Lesson 3 A Survey of Probability Concepts Kafu

Outline Definitions Learning Exercise #1 Definitions Learning Exercise #2 and #3 Basic rules of Probability Independence Bayes’ Theorem Probabilities for throwing n dice Some principles of counting Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 2

Definition: Probability A probability is a measure of the likelihood that an event in the future will happen. n It can only assume a value between 0 and 1. n A value near zero means the event is not likely to happen. A value near one means it is likely. n There are three definitions of probability: classical, empirical, and subjective. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data

Definition: Classical probability n The classical definition applies when there are n equally likely outcomes. n For example, in the tossing of a single perfectly cubical die, made of completely homogeneous material, the equally likely events are the appearance of any of the specific number of dots (from 1 to 6) on its upper face. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 4

Definition: Empirical or statistical probability n The empirical or statistical definition applies when the number of times the event happens is divided by the number of observations, based on data. Throughout her teaching career Professor Jones has awarded 186 A’s out of 1, 200 students. What is the probability that a student in her section this semester will receive an A? This number may be interpreted as “unconditional probability”. In most cases, we are interested in the probability of earning an A for a selected student who study 10 hours or more per week. We call this “conditional probability”. P(A | study 10 or more hours per week) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 5

Definition: Subjective probability n Subjective probability is based on whatever information is available, based on subjective feelings. n Estimating the probability mortgage rates for home loans will top 8 percent this year. n Estimating the probability that HK’s economic growth will be 3% this year. n Estimating the probability that HK will solve its deficit problem in 2006. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 6

Example 1 (to be used to illustrate the definitions) n A fair die is rolled once. n Peter is concerned with whether the resulted number is even, i. e. , 2, 4, 6. n Paul is concerned with whether the resulted number is less than or equal to 3, i. e. , 1, 2, 3. n Mary is concerned with whether the resulted number is 6. n Sonia is concerned with whether the resulted number is odd, i. e. , 1, 3, 5. n A fair die is rolled twice. n John is concerned with whether the resulted number of first roll is even, i. e. , 2, 4, 6. n Sarah is concerned with whether the resulted number of second roll is even, i. e. , 2, 4, 6. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data

Definitions: Experiment and outcome n An experiment is the observation of some activity or the act of taking some measurement. n The experiment is rolling the one die in the first example, and rolling one die twice in the second example. n An outcome is the particular result of an experiment. n The possible outcomes are the numbers 1, 2, 3, 4, 5, and 6 in the first example. n The possible outcomes are number pairs (1, 1), (1, 2), …, (6, 6), in the second example. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 8

Definition: Event n An event is the collection of one or more outcomes of an experiment. n For Peter: the occurrence of an even number, i. e. , 2, 4, 6. n For Paul: the occurrence of a number less than or equal to 3, i. e. , 1, 2, 3. n For Mary: the occurrence of a number 6. n For Sonia: the occurrence of an odd number, i. e. , 1, 3, 5. n For John: the occurrence of (2, 1), (2, 2), (2, 3), …, (2, 6), (4, 1), …, (4, 6), (6, 1), …, (6, 6) [John does not care about the result of the second roll]. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 9

Learning exercise 1: University Demographics n Current enrollments by college and by sex appear in the following table. College Ag-For Arts-Sci Bus-Econ Female 500 1500 400 Male 900 1200 1400 2700 Totals Educ 1000 Engr Law Undecl Totals 200 100 800 4500 500 1300 200 900 5500 900 1500 300 1700 10000 n If we select a student at random, what is the probability that the student is : n A female or male, i. e. , P(Female or Male). n Not from Agricultural and Forestry, i. e. , P(not-Ag-For) n A female given that the student is known to be from Bus. Econ, i. e. , P(Female |Bus. Econ). n A female and from Bus. Econ, i. e. , P(Female and Bus. Econ). n From Bus. Econ, i. e. , P(Bus. Econ). Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 10

Learning exercise 1: University Demographics College Ag-For Arts-Sci Bus-Econ Educ Engr Female 500 1500 400 1000 200 100 800 4500 Male 900 1200 500 1300 200 900 5500 1400 2700 900 1500 300 1700 10000 Totals Law Undecl Totals P(Female or Male) =(4500 + 5500)/10000 = 1 P(not-Ag-For) =(10000 – 1400) /10000 = 0. 86 P(Female | Bus Econ) = 400 /900 = 0. 44 P(Female and Bus. Econ) = 400 /10000 = 0. 04 P(Bus. Econ) = 900 /10000 = 0. 09 P(Female and Bus. Econ) = P(Bus. Econ) P(Female | Bus Econ) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 11

Definition: Mutually Exclusive events n Events are mutually exclusive if the occurrence of any one event means that none of the others can occur at the same time. n Peter’s event and Paul’s event are not mutually exclusive – both contains 2. n Peter’s event and Mary’s event are not mutually exclusive – both contains 6. n Paul’s event and Mary’s event are mutually exclusive – no common numbers. n Peter’s event and Sonia’s event are mutually exclusive – no common numbers. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 12

Definition: Exhaustive events n Events are collectively exhaustive if at least one of the events must occur when an experiment is conducted. n Peter’s event (even numbers) and Sonia’s event (odd numbers) are collectively exhaustive. n Peter’s event (even numbers) and Mary’s event (number 6) are not collectively exhaustive. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 13

Conditional Probability n A conditional probability is the probability of a particular event occurring, given that another event has occurred. n The probability of the event A given that the event B has occurred is written P(A|B). n P(female | Bus. Econ) n P(girl | test says girl) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 14

Learning exercise 2: Predicting Sex of Babies n Many couples take advantage of ultrasound exams to determine the sex of their baby before it is born. Some couples prefer not to know beforehand. In any case, ultrasound examination is not always accurate. About 1 in 5 predictions are wrong. n In one medical group, the proportion of girls correctly identified is 9 out of 10, i. e. , applying the test to 100 baby girls, 90 of the tests will indicate girls. and n the number of boys correctly identified is 3 out of 4. i. e. , applying the test to 100 baby boys, 75 of the tests will indicate boys. n The proportion of girls born is 48 out of 100. n What is the probability that a baby predicted to be a girl actually turns out to be a girl? Formally, find P(girl | test says girl). Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 15

Learning exercise 2: Predicting Sex of Babies n P(girl | test says girl) n In one medical group, the proportion of girls correctly identified is 9 out of 10 and n the number of boys correctly identified is 3 out of 4. n The proportion of girls born is 48 out of 100. n Think about the next 1000 births handled by this medical group. n 480 = 1000*0. 48 are girls n 520 = 1000*0. 52 are boys n Of the girls, 432 (=480*0. 9) tests indicate that they are girls. n Of the boys, 130 (=520*0. 25) tests indicate that they are girls. n In total, 562 (=432+130) tests indicate girls. Out of these 562 babies, 432 are girls. n Thus P(girl | test says girl ) = 432/562 = 0. 769 Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 16

Learning exercise 2: Predicting Sex of Babies With the information given, we can fill in the following table in sequence from (1) to (9), with the initial assumption of 1000 babies in total. For the question at hand, i. e. , P(girl | test says girl ), we only need to fill in the cells from (1) to (6). Test says girl Test says boy Totals Girl (4) 432 (7) 48 (2) 480 Boy (5) 130 (8) 390 (3) 520 Totals (6) 562 (9) 438 (1) 1000 P(girl | test says girl ) = 432/562 = 0. 769 Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 17

Learning exercise 2: Predicting Sex of Babies 1000*P(girls) n 480 = 1000*0. 48 are girls n 520 = 1000*0. 52 are boys 1000*P(boys) n Of the girls, 432 (=480*0. 9) tests indicate that they are girls. 1000*P(girls)*P(test says girls|girls) n Of the boys, 130 (=520*0. 25) tests indicate that they are girls. 1000*P(boys)*P(test says girls | boys) n In total, 562 tests indicate girls. 1000*[P(girls)*P(test says girls|girls) + P(boys)*P(test says girls|boys)] n Out of these 562 babies, 432 are girls. n Thus P(girls | test syas girls ) = 432/562 = 0. 769 1000*P(girls)*P(test says girls|girls) 1000*[P(girls)*P(test says girls|girls) + P(boys)*P(test says girls|boys)] Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 18

Learning exercise 3: Putting in Extra Trunk Lines n Given recent flooding (or other condition more appropriate to your area) between Town A and Town B, the local telephone company is assessing the value of adding an independent trunk line between the two towns. The second line will fail independently of the first because it will depend on different equipment and routing (we assume a regional disaster is highly unlikely). n Under current conditions, the present line works 98 out of 100 times someone wishes to make a call. If the second line performs as well, what is the chance that a caller will be able to get through? Formally, n P( Line 1 works ) = 98/100 n P( Line 2 works ) = 98/100 n Find P( Line 1 or Line 2 works ). Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 19

Learning exercise 3: Putting in Extra Trunk Lines n P( Line 1 works ) = 98/100 n P( Line 2 works ) = 98/100 n Find P( Line 1 or Line 2 works ) = 1 – P(Both line 1 and Line 2 fail) = 1 – P(Line 1 fails)*P(line 2 fails) = 1 – 0. 02*0. 02 = 0. 9996. Line 2 works Ka-fu Wong © 2004 Line 2 fails Line 1 works 0. 98*0. 02 Line 1 fails 0. 02*0. 98 0. 02*0. 02 ECON 1003: Analysis of Economic Data 20

Learning exercise 3: Putting in Extra Trunk Lines n P( individual line works ) = 50/100 n Find the number of lines we must install so that P(at least one of the lines works ) is larger than 0. 9. Suppose we install k independent lines P( at least one of the k lines works ) = 1 – P(all the k lines fail) = 1 – P(Line 1 fails)*P(line 2 fails)*… *P(line k fails) = 1 – 0. 5 k We want to find the smallest integer k such that 1 – 0. 5 k >0. 9 or 0. 1> 0. 5 k Log 0. 1< k log(0. 5) or k>log 0. 1 / log(0. 5) = 3. 32 Hence k=4. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 21

Basic Rules of Probability If two events A and B are mutually exclusive, the special rule of addition states that the probability of A or B occurring equals the sum of their respective probabilities: P(A or B) = P(A) + P(B) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 22

EXAMPLE 3 n New England Commuter Airways recently supplied the following information on their

EXAMPLE 3 n If A is the event that a flight arrives early, then P(A) = 100/1000 =. 10. n If B is the event that a flight arrives late, then P(B) = 75/1000 =. 075. n The probability that a flight is either early or late is: P(A or B) = P(A) + P(B) =. 10 +. 075 =. 175. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 24

The Complement Rule n The complement rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1. n If P(A) is the probability of event A and P(~A) is the complement of A, P(A) + P(~A) = 1 or P(A) = 1 - P(~A). Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 25

The Complement Rule continued n A Venn diagram illustrating the complement rule would appear

EXAMPLE 4 n Recall EXAMPLE 3. Use the complement rule to find the probability of an early (A) or a late (B) flight n P(A or B) = 1 - P(C or D) n If C is the event that a flight arrives on time, then P(C) = 800/1000 =. 8. n If D is the event that a flight is canceled, then P(D) = 25/1000 =. 025. P(A or B) = 1 - P(C or D) = 1 - [. 8 +. 025] =. 175 Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 27

EXAMPLE 4 continued n P(A or B) = 1 - P(C or D) =

The General Rule of Addition n If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula: P(A or B) = P(A) + P(B) - P(A and B) B A and B A Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 29

EXAMPLE 5 n In a sample of 500 students, 320 said they had a stereo, 175 said they had a TV, and 100 said they had both: Both Stereo 100 320 220 Ka-fu Wong © 2004 TV 175 75 ECON 1003: Analysis of Economic Data 30

EXAMPLE 5 continued n In a sample of 500 students, 320 said they had a stereo, 175 said they had a TV, and 100 said they had both. n If a student is selected at random, what is the probability that the student has only a stereo, only a TV, and both a stereo and TV? P(student has a stereo) = 320/500 =. 64. P(student has a TV) = 175/500 =. 35. P(student has both a stereo and TV) = 100/500 =. 20. P(student as only a stereo) = 220/500 =. 44. P(student has only a TV) = 75/500 =. 15. P(student as only a stereo) = P(student has a stereo) - P(student has both a stereo and TV) P(student has only a TV) = P(student has a TV) - P(student has both a stereo and TV) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 31

EXAMPLE 5 continued n In a sample of 500 students, 320 said they had a stereo, 175 said they had a TV, and 100 said they had both. P(S) = 320/500 =. 64. P(T) = 175/500 =. 35. P(S and T) = 100/500 =. 20. n If a student is selected at random, what is the probability that the student has either a stereo or a TV in his or her room? P(either S or T) = P(only S) + P (only T) = [P(S) - P(S and T)] +[P(T) - P(S and T)] =. 64 -. 20 +. 35 -. 20 =. 59. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 32

Joint Probability n A joint probability measures the likelihood that two or more events will happen concurrently. n An example would be the event that a student has both a stereo and TV in his or her dorm room. n In an experiment of throwing only one die, let x denote the outcome, P(x=5, and x=6) =0 because the two outcomes are mutually exclusive. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 33

Special Rule of Multiplication n The special rule of multiplication requires that two events A and B are independent. n Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other. n This rule is written: P(A and B) = P(A)P(B) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 34

EXAMPLE 6 n Chris owns two stocks, IBM and General Electric (GE). The probability that IBM stock will increase in value next year is. 5 and the probability that GE stock will increase in value next year is. 7. Assume the two stocks are independent. What is the probability that both stocks will increase in value next year? P(IBM and GE) = (. 5)(. 7) =. 35. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 35

EXAMPLE 6 continued n The probability that IBM stock will increase in value next year is. 5 and the probability that GE stock will increase in value next year is. 7. n What is the probability that at least one of these stocks increase in value during the next year? (This means that either one can increase or both. ) Approach 1: P(at least one) = P(GE only) + P(IBM only) + P(both) = (. 5)(. 3) + (. 5)(. 7) +(. 7)(. 5) =. 85. Approach 2: P(at least one) = P(GE) + P(IBM) – P(both) = 0. 5 + 0. 7 – 0. 35 =. 85. Approach 3: P(at least one) = 1 - P(both GE and IBM do not increase) = 1 – 0. 15 =. 85. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 36

General Multiplication Rule n The general rule of multiplication is used to find the joint probability that two events will occur. n It states that 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. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 37

General Multiplication Rule n The joint probability, P(A and B) is given by the following formula: P(A and B) = P(A)P(B|A) or P(A and B) = P(B)P(A|B) n Examples: n P(test says girl and girl) = P(girls) * P(test says girls | girls) n P(test says boy and boy) = P(boys) * P(test says boys | boys) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 38

Independence -- an illustration n Consider whether the decision of a young man going to party depends on whether his girlfriend goes to the same party. n Assume the probability of the young man going to party is 0. 7 (i. e. , he goes to 70 out of 100 parties on average). n If he tends to go to whichever party his girlfriend goes, his party behavior depends on his girlfriend’s. That is, the probability of going to a party conditional on his girlfriend’s presence is larger than 0. 7 (extreme case being 1. 0). n If he tends to avoid going to whichever party his girlfriend goes, his party behavior also depends on his girlfriend’s. That is, the probability of going to a party conditional on his girlfriend’s presence is less than 0. 7 (extreme case being 0. 0). n If in making the party decision, he never considers whether his girlfriend is going to a party, his party behavior does not depends on his girlfriend’s. That means, the probability of going to a party conditional on his girlfriend’s presence is 0. 7. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 39

Independence -- an illustration n Define events: n A: a young man goes to a party n B: his girlfriend goes to the same party. n Assume P(A) =0. 7 n His party behavior does not depend on his girlfriend’s only if P(A|B) =P(A) = 0. 7. And, event A is said to be independent of event B. n P(the young man and his girlfriends shows up in a party) = P(A & B) = P(B)*P(A|B). n If he always goes to whichever party his girlfriend goes, P(A|B) = 1. Hence, P(A & B) = P(B)*P(A|B) = P(B). n If he always avoid to whichever party his girlfriend goes, P(A|B) = 0. Hence, P(A & B) = P(B)*P(A|B) = 0. n If in making the party decision, he never considers whether his girlfriend is going to a party, P(A|B) = 0. 7. Hence, P(A & B) = P(B)*P(A|B) = P(B)*P(A) = 0. 7. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 40

Independence n Event A is independent of B n P(A|B) = P (A) n Event B is independent of A n P(B|A) = P (B) implies P(A&B) = P(B|A) * P(A) = P(B) * P(A) n P(A|B) = P (A) implies P(A&B) = P(A|B) * P(B) = P(A) * P(B) n Thus, if P(A&B) = P(B) * P(A), we must have either P(A|B) = P (A) or P(B|A) = P (B). Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 41

EXAMPLE 7 The Dean of the School of Business at Owens University collected the

EXAMPLE 7 continued If a student is selected at random, what is the probability that the student is a female (F) accounting major (A) P(A and F) = 110/1000. Given that the student is a female, what is the probability that she is an accounting major? Approach 1: P(A|F) = P(A and F)/P(F) = [110/1000]/[400/1000] =. 275 Ka-fu Wong © 2004 Approach 2: P(A|F) = 110/400 =. 275 ECON 1003: Analysis of Economic Data 43

Tree Diagrams n A tree diagram is useful for portraying conditional and joint probabilities. It is particularly useful for analyzing business decisions involving several stages. n EXAMPLE 8: 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 showing this information. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 44

EXAMPLE 8 continued 6/11 7/12 5/12 R 1 5/11 B 2 7/11 R 2

EXAMPLE 8 continued The tree diagram is very illustrative about the relation between joint probability and conditional probability Let A (B) be the event of a red chip in the first (second) draw. 6/11 P(B|A) = 6/11 R 2 P(A) = 7/12 5/12 R 1 5/11 B 2 7/11 R 2 B 1 4/11 Ka-fu Wong © 2004 P(A and B) = P(A)*P(B|A) = 6/11 * 7/12 B 2 ECON 1003: Analysis of Economic Data 46

Bayes’ Theorem n Bayes’ Theorem is a method for revising a probability given additional information. n If A 1 and A 2 are mutually exclusive and exhaustive, and A 1 and A 2 together covers event B, we have : Bayes Theorem is an essential tool to understand options and real options in finance. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 47

Bayes’ Theorem n A 1 and A 2 are mutually exclusive means P (A 1 or A 2 ) = P(A 1) + P(A 2 ) P (A 1 and A 2 ) = 0 n A 1 and A 2 are exhaustive means P (A 1 or A 2 ) = P(A 1) + P(A 2 ) = 1 n A 1 and A 2 together covers event B means P(B) = P(A 1 & B) + P(A 2 & B) (see the Venn diagram illustration) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 48

Venn Diagram illustration of Bayes’ theorem A 1 B A 1 & B Ka-fu

Bayes’ Theorem n Bayes’ Theorem can be derived based on simple manipulation of the general multiplication rule. P(A 1|B) = P(A 1 & B) /P(B) = [P(A 1) P(B|A 1)] / [P(A 1 & B) + P(A 2 & B)] = [P(A 1) P(B|A 1) ]/ [P(A 1) P(B|A 1) + P(A 2) P(B|A 2)] Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 50

EXAMPLE 9 n Duff Cola Company recently received several complaints that their bottles are under-filled. A complaint was received today but the production manager is unable to identify which of the two Springfield plants (A or B) filled this bottle. The following table summarizes the Duff production experience. n What is the probability that the under-filled bottle came from plant A? Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 51

Example 9 continued n What is the probability that the under-filled bottle came from plant A? The likelihood the bottle was filled in Plant A is reduced from . 55 to. 4783. Without the information about U, the manager will say the under-filled bottle is likely from plant A. With the additional information about U, the manager will say the under-filled bottle is likely from plant B. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 52

Probabilities for throwing n Dice n Suppose we roll n regular balanced six-sided dice. If the event X is the sum of the n values which appear, what are the probabilities associated for each value of X, for the possible values X = n, . . . , 6 n? n In the case n =1, these probabilities are all 1/6. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 53

Probabilities for throwing n Dice For two dice , it is easiest to consider a table of possible outcomes: (1, 1) (1, 2 (1, 3) (1, 4) ) (1, 5 ) (1, 6) (2, 1) (2, 2 (2, 3) (2, 4) ) (2, 5 ) (2, 6) (3, 1) (3, 2 (3, 3) (3, 4) ) (3, 5 ) (3, 6) (4, 1) (4, 2 (4, 3) (4, 4) ) (4, 5 ) (4, 6) (5, 1) (5, 2 (5, 3) (5, 4) ) (5, 5 ) (5, 6) Next, we can consider the sums associated with these outcomes: 2 3 4 5 6 7 8 9 10 6 7 8 9 10 11 12 (6, 1) (6, 2 (6, 3) (6, 4) (6, 5 (6, 6) ) ) Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 54

Probabilities for throwing n Dice Next, we can consider the sums associated with these outcomes: Since there are 36 outcomes, each equally likely, we see that the probabilities are: X # P(X) 2 1 1/36 8 5 5/36 3 2 2/36 9 4 4/36 4 3 3/36 10 3 3/36 2 3 4 5 6 7 8 9 10 6 7 8 9 10 11 5 4 4/36 11 2 2/36 7 8 9 10 11 12 6 5 5/36 12 1 1/36 7 6 6/36 Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 55

Probabilities for throwing n Dice n If there are more than two dice, it is difficult to make tables. The number of outcomes will be 6 n, so to calculate the probabilities it is sufficient to count the number of times each sum occurs among the 6 n possible outcomes. This is easily done by considering the generating function for the number of times each sum appears: f(x) = (x + x 2 + x 3 + x 4 + x 5+x 6)n n In our example, n=2 (x + x 2 + x 3 + x 4 + x 5+x 6)2 =x 2 + 2 x 3 +3 x 4 + 4 x 5+5 x 6+6 x 7+5 x 8+4 x 9+3 x 10+2 x 11+x 12 n The coefficient of x 3 is number of times the sum of 3 occurs. Note: The formula is for reference only. No need to memorize it. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 56

Some Principles of Counting n The multiplication formula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both. n Example 10: Dr. Delong has 10 shirts and 8 ties. How many shirt and tie outfits does he have? (10)(8) = 80 Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 57

Some Principles of Counting n A permutation is any arrangement of r objects selected from n possible objects. n Note: The order of arrangement is important in permutations. n! = n (n-1) (n-2) … 2 1 n=1, n!=1 n=2, n!=2 1=2 n=3, n!=3 2 1=6 n=4, n!=4 3 2 1=24 n=5, n!=5 4 3 2 1=120 Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 58

Some Principles of Counting n A combination is the number of ways to choose

EXAMPLE 11 n There are 12 players on the Carolina Forest High School basketball team. Coach Thompson must pick five players among the twelve on the team to comprise the starting lineup. How many different groups are possible? Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 60

Example 11 continued n Suppose that in addition to selecting the group, he must also rank each of the players in that starting lineup according to their ability. Ka-fu Wong © 2004 ECON 1003: Analysis of Economic Data 61