1 1 Slide © 2003 Thomson/South-Western

2 2 Slide © 2003 Thomson/South-Western Chapter 4 Introduction to Probability n Experiments, Counting Rules, and Assigning Probabilities Assigning Probabilities n Events and Their Probability n Some Basic Relationships of Probability n Conditional Probability n Bayes’ Theorem

3 3 Slide © 2003 Thomson/South-Western Probability n Probability is a numerical measure of the likelihood that an event will occur. n Probability values are always assigned on a scale from 0 to 1. n A probability near 0 indicates an event is very unlikely to occur. n A probability near 1 indicates an event is almost certain to occur. n A probability of 0.5 indicates the occurrence of the event is just as likely as it is unlikely.

4 4 Slide © 2003 Thomson/South-Western Probability as a Numerical Measure of the Likelihood of Occurrence 01.5 Increasing Likelihood of Occurrence Probability: The occurrence of the event is just as likely as it is unlikely. just as likely as it is unlikely.

5 5 Slide © 2003 Thomson/South-Western An Experiment and Its Sample Space n An experiment is any process that generates well- defined outcomes. n The sample space for an experiment is the set of all experimental outcomes. n A sample point is an element of the sample space, any one particular experimental outcome.

6 6 Slide © 2003 Thomson/South-Western 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 Investment Gain or Loss in 3 Months (in $000) in 3 Months (in $000) Markley Oil Collins Mining Markley Oil Collins Mining

7 7 Slide © 2003 Thomson/South-Western A Counting Rule for Multiple-Step Experiments n 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 ). n A helpful graphical representation of a multiple-step experiment is a tree diagram.

8 8 Slide © 2003 Thomson/South-Western Example: Bradley Investments n 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

9 9 Slide © 2003 Thomson/South-Western Example: Bradley Investments n Tree Diagram Markley Oil Collins Mining Experimental Markley Oil Collins Mining Experimental (Stage 1) (Stage 2) Outcomes (Stage 1) (Stage 2) Outcomes Gain 5 Gain 8 Gain 10 Gain 8 Lose 20 Lose 2 Even (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

10 Slide © 2003 Thomson/South-Western 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. n Number of combinations of N objects taken n at a time where N ! = N ( N - 1)( N - 2)... (2)(1) n ! = n ( n - 1)( n - 2)... (2)(1) n ! = n ( n - 1)( n - 2)... (2)(1) 0! = 1 0! = 1 Counting Rule for Combinations

11 Slide © 2003 Thomson/South-Western Counting Rule for Permutations 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. n Number of permutations of N objects taken n at a time

12 Slide © 2003 Thomson/South-Western Assigning Probabilities n Classical Method Assigning probabilities based on the assumption of equally likely outcomes. n Relative Frequency Method Assigning probabilities based on experimentation or historical data. n Subjective Method Assigning probabilities based on the assignor’s judgment.

13 Slide © 2003 Thomson/South-Western Classical Method If an experiment has n possible outcomes, this method would assign a probability of 1/ n to each outcome. n Example Experiment: Rolling a die Experiment: Rolling a die Sample Space: S = {1, 2, 3, 4, 5, 6} Sample Space: S = {1, 2, 3, 4, 5, 6} Probabilities: Each sample point has a 1/6 chance Probabilities: Each sample point has a 1/6 chance of occurring. of occurring.

14 Slide © 2003 Thomson/South-Western Example: Lucas Tool Rental n Relative Frequency Method Lucas would like to assign probabilities to the number of floor polishers it rents per day. Office records show the following frequencies of daily rentals for the last 40 days. Number of Number Number of Number Polishers Rentedof Days Polishers Rentedof Days

15 Slide © 2003 Thomson/South-Western n 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 Number of Number Polishers Rentedof Days Probability = 4/ = 4/ = 6/ = 6/ etc etc Example: Lucas Tool Rental

16 Slide © 2003 Thomson/South-Western Subjective Method n When economic conditions and a company’s circumstances change rapidly it might be inappropriate to assign probabilities based solely on historical data. n 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. n The best probability estimates often are obtained by combining the estimates from the classical or relative frequency approach with the subjective estimates.

17 Slide © 2003 Thomson/South-Western Example: Bradley Investments Applying the subjective method, an analyst Applying the subjective method, an analyst made the following probability assignments. Exper. Outcome Net Gain/Loss Probability Exper. Outcome Net Gain/Loss Probability ( 10, 8) $18,000Gain.20 ( 10, 8) $18,000Gain.20 ( 10, -2) $8,000Gain.08 ( 10, -2) $8,000Gain.08 ( 5, 8) $13,000Gain.16 ( 5, 8) $13,000Gain.16 ( 5, -2) $3,000Gain.26 ( 5, -2) $3,000Gain.26 ( 0, 8) $8,000Gain.10 ( 0, 8) $8,000Gain.10 ( 0, -2) $2,000Loss.12 ( 0, -2) $2,000Loss.12 (-20, 8) $12,000Loss.02 (-20, 8) $12,000Loss.02 (-20, -2) $22,000Loss.06 (-20, -2) $22,000Loss.06

18 Slide © 2003 Thomson/South-Western Events and Their Probability n An event is a collection of sample points. n The probability of any event is equal to the sum of the probabilities of the sample points in the event. n 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.

19 Slide © 2003 Thomson/South-Western Example: Bradley Investments n Events and Their Probabilities Event M = Markley Oil Profitable M = {(10, 8), (10, -2), (5, 8), (5, -2)} M = {(10, 8), (10, -2), (5, 8), (5, -2)} P( M ) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2) P( M ) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2) = = =.70 =.70 Event C = Collins Mining Profitable Event C = Collins Mining Profitable P( C ) =.48 (found using the same logic) P( C ) =.48 (found using the same logic)

20 Slide © 2003 Thomson/South-Western Some Basic Relationships of Probability n 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 Complement of an Event Union of Two Events Union of Two Events Intersection of Two Events Intersection of Two Events Mutually Exclusive Events Mutually Exclusive Events

21 Slide © 2003 Thomson/South-Western Complement of an Event n The complement of event A is defined to be the event consisting of all sample points that are not in A. n The complement of A is denoted by A c. n The Venn diagram below illustrates the concept of a complement. Event A AcAcAcAc Sample Space S

22 Slide © 2003 Thomson/South-Western n 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 is denoted by A  B  n The union of A and B is illustrated below. Sample Space S Event A Event B Union of Two Events

23 Slide © 2003 Thomson/South-Western Example: Bradley Investments n Union of Two Events Event M = Markley Oil Profitable Event C = Collins Mining Profitable Event C = Collins Mining Profitable M  C = Markley Oil Profitable M  C = Markley Oil Profitable or Collins Mining Profitable or Collins Mining Profitable 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)} P( M  C) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2) P( M  C) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2) + P(0, 8) + P(-20, 8) = = =.82 =.82

24 Slide © 2003 Thomson/South-Western 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 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  The intersection is denoted by A  n The intersection of A and B is the area of overlap in the illustration below. Sample Space S Event A Event B Intersection

25 Slide © 2003 Thomson/South-Western n Intersection of Two Events Event M = Markley Oil Profitable Event C = Collins Mining Profitable Event C = Collins Mining Profitable M  C = Markley Oil Profitable M  C = Markley Oil Profitable and Collins Mining Profitable and Collins Mining Profitable M  C = {(10, 8), (5, 8)} M  C = {(10, 8), (5, 8)} P( M  C) = P(10, 8) + P(5, 8) P( M  C) = P(10, 8) + P(5, 8) = = =.36 =.36 Example: Bradley Investments

26 Slide © 2003 Thomson/South-Western Addition Law n The addition law provides a way to compute the probability of event A, or B, or both A and B occurring. n The law is written as: P( A  B ) = P( A ) + P( B ) - P( A  B  P( A  B ) = P( A ) + P( B ) - P( A  B 

27 Slide © 2003 Thomson/South-Western Example: Bradley Investments n Addition Law Markley Oil or Collins Mining Profitable Markley Oil or Collins Mining Profitable We know: P( M ) =.70, P( C ) =.48, P( M  C ) =.36 We know: P( M ) =.70, P( C ) =.48, P( M  C ) =.36 Thus: P( M  C) = P( M ) + P( C ) - P( M  C ) Thus: P( M  C) = P( M ) + P( C ) - P( M  C ) = = =.82 =.82 This result is the same as that obtained earlier using This result is the same as that obtained earlier using the definition of the probability of an event. the definition of the probability of an event.

28 Slide © 2003 Thomson/South-Western Mutually Exclusive Events n 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 Sample Space S Event B Event A

29 Slide © 2003 Thomson/South-Western Mutually Exclusive Events n Addition Law for Mutually Exclusive Events P( A  B ) = P( A ) + P( B ) P( A  B ) = P( A ) + P( B )

30 Slide © 2003 Thomson/South-Western Conditional Probability n The probability of an event given that another event has occurred is called a conditional probability. n The conditional probability of A given B is denoted by P( A | B ). n A conditional probability is computed as follows:

31 Slide © 2003 Thomson/South-Western Example: Bradley Investments n Conditional Probability Collins Mining Profitable given Collins Mining Profitable given Markley Oil Profitable Markley Oil Profitable

32 Slide © 2003 Thomson/South-Western Multiplication Law n The multiplication law provides a way to compute the probability of an intersection of two events. n The law is written as: P( A   B ) = P( B )P( A | B ) P( A   B ) = P( B )P( A | B )

33 Slide © 2003 Thomson/South-Western Example: Bradley Investments n Multiplication Law Markley Oil and Collins Mining Profitable We know: P( M ) =.70, P( C | M ) =.51 We know: P( M ) =.70, P( C | M ) =.51 Thus: P( M  C) = P( M )P( M|C ) Thus: P( M  C) = P( M )P( M|C ) = (.70)(.51) = (.70)(.51) =.36 =.36 This result is the same as that obtained earlier using the definition of the probability of an event. This result is the same as that obtained earlier using the definition of the probability of an event.

34 Slide © 2003 Thomson/South-Western Independent Events n Events A and B are independent if P( A | B ) = P( A ).

35 Slide © 2003 Thomson/South-Western Independent Events n Multiplication Law for Independent Events P( A  B ) = P( A )P( B ) P( A  B ) = P( A )P( B ) n The multiplication law also can be used as a test to see if two events are independent.

36 Slide © 2003 Thomson/South-Western Example: Bradley Investments n Multiplication Law for Independent Events Are M and C independent?  Does  P( M  C ) = P( M)P(C) ? We know: P( M  C ) =.36, P( M ) =.70, P( C ) =.48 We know: P( M  C ) =.36, P( M ) =.70, P( C ) =.48 But: P( M)P(C) = (.70)(.48) =.34 But: P( M)P(C) = (.70)(.48) =  so  M and C are not independent..34  so  M and C are not independent.

37 Slide © 2003 Thomson/South-Western End of Chapter 4