Statistics for Business and Economics (14e) Ch4.ppt

Statistics for Business and Economics (14e),Anderson, Sweeney, Williams, Camm, Cochran, Fry, Ohlmann 2020 Cengage Learning, 2020 Cengage. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website or school-approved learning management system for classroom use.,1,Chapter 4 - Introduction to Probability,4.1 - Random Experiments, Counting Rules, and Assigning Probabilities 4.2 - Events and Their Probability 4.3 - Some Basic Relationships of Probability 4.4 - Conditional Probability 4.5 - Bayes Theorem,2,Uncertainties,Managers often base their decisions on an analysis of uncertainties such as the following: What are the chances that the sales will decrease if we increase prices? What is the likelihood a new assembly method will increase productivity? What are the odds that a new investment will be profitable?,3,Probability,Probability is a numerical measure of the likelihood that an event will occur. Probability values are always assigned on a scale from 0 to 1. A probability near zero indicates an event is quite unlikely to occur. A probability near one indicates an event is almost certain to occur.,4,Statistical Experiments,In statistics, the notion of an experimental differs somewhat from that of an experiment in the physical sciences. In statistical experiments, probability determines outcomes. Even though the experiment is repeated exactly the same way, an entirely different outcome may occur. For this reason, statistical experiments are sometimes called random experiments.,5,Random Experiment and Its Sample Space (1 of 2),A Random experiment is a process that generates well-defined experimental outcomes. The sample space for an experiment is the set of all experimental outcomes. An experimental outcome is also called a sample point.,6,Random Experiment and Its Sample Space (2 of 2),Example: Bradley Investments Bradley has invested in two stocks, Markley Oil and Collins Mining. Bradley has determined that the possible outcomes of these investments three months from now are as follows: Investment Gain or Loss in 3 Months (in $1000s):,7,A Counting Rule for Multiple-Step Experiments,If an experiment consists of a sequence of k steps in which there are n1 possible results for the first step, n2 possible results for the second step, and so on, then the total number of experimental outcomes is given by (n1)(n2) . . . (nk). A helpful graphical representation of a multiple-step experiment is a tree diagram. Markley Oil: n1 = 4 Collins Mining: n2 = 2 Total number of experimental outcomes: (4)(2) = 8.,8,Tree Diagram (1 of 2),Example: Bradley Investments,9,Counting Rule for Combinations,Number of Combinations of N Objects Taken n at a Time A second useful counting rule enables us to count the number of experimental outcomes when n objects are to be selected from a set of N objects.,10,Counting Rule for Permutations,Number of Permutations of N Objects Taken n at a Time A third useful counting rule enables us to count the number of experimental outcomes when n objects are to be selected from a set of N objects, where the order of selection is important.,11,Assigning Probabilities (1 of 2),Basic Requirements for Assigning Probabilities 1.The probability assigned to each experimental outcome must be between 0 and 1, inclusively.,where Ei is the i th experimental outcome and P(Ei) is its probability 2. The sum of the probabilities for all experimental outcomes must equal 1.,where n is the number of experimental outcomes.,12,Assigning Probabilities (2 of 2),Classical Method Assigning probabilities based on the assumption of equally likely outcomes Relative Frequency Method Assigning probabilities based on experimental or historical data Subjective Method Assigning probability based on judgment.,13,Classical Method,Example: Rolling a Die If an experiment has n possible outcomes, the classical method would assign a probability of 1/n to each outcome. Experiment: Rolling a die Sample Space: S = 1, 2, 3, 4, 5, 6 Probabilities: Each sample point has a 1/6 chance of occurring,14,Relative Frequency Method,Example: Lucas Tool Rental Lucas 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 rental for the last 40 days. Each probability assignment is given by dividing the frequency (number of days) by the total frequency (total number of days).,15,Subjective Method (1 of 2),When economic conditions or a companys circumstances change rapidly it might be inappropriate to assign probabilities based solely on historical data. We can use any data available as well as our experience and intuition, but ultimately a probability value should express our degree of belief that the experimental outcome will occur. The best probability estimates often are obtained by combining the estimates from the classical or relative frequency approach with the subjective estimate.,16,Subjective Method (2 of 2),Example: Bradley Investments An analyst made the following probability estimates.,17,Events and Their Probabilities (1 of 2),An event is a collection of sample points. The probability of any event is equal to the sum of the probabilities of the sample points in the event. If we can identify all the sample points of an experimental and assign a probability to each, we can compute the probability of an event. Event M = Markley Oil is Profitable M = (10, 8), (10, 2), (5, 8), (5, 2) P(M) = P(10, 8) + P(10, 2) + P(5, 8) + P(5, 2) = 0.20 + 0.08 + 0.16 + 0.26 = 0.70,18,Events and Their Probabilities (2 of 2),Example: Bradley Investments Event C = Collins Mining is Profitable C = (10, 8), (5, 8), (0, 8), (20, 8) P(C) = P(10, 8) + P(5, 8) + P(0, 8) + P(20, 8) = 0.20 + 0.16 + 0.10 + 0.02 = 0.48,19,Some Basic Relationships of Probability,There are some basic probability relationships that can be used to compute the probability of an event without knowledge of all the sample point probabilities. Complement of an Event Union of Two Events Intersection of Two Events Mutually Exclusive Events,20,Complement of an Event,The complement of event A is defined to be the event consisting of all sample points that are not in A. The complement of A is denoted by AC.,21,Union of Two Events (1 of 2),The union of events A and B is the event containing all sample points that are in A and B or both. The union of events A and B is denoted by A B.,22,Union of Two Events (2 of 2),Example: Bradley Investments Event M = Markley Oil is Profitable Event C = Collins Mining is Profitable M C = Markley Oil is Profitable OR Collins Mining is Profitable (or both are profitable). M C = (10, 8), (10, 2), (5, 8), (5, 2), (0, 8), (20, 8) P(M C) = P(10, 8) + P(10, 2) + P(5, 8) + P(5, 2) + P(0, 8) + P(20, 8) = 0.20 + 0.08 + 0.16 + 0.26 + 0.10 + 0.02 = 0.82,23,Intersection of Two Events (1 of 2),The intersection of events A and B is the set of all sample points that are in both A and B,24,Intersection of Two Events (2 of 2),Example: Bradley Investments,31,Addition Law,The addition law provides a way to compute the probability of event A, or B, or both A and B occurring. The law is written as:,26,Mutually Exclusive Events,Two events are said to be mutually exclusive if the event have no sample points in common Two events are mutually exclusive if, when one event occurs, the other cannot occur.,27,Conditional Probability (1 of 2),The probability of an event given that another event has occurred is called a conditional probability. The conditional probability of A given B has already occurred denoted by ). A conditional probability is computed as follows:,28,Conditional Probability (2 of 2),Example: Bradley Investments,29,Multiplication Law,The multiplication law provides a way to compute the probability of the intersection of two events. The law is written as:,30,Joint Probability Table,Joint probabilities appear in the body of the table Marginal probabilities appear in the margins of the table.,31,Independent Events,If the probability of event A is not changed by the existence of event B, we would say that events A and B and are independent. Two events A and B are independent if:,32,Multiplication Law for Independent Events,The multiplication law also can be used as a test to see if two events are independent. The law is written as:,33,Mutual Exclusiveness and Independence,Do not confuse the notion of mutually exclusive events with that of independent events. Two events with nonzero probability cannot both mutually exclusive and independent. If one mutually exclusive event is known to occur, the other cannot; occur thus, the probability of the other even occurring is reduced to zero (and therefore dependent). Two events that are not mutually exclusive, might or night not be independent.,34,Bayes Theorem (1 of 2),Often we begin probability analysis with initial or prior probabilities. Then, from a sample, special report, or a product test we obtain some additional information. Given this information, we calculate revised or posterior probabilities. Bayes theorem provides the means for revising the prior probabilities.,35,Bayes Theorem (2 of 2),A proposed shopping center will provide strong competition for downtown businesses like L. S. Clothiers. If the shopping center is built, the owner of L. S. Clothiers feels it would be best to relocate to the shopping center. The shopping center cannot be built unless a zoning change is approved by the town council. The planning board must first make a recommendation, for or against the zoning change, to the council. Let: A1 = town council approves the zoning change A2 = town council disapproves the change Using subjective judgment:,36,New Information,The planning board has recommended against the zoning change. Let B denote the event of a negative recommendation by the planning board. Given that B has occurred, should L. S. Clothiers revise the probabilities that the town council will approve or disapprove the zoning change? Past history with the planning board and the town council indicates the following: Hence:,37,Tree Diagram (2 of 2),Example: L. S. Clothiers,38,Bayes Theorem,To find the posterior probability that event Ai will occur given that event B has occurred, we apply Bayes theorem.,Bayes theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space.,39,Posterior Probabilities (1 of 2),Example: L. S. Clothiers Given the planning boards recommendation not to approve the zoning change, we revise the prior probabilities as follows:,40,Posterior Probabilities (2 of 2),The planning boards recommendation is good news for L. S. Clothiers. The posterior probability of the town council approving the zoning change is 0.34 compared to a prior probability of 0.70.,41,Bayes Theorem: Tabular Approach (1 of 6),Step 1: Prepare the following three columns: Column 1 - The mutually exclusive events for which posterior probabilities are desired. Column 2 - The prior probabilities for the events. Column 3 - The conditional probabilities of the new information given each event.,42,Bayes Theorem: Tabular Approach (2 of 6),Example: L. S. Clothiers, Step 1,43,Bayes Theorem: Tabular Approach (3 of 6),Step 2: Prepare the fourth column Column 4: Compute the joint probabilities for each event and the new information B by using the multiplication law. Multiply the probabilities in column 2 by the corresponding conditional probabilities in column 3. That is,44,Bayes Theorem: Tabular Approach (4 of 6),Step 2 (continued) We see that there is a 0.14 probability of the town council approving the zoning change and a negative recommendation by the planning board. There is a 0.27 probability of the town council disapproving the zoning change and a negative recommendation by the planning board. Step 3 Sum the joint probabilities in Column 4. The sum is the probability of the new information, P(B). The sum 0.14 + 0.27 shows an overall probability of 0.41 of a negative recommendation by the planning board.,45,Bayes Theorem: Tabular Approach (5 of 6),Example: L.S Clothiers, Step 3,46,Bayes Theorem: Tabular Approach (6 of 6),Step 4 Prepare the fifth column: Column 5 Compute the posterior probabilities using the basic relationship of conditional probability.,47,End of Chapter 4,48,