4 Spring 2008
Chapter 4: Experiment, outcomes, and sample space 2 robability and statistics are related in an important way. It is used to allow us to evaluate the reliability of our conclusions about the population when we have only sample information. Data are obtained by observing either uncontrolled events in natureorcontrolled situation i in lb laboratory. We usethe term experiment to describe either method of data collection. The observation or measurement generated by an experiment may or may not produce a numerical value. Here are some examples: Recording a test grade Interviewing a householder to obtain his or her opinion in certain issue.
Chapter 4: Experiment, outcomes, and sample space 3 Table 1: Examples of experiments, outcomes, and sample spaces Experiment Toss a coin once Roll a die once lay a lottery Select a student Toss a coin twice Outcomes Head, Tail 1, 2, 3, 4, 5, 6 Win, Lose M, F HH, HT, TH, TT Sample Space S = {Head, Tail} S = {1, 2, 3, 4, 5, 6} S = {Win, Lose} S = {M,F} S = {HH, HT, TH, TT} Venn diagram is a picture that depicts all possible outcomes for an experiment while in tree diagram, each outcome is represented a branch of a tree.
Chapter 4: Experiment, outcomes, and sample space 4 HH TH Venn Diagram HT TT HH H HT TH T Tree Diagram simple event is the outcome that is observed on a single repetition of the experiment. It is often denoted d by E with a subscript. TT Example: Toss a die and observe the number that appears on the upper face. List the simple event in the experiment. Solution:
Chapter 4: Experiment, outcomes, and sample space 5 We can now define an event or compound event as a collection of simple events, often denoted a capital letter. Example: Tossing a die continued We can define the events and as follow, : Observe an odd number : Observe a number less than 4 Example: Draw a tree diagram for three tosses of a coin. List all outcomes for this experiment in a sample space S. Solution
Chapter 4: Experiment, outcomes, and sample space 6 Example: Suppose we randomly select two persons from members of a club and observe whether the person selected is a man or woman. Write all the outcomes from experiment. Draw the Venn and tree diagrams for this experiment. Solution: Two events are mutually exclusive if, when one event occurs, the other cannot, and vice versa. The set of all simple events is called the sample space, S.
Chapter 4: Calculating robability 7 robability is a numerical measure of the likelihood that an event will occur. Two properties of probability bilit The probability of an even always lies in the range 0 to 1. 0 E i 1 0 1 The sum of the probabilities of all simple events for an experiment, denoted by ΣEi, is always 1. E i = E 1 + E 2 + E 3 +.... = 1
Chapter 4: Calculating robability 8 Three conceptual approaches to probability 1. Classical probability Two or more events that have the same probability of occurrence are said to be equally likely events. The probability of a simple event is equal to 1 divided by the total number of all final outcomes for an equally likely experiment. Classical probability rule to find probability 1 E i = Total number of outcomes Number of outcomes favorable = Total number of outcomes to
Chapter 4: Calculating robability 9 Example: Find the probability of obtaining a head and the probability of obtaining a tail for one toss of a coin. Solution
Chapter 4: Calculating robability 10 Example: Find the probability of obtaining an even number in one roll of a die. Solution
Chapter 4: Calculating robability 11 Example: candy dish contains one green and two red candies. You close your eyes, choose two candies one at a time from the dish, and record their colors. What is the probability that both candies are red? Solution
Chapter 4: Calculating robability 12 Calculating the probability of an event: List all the simple events in the sample space. ssign an appropriate probability to each simple event. Determine which simple events result in the event of interest. Sum the probabilities of the simple events that result in the event of interest. lways 1. Include all simple events in the sample space. 2. ssign realistic probabilities to the simple events.
Chapter 4: Calculating robability 13 Example: six years boy has a safe box that contains four banknotes: One Dinar, Five Dinar, Ten Dinar, Twenty Dinar. His sister which is a three years old girl randomly grabbed three banknotes from the safe box to buy a 30 KD toy. Find the odds probability that this girl can buy the toy. Solution
Chapter 4: Calculating robability 14 2. Relative frequency concept of probability Suppose we want to know the following probabilities: The next car coming out of an auto factory is a lemon randomly selected family owns a home randomly selected woman has never smoked The outcomes above are neither equally likely nor fixed for each sample. The variation goes to zero as n becomes larger and larger If an experiment is repeated n times and an event is observed f times, then = f n
Chapter 4: Calculating robability 15 Example: In a group of 500 women, 80 have played golf at least once. Suppose one of these 500 woman is selected. What is the probability that she has played golf at least once Solution
Chapter 4: Calculating robability 16 Example: Ten of the 500 randomly selected cars manufactured at a certain auto factory are found to be lemons. What is the probability that the next car manufactured at that factory is a lemon? Solution:
Chapter 4: Calculating robability 17 Example: Lucca Tool Rental would like to assign probabilities to the number of car polishers it rents each day. Office records show the following frequencies of daily rentals for the last 40 days. Number of olishers Rented Number of Days 0 4 1 6 2 18 3 10 4 2
Chapter 4: Calculating robability 18 Solution Number of olishers Rented Number of Days robability
Chapter 4: Calculating robability 19 Law of large numbers: If an experiment is repeated again and again, the probability of an event obtained from the relative frequency approaches the actual probability. 3. Subjective probability Suppose we want to know the following probabilities: student who is taking a statistics class will get an grade KSE price index will be higher at the end of the day The razilian team will win world cup 2006 Subjective probability is the probability assigned to an event based Subjective probability is the probability assigned to an event based on subjective judgment, experience, information, and belief.
Chapter 4: Counting Rule 20 Suppose that an experiment involves a large number N of simple events and you know that all the simple events are equally likely. Then each simple event has probability 1/N and the probability of an event can be calculated as Where n The mn rule n = N is the number of simple events that result in event Consider an experiment that is performed in two stages. If the first stage can be performed in m ways and for each of these ways, the second stage can be accomplished in n ways, then there mn ways to accomplish the experiment
21 Chapter 4: Counting Rule Example: Suppose you want to order a car in one of three styles and in one of four paint colors. To find out how many options are available, you can think of first picking one of the 1 m = 3 styles and then one of n = 4 colors. Using the mn rule, as shown in the figure below, you have mn= 34 = 12 possible options. Style 2 3 Color 1 2 3 4 1 2 3 4 1 2 3 4
Chapter 4: Counting Rule 22 The extended mn rule If an experiment is performed in k stages, with n1 ways to accomplish the first stage, n2 to accomplish the second stage, and nk ways to accomplish the kth stage, the number of ways to accomplish the experiment is n 1 n2 n3... n k Example: bus driver can take three routes from city to city, four routes from city to city C, and three routes from city C to city D. For traveling from to D, the di driver must di drive fromtotoctod,howmanypossibleroutesfromtod are available Example: medical technician records a person s blood type and Rh factor. Calculate the total outcome of this experiment using the mn method and using a tree diagram.
Chapter 4: Counting Rule 23 Example: hmad has invested in two stocks, RC Oil and Coal Mining. hmad has determined that the possible outcomes of these investments three months from now are as follows. Investment Gain or Loss in 3 Months in 000 rc Oil Coal Miningi 10 8 5-2 0-20
Chapter 4: Counting Rule 24 Solution hmad investments e scan be viewed e as a two step sepexperiment. e It involves two stocks, each with a set of experimental outcomes. RC Oil: n1 = 4 Coal Mining: n2 = 2 Total Number of Experimental Outcomes: n1n2 = 42 = 8
Chapter 4: Counting Rule 25 RC Oil Stage 1 Coal Mining Stage 2 Experimental Outcomes Gain 8 10, 8 Gain 18,000 Gain 10 Gain 8 Lose 2 10, -2 Gain 8,000 5, 8 Gain 13,000 Gain 5 Even Lose 2 Gain 8 Lose 20 Lose 2 Gain 8 Lose 2 5, -2 Gain 3,000 0, 8 Gain 8,000 0, -2 Lose 2,000-20, 8 Lose 12,000-20, -2 Lose 22,000
Chapter 4: Counting Rule 26 counting rule for permutations orderings The number of ways we can arrange n distinct objects, taking them r at a time, is n r = n r n! = n r! wheren! = n n 1 n counting rule for combination 2... 321 and 0! = 1 The number of distinct combinations of n distinct objects that can be formed, taking them r at a time, is n C r = C n r = n r = n! r! n r!
Chapter 4: Counting Rule 27 ermutations: Given that position order is important, if one has 5 different objects e.g.,, C, D, and E, how many unique ways can they be placed in 3 positions e.g. DE, ED, DE, DE, ED, ED, C, C, C, C etc. C bi i If h 5 diff bj C D Combinations: If one has 5 different objects e.g.,, C, D, and E, how many ways can they be grouped as 3 objects when position does not matter e.g. C, D, E, CD, CE, DE are correct but C is not ok as is equal to C
Chapter 4: Marginal & Conditional robabilities 28 Suppose all 100 employees of a company were asked whether they are in favor of or against paying high salaries to CEOs of U.S. companies. The following table gives a two way classification of their responses. In favor gainst Total Male 15 45 60 Female 4 36 40 Total 19 81 100 Marginal probability is the probability of a single event without consideration of any other event
Chapter 4: Marginal & Conditional robabilities 29 Suppose one employee is selected, he/she maybe classified either on the bases of gender alone or on the bases of opinion. The probability of each of the following event is called marginal probability Suppose the employee selected is known to be male. What is the probability that he is in favor?
Chapter 4: Marginal & Conditional robabilities 30 This probability is called conditional probability and is written as the probability that the employee selected is in favor given that he is a male. in favor male This event has already occurred The event whose probability is to be determined Read as given Conditional probability is the probability that an even will occur given that another event has already occurred. If and are two events, then the conditional probability of given is written as
Chapter 4: Marginal & Conditional robabilities 31 Example: Find the conditional probability in favor male for the data on 100 employees Solution
Chapter 4: Marginal & Conditional robabilities 32 Example: Find the conditional probability female in favor for the data on 100 employees Solution Male Female In favor gainst Ttl Total 15 45 60 4 36 40 Total number of in favor Females who are in favor Total 19 81 100 Number of females who are in favor female in favor = Total number of employees who are in favor 4 = =.2105 19
Chapter 4: Marginal & Conditional robabilities 33 Example: Consider the experiment of tossing a fair die. Denote by and the following events: ={Observing an even number}, ={Observing a number of dots less than or equal to 3}. Find the probability of the event, given the event. Solution
Chapter 4: Mutually Exclusive Events 34 Events that cannot occur together are called mutually exclusive events. Example: Consider the following events for one roll of a die : an even number is observed = {2,4,6} :anodd number is observed = {1,3,5} C: a number less than 5 is observed = {1,2,3,4} and are mutually exclusive events but and C are not. How about and C? Simple events are mutually exclusive always.
Chapter 4: Independent vs. Dependent events 35 Two events are said to be independent if the occurrence of one does not affect the probability of the other one. and are said to be independent d events if either = or = Example: Refer to the information on 100 employees. re events female F and in favor independent? Solution:
Chapter 4: Independent vs. Dependent events 36 What is the difference between mutually exclusive and independent events? It is common to get confused or not to tell the difference bt between these two terminologies. When two events are mutually exclusive, they cannot both happen. Once the event e has occurred, event e cannot occur, so that = 0, or vice versa. The occurrence of event certainly affects the probability that event can occur. Therefore, Mutually exclusive events must be dependent. Independent events are never mutually exclusive. ut, dependent events may or may not be mutually exclusive
Chapter 4: Independent vs. Dependent events 37 Example: sample of 420 people were asked if they smoke or not and whether they are graduate or not. The following two way classification table gives their responses Smoker Nonsmoker College graduate 35 130 Not a college graduate 80 175 If an person is selected at random from this sample, find the probability that this person is a College graduate G. Nonsmoker NS. Smoker S given the person is not a college graduate NG. College graduate G given the person is a nonsmoker NS. re the events Smoker and college graduate independent? Why?
Chapter 4: Complementary Events 38 Two mutually exclusive events that taken together include alltheoutcomeforanexperimentsamplespace,sare called complementary events. Consider the following Venn diagram: Venn diagram of two complementary evens The complement of event, denoted by and read as bar or complement, is the event that includes all the outcomes for an experiment that are not in Since two complementary e events, taken together, e include all the sample space S, the sum of probabilities of all outcomes is 1 + = 1 = 1
Chapter 4: Complementary Events 39 Example: In a lot of five machines, two are defective. If one of machines is randomly selected, what are the complementary events for this experiment and what are their probabilities? Solution:
Chapter 4: Intersection of Events & Multiplication Rule 40 Intersection of events Let and be two events defined in a sample space. The intersection of and represents the collection of all outcomes that are common to both and is denoted by any of the followings and,, or Example: Let = the event that a person owns a C Let = the event that a person owns a mobile and Intersection of events and
Chapter 4: Intersection of Events & Multiplication Rule 41 Multiplication rule Sometimes we may need to find the probability of two events happening together. The probability of the intersection of two events and is called their joint probability. It is written or It can be obtained by multiplying the marginal probability of one event with the conditional probability of the second one. Multiplication rule: The probability of the intersection of two events and is = = =
Chapter 4: Intersection of Events & Multiplication Rule 42 Example: The following table gives the classification of all employees of a company by gender and college degree. College graduate G Not a college graduate N Total Male M 7 20 27 Female F 4 9 13 Total 11 29 40 If one employee is selected at random, what is the probability that the employee is a female and a college graduate? Solution
43 Chapter 4: Intersection of Events & Multiplication Rule
Chapter 4: Intersection of Events & Multiplication Rule 44 Example: boxcontains20dvd,4ofwhicharedefective.if two DVDs are selected at random without replacement, what is the probability that both are defective? Solution:
Chapter 4: Intersection of Events & Multiplication Rule 45 If events and are independent, their joint probability simplifies from = TO = Sometimes we know the joint probability bilit of two events and, in this case, the conditional probability of given or given is = = given that 0 and 0
Chapter 4: Intersection of Events & Multiplication Rule 46 Example: ccording to a survey, 60% of all homeowners owe money on home mortgages. 36% owe money on both home mortgages and car loans. Find the conditional probability that a homeowner selected at random owes money on a car loan given that he owes money on a home mortgage. Solution:
Chapter 4: Intersection of Events & Multiplication Rule 47 Example: computer company has two quality control inspectors, Mr. Smith and Mr. Robertson, who independently inspect each computer before it is shipped to a client. The probability that Mr. Smith fails to detect a defective C is.02 while it is.01 for Mr. Robertson. Find the probability that both inspectors will fail to detect a defective C. Solution:
Chapter 4: Intersection of Events & Multiplication Rule 48 Example: The probability that a patient is allergic to enicillin is.2. Suppose this drug is administrated to three patients. Find a The probability that all three of them are allergic to it b t least one of them is not allergic Solution:
Chapter 4: Union of Events & ddition Rule 49 Union of events Let and be two events defined in a sample space S. The union of events and is the collection of all outcomes that belong either to and or to both and and is denoted by or or S
Chapter 4: Union of Events & ddition Rule 50 Example: company has 1000 employees. Of them, 400 are females and 740 are labor union members. Of the 400 females, 250 are union members. Describe the union of events female and union member Solution
Chapter 4: Union of Events & ddition Rule 51 ddition rule The probability of the union of two events and is = + Example: university president has proposed that all students must take a course in ethics as a requirement for graduation. Three hundred faculty members and students from this university were asked about their opinion on this issue and it shown in the following table. Opinion Favor Oppose Neutral Total Faculty 45 15 10 70 Student 90 110 30 230 Total 135 125 40 300 Find the probability that a person selected is a faculty member or in favor.
Chapter 4: Union of Events & ddition Rule 52 Solution: Opinion Favor Oppose Neutral C Total Faculty F Student S Total
Chapter 4: Union of Events & ddition Rule 53 Example: There are a total of 7225 thousand persons with multiple jobs in the US. Of them, 4115 thousand are male, 1742 thousand are single, and 905 thousand are male and single. What is the probability that a selected person is a male or single? Solution Male M Female F Total Single Married Total
Chapter 4: Union of Events & ddition Rule 54 The probability of the union of two mutually exclusive events and is = + Example: university president has proposed p that all students must take a course in ethics as a requirement for graduation. Three hundred faculty members and students from this university were asked about their opinion on this issue and it shown in the following table. Faculty Student Ttl Total Favor 45 90 135 Opinion Oppose 15 110 125 Neutral 10 30 40 Total Find the probability that a person selected is in favor or is neutral 70 230 300
Chapter 4: Union of Events & ddition Rule 55 Solution:
Chapter 4: Union of Events & ddition Rule 56 Example: Eighteen percent of the working lawyers in the United States are female. Two lawyers are selected at random and it is observed whether they are male or female. a Draw a tree diagram for this experiment b Find the probability that at least one of the two lawyers is a female. Solution:
57 Chapter 4: ayes Theorem ddition Law + = Multiplication Law = Conditional robability = = =
Chapter 4: ayes Theorem 58 Example: Manufacturing firm that receives shipment of parts from two different suppliers. Currently, 65 percent of the parts purchased by the company are from supplier 1 and the remaining 35 percent are from supplier 2. Historical Data suggest the quality rating of the two supplier are shown in the table: Good arts ad arts Supplier 1 98 2 Supplier 1 95 5 a Draw a tree diagram for this experiment with the probability of all outcomes b Given the information the part is bad, What is the probability the part came from supplier 1?
Chapter 4: ayes Theorem 59 Solution:
60 Chapter 4: ayes Theorem 1 1 ayes theorem two-event case 2 2 1 1 1 1 1 + = 2 2 1 1 2 2 2 + = ayes theorem... 2 2 1 1 n n i i i + + + = 2 2 1 1 n n