Chapter 4 Introduction to robability Experiments, Counting Rules, and Assigning robabilities Events and Their robability Some Basic Relationships of robability Conditional robability Bayes Theorem robability robability is a numerical measure of the likelihood that an event will occur. robability values are always assigned on a scale from 0 to 1. A probability near 0 indicates an event is very unlikely to occur. A probability near 1 indicates an event is almost certain to occur. A probability of 0.5 indicates the occurrence of the event is just as likely as it is unlikely. Slide 1 Slide 2 robability as a Numerical Measure of the Likelihood of Occurrence An Experiment and Its Sample Space An experiment is any process that generates welldefined outcomes. robability: Increasing Likelihood of Occurrence 0.5 1 The occurrence of the event is just as likely as it is unlikely. The sample space for an experiment is the set of all experimental outcomes. A sample point is an element of the sample space, any one particular experimental outcome. Slide 3 Slide 4 1
Assigning probabilities to Events Random experiment a random experiment is a process or course of action, whose outcome is uncertain. erforming the same random experiment repeatedly, may result in different outcomes, therefore, the best we can do is talk about the probability of occurrence of a certain outcome. To determine the probabilities we need to define the possible outcomes first. Slide 5 Determining the outcomes. Build an exhaustive list of all possible outcomes. Our objective is to Make sure the listed outcomes determine are mutually (A, the exclusive. probability that A list of outcomes that meet the event two Aconditions will occur. above, is called a sample space. Simple events The individual outcomes are called simple events. Simple events cannot be further decomposed into constituent outcomes. Sample Space a sample space of a random experiment is a list of all possible outcomes of the experiment. The outcomes must be mutually exclusive and exhaustive. Event An event is any collection of one or more simple events Slide 6 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 $000 Markley Oil Collins Mining 10 8 5-2 0-20 A Counting Rule for Multiple-Step Experiments If an experiment consists of a sequence of k steps in which there are n 1 possible results for the first step, n 2 possible results for the second step, and so on, then the total number of experimental outcomes is given by (n 1 (n 2... (n k. A helpful graphical representation of a multiple-step experiment is a tree diagram. Slide 7 Slide 8 2
A Counting Rule for Multiple-Step Experiments Bradley Investments can be viewed as a two-step experiment; it involves two stocks, each with a set of experimental outcomes. Markley Oil: n 1 = 4 Collins Mining: n 2 = 2 Total Number of Experimental Outcomes: n 1 n 2 = (4(2 = 8 Slide 9 Tree Diagram Markley Oil Collins Mining Experimental (Stage 1 (Stage 2 Outcomes Gain 10 Lose 20 Gain 5 Even Gain 8 Lose 2 Gain 8 Lose 2 Gain 8 Lose 2 Gain 8 Lose 2 (10, 8 Gain $18,000 (10, -2 Gain $8,000 (5, 8 Gain $13,000 (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 Slide 10 Counting Rule for Combinations Another 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. Number of combinations of N objects taken n at a time N N Cn N =! = n n!( N n! Counting Rule for ermutations 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. Number of permutations of N objects taken n at a time n N N n N! =! = n ( N n! where N! = N(N -1(N - 2... (2(1 n! = n(n - 1( n - 2... (2(1 0! = 1 Slide 11 Slide 12 3
Assigning robabilities Classical Method Assigning probabilities based on the assumption of equally likely outcomes. Relative Frequency Method Assigning probabilities based on experimentation or historical data. Subjective Method Assigning probabilities based on the assignor s judgment. Classical Method If an experiment has n possible outcomes, this method would assign a probability of 1/n to each outcome. robability of an event = Number of favorable outcomes Total number of possible outcomes Example Experiment: Rolling a die Sample Space: S = {1, 2, 3, 4, 5, 6} robabilities: Each sample point has a 1/6 chance of occurring. Slide 13 Slide 14 Example: Lucas Tool Rental Relative Frequency Method Lucas would like to assign probabilities to the number of floor polishers it rents per day. n( E ( E = lim n n Office records show the following frequencies of daily rentals for the last 40 days. Number of Number olishers Rented of Days 0 4 1 6 2 18 3 10 4 2 Slide 15 Example: Lucas Tool Rental Relative Frequency Method The probability assignments are given by dividing the number-of-days frequencies by the total frequency (total number of days. Number of Number olishers Rented of Days robability 0 4.10 = 4/40 1 6.15 = 6/40 2 18.45 etc. 3 10.25 4 2.05 40 1.00 Slide 16 4
Subjective Method When economic conditions and a company s 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 estimates. Applying the subjective method, an analyst made the following probability assignments. Exper. Outcome Net Gain/Loss robability ( 10, 8 $18,000 Gain.20 ( 10, -2 $8,000 Gain.08 ( 5, 8 $13,000 Gain.16 ( 5, -2 $3,000 Gain.26 ( 0, 8 $8,000 Gain.10 ( 0, -2 $2,000 Loss.12 (-20, 8 $12,000 Loss.02 (-20, -2 $22,000 Loss.06 Slide 17 Slide 18 Given a sample space S={E 1,E 2,,E n }, the following characteristics for the probability (E i of the simple event E i must hold: 1. 2. Assigning probabilities ( 0 E n i ( E 1 for i = 1 each i Events and Their robability 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 experiment and assign a probability to each, we can compute the probability of an event. = i 1 robability of an event: The probability (A of event A is the sum of the probabilities assigned to the simple events contained in A. Slide 19 Slide 20 5
Events and Their robabilities Event M = Markley Oil rofitable M = {(10, 8, (10, -2, (5, 8, (5, -2} (M = (10, 8 + (10, -2 + (5, 8 + (5, -2 =.2 +.08 +.16 +.26 =.70 Event C = Collins Mining rofitable (C =.48 (found using the same logic Some Basic Relationships of robability There are some basic probability relationships that can be used to compute the probability of an event without knowledge of al the sample point probabilities. Complement of an Event Union of Two Events Intersection of Two Events Mutually Exclusive Events Slide 21 Slide 22 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 A c (or ~A. The Venn diagram below illustrates the concept of a complement. Sample Space S 5-14 The Complement Rule The complement rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1. If (A is the probability of event A and (~A is the complement of A, (A + (~A = 1 OR (A = 1-(~A. Event A A c Slide 23 Slide 24 6
Union of Two Events The union of events A and B is the event containing all sample points that are in A or B or both. The union is denoted by A B. The union of A and B is illustrated below. Sample Space S Union of Two Events Event M = Markley Oil rofitable Event C = Collins Mining rofitable M C = Markley Oil rofitable or Collins Mining rofitable Event A Event B M C = {(10, 8, (10, -2, (5, 8, (5, -2, (0, 8, (-20, 8} (M C = (10, 8 + (10, -2 + (5, 8 + (5, -2 + (0, 8 + (-20, 8 =.20 +.08 +.16 +.26 +.10 +.02 =.82 Slide 25 Slide 26 Intersection of Two Events The intersection of events A and B is the set of all sample points that are in both A and B. The intersection is denoted by A Β. The intersection of A and B is the area of overlap in the illustration below. Intersection Sample Space S Event A Event B Intersection of Two Events Event M = Markley Oil rofitable Event C = Collins Mining rofitable M C = Markley Oil rofitable and Collins Mining rofitable M C = {(10, 8, (5, 8} (M C = (10, 8 + (5, 8 =.20 +.16 =.36 Slide 27 Slide 28 7
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: (A B = (A + (B - (A B Addition Law Markley Oil or Collins Mining rofitable We know: (M =.70, (C =.48, (M C =.36 Thus: (M C = (M + (C - (M C =.70 +.48 -.36 =.82 This result is the same as that obtained earlier using the definition of the probability of an event. Slide 29 Slide 30 Mutually Exclusive Events Two events are said to be mutually exclusive if the events have no sample points in common. That is, two events are mutually exclusive if, when one event occurs, the other cannot occur. Sample Space S Mutually Exclusive Events Addition Law for Mutually Exclusive Events (A B = (A + (B Event A Event B Slide 31 Slide 32 8
Conditional robability The probability of an event given that another event has occurred is called a conditional probability. The conditional probability of A given B is denoted by (A B. A conditional probability is computed as follows: ( A B ( AB = ( B Conditional robability Collins Mining rofitable given Markley Oil rofitable ( C M. 36 ( CM = = =. 51 ( M. 70 Slide 33 Slide 34 Multiplication Law The multiplication law provides a way to compute the probability of an intersection of two events. The law is written as: (A B = (B(A B Multiplication Law Markley Oil and Collins Mining rofitable We know: (M =.70, (C M =.51 Thus: (M C = (M(M C = (.70(.51 =.36 This result is the same as that obtained earlier using the definition of the probability of an event. Slide 35 Slide 36 9
5-24 Special Rule of Multiplication 5-25 EXAMLE 6 The special rule of multiplication requires that two events A and B are independent. Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other. The special rule is written: Chris owns two stocks which are independent of each other. The probability that stock A increases in value next year is.5. The probability that stock B will increase in value next year is.7. What is the probability that both stocks increase in value next year? (A and B = (.5(.7 =.35. (A and B = (A*(B. Slide 37 Slide 38 5-26 EXAMLE 6 continued Independent Events What is the probability that at least one of these stocks increase in value during the next year (this implies that either one can increase or both? Thus, (at least one = (.5(.3 + (.5(.7 + (.7(.5 =.85. Events A and B are independent if (A B = (A. Slide 39 Slide 40 10
Independent Events 5-23 Joint robability Multiplication Law for Independent Events (A B = (A(B The multiplication law also can be used as a test to see if two events are independent. Joint robability is a probability that measures the likelihood that two or more events will happen concurrently. An example would be the event that a student has both a stereo and TV in his or her dorm room. Slide 41 Slide 42 5-28 General Multiplication Rule Multiplication Law for Independent Events Are M and C independent? Does (M C = (M(C? We know: (M C =.36, (M =.70, (C =.48 But: (M(C = (.70(.48 =.34.34.36, so M and C are not independent. The general rule of multiplication is used to find the joint probability that two events will occur, as it states: for two events A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of B given that A has occurred. Slide 43 Slide 44 11
5-29 General Multiplication Rule 5-30 EXAMLE 7 The joint probability, (A and B is given by the following formula: (A and B = (A*(B A OR (A and B = (B*(A B The Dean of the School of Business at Miami collected the following information about undergraduate students in her college: MAJOR Male Female Total Accounting 170 110 280 Finance 120 100 220 Marketing 160 70 230 Management 150 120 270 Total 600 400 1000 Slide 45 Slide 46 5-30 5-32 Tree Diagrams If a student is selected at random, what is the probability that the student is a female accounting major? (A and F = 110/1000. Given that the student is a female, what is the probability that she is an accounting major? (A F = [(A and F]/[(F] = [110/1000]/[400/1000] =.275. A tree diagram is very useful for portraying conditional and joint probabilities and is particularly useful for analyzing business decisions involving several stages. EXAMLE : In a bag containing 7 red chips and 5 blue chips you select 2 chips one after the other without replacement. Construct a tree diagram for this information. Slide 47 Slide 48 12
Bayes Theorem Example: L. S. Clothiers 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. rior robabilities New Information Application of Bayes Theorem osterior robabilities 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. 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: A 1 = town council approves the zoning change A 2 = town council disapproves the change rior robabilities Using subjective judgment: (A 1 =.7, (A 2 =.3 Slide 49 Slide 50 Example: L. S. Clothiers 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? Conditional robabilities ast history with the planning board and the town council indicates the following: (B A 1 =.2 (B A 2 =.9 Example: L. S. Clothiers Tree Diagram (B A 1 =.2 (A 1 B =.14 (A 1 =.7 (B c A 1 =.8 (A 1 B c =.56 (B A 2 =.9 (A 2 B =.27 (A 2 =.3 (B c A 2 =.1 (A 2 B c =.03 Slide 51 Slide 52 13
Bayes Theorem To find the posterior probability that event A i will occur given that event B has occurred we apply Bayes theorem. ( A i ( B A i ( A i B = ( A ( B A + ( A ( B A +... + ( A ( B A 1 1 2 2 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. n n Example: L. S. Clothiers osterior robabilities Given the planning board s recommendation not to approve the zoning change, we revise the prior probabilities as follows. ( A 1 ( B A 1 (. 7 (. 2 ( A 1 B = = ( A ( B A + ( A ( B A (. 7 (. 2 + (. 3 (. 9 1 1 2 2 =.34 Conclusion The planning board s recommendation is good news for L. S. Clothiers. The posterior probability of the town council approving the zoning change is.34 versus a prior probability of.70. Slide 53 Slide 54 Tabular Approach Step 1 repare 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. Tabular Approach (1 (2 (3 (4 (5 rior Conditional Events robabilities robabilities A i (A i (B A i A 1.7.2 A 2.3.9 1.0 Slide 55 Slide 56 14
Tabular Approach Step 2 In column 4, compute the joint probabilities for each event and the new information B by using the multiplication law. Multiply the prior probabilities in column 2 by the corresponding conditional probabilities in column 3. That is, (A i IB = (A i (B A i. Tabular Approach (1 (2 (3 (4 (5 rior Conditional Joint Events robabilities robabilities robabilities A i (A i (B A i (A i I B A 1.7.2.14 A 2.3.9.27 1.0 Slide 57 Slide 58 Tabular Approach Step 3 Sum the joint probabilities in column 4. The sum is the probability of the new information (B. We see that there is a.14 probability of the town council approving the zoning change and a negative recommendation by the planning board. There is a.27 probability of the town council disapproving the zoning change and a negative recommendation by the planning board. The sum.14 +.27 shows an overall probability of.41 of a negative recommendation by the planning board. Tabular Approach (1 (2 (3 (4 (5 rior Conditional Joint Events robabilities robabilities robabilities A i (A i (B A i (A i I B A 1.7.2.14 A 2.3.9.27 1.0 (B =.41 Slide 59 Slide 60 15
Tabular Approach Step 4 In column 5, compute the posterior probabilities using the basic relationship of conditional probability. ( A i B = ( A i B ( B Note that the joint probabilities (A i IB are in column 4 and the probability (B is the sum of column 4. Tabular Approach (1 (2 (3 (4 (5 rior Conditional Joint osterior Events robabilities robabilities robabilities robabilities A i (A i (B A i (A i I B (A i B A 1.7.2.14.3415 A 2.3.9.27.6585 1.0 (B =.41 1.0000 Slide 61 Slide 62 End of Chapter 4 Slide 63 16