Chapter 4 Introduction to Probability n n n

Chapter 4 Introduction to Probability n n n Experiments, Counting Rules, and Assigning Probabilities Events and Their Probabilities Some Basic Relationships of Probability Conditional Probability Bayes’ Theorem Slide 1

Probability n n n 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 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 2

An Experiment and Its Sample Space n An experiment is any process that generates welldefined 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. Slide 3

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 $000) Markley Oil Collins Mining 10 8 5 -2 0 -20 Slide 4

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). . . (nk ). Slide 5

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 Slide 6

Example: Bradley Investments n Tree Diagram Markley Oil Collins Mining (Stage 1) (Stage 2) Gain 8 Experimental Outcomes (10, 8) Gain (10, -2) Gain Lose 2 Gain 10 (5, 8) Gain 8 (5, -2) Gain Lose 2 Gain 5 Gain 8 (0, 8) Gain Even (0, -2) Lose 20 Gain 8 (-20, 8) Lose (-20, -2) Lose 2 $18, 000 $13, 000 $8, 000 $2, 000 $12, 000 $22, 000 Slide 7

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. n The number of combinations of N objects taken n at a time is where N ! = N (N - 1)(N - 2). . . (2)(1) n! = n(n - 1)(n - 2). . . (2)(1) 0! = 1 Slide 8

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 The number of permutations of N objects taken n at a time is Slide 9

Assigning Probabilities to Experimental Outcomes n n n 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. Slide 10

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 Sample Space: S = {1, 2, 3, 4, 5, 6} Probabilities: Each sample point has a 1/6 chance of occurring. Slide 11

Example: Lucas Tool Rental 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 Polishers Rented of Days 0 4 1 6 2 18 3 10 4 2 n Slide 12

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 Polishers Rented of Days Probability 0 4. 10 = 4/40 1 6. 15 = 6/40 2 18. 45 etc. 3 10. 25 4 2. 05 40 1. 00 n Slide 13

Subjective Method n n n 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. Slide 14

Example: Bradley Investments Applying the subjective method an analyst made the following probability assignments. Exper. Outcome Net Gain/Loss Probability (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 15

Events and Their Probabilities n n 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. Slide 16

Example: Bradley Investments n Events and Their Probabilities Event M = Markley Oil Profitable M = {(10, 8), (10, -2), (5, 8), (5, -2)} P(M) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2) =. 2 +. 08 +. 16 +. 26 =. 70 Event C = Collins Mining Profitable P(C ) =. 48 (found using the same logic) Slide 17

Conditional Probability n n n 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 P(A |B). A conditional probability is computed as follows: Slide 18

Example: Bradley Investments n Conditional Probability Collins Mining Profitable Given Markley Oil Profitable Slide 19

Multiplication Law n n The multiplication law provides a way to compute the probability of an intersection of two events. The law is written as: P(A B) = P(B)P(A |B) Slide 20

Example: Bradley Investments n Multiplication Law Markley Oil and Collins Mining Profitable We know: P(M) =. 70, P(C |M) =. 51 Thus: P(M C) = P(M)P(M|C ) = (. 70)(. 51) =. 36 This result is the same as that obtained earlier using the definition of the probability of an event. Slide 21

Multiplication Law for Independent Events n n n Events A and B are independent if P(A |B) = P(A ). Multiplication Law for Independent Events: P(A B) = P(A )P(B) The multiplication law also can be used as a test to see if two events are independent. Slide 22

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 But: P(M)P(C) = (. 70)(. 48) =. 34 so M and C are not independent. Slide 23