Chapter 4Chapter 4 Introduction to Probability Introduction to Probability

Experiments, Counting Rules, Experiments, Counting Rules, and Assigning Probabilitiesand Assigning Probabilities

Events and Their ProbabilityEvents and Their Probability Some Basic RelationshipsSome Basic Relationships of Probabilityof Probability Conditional ProbabilityConditional Probability BayesBayes’’ Theorem Theorem

Probability as a Numerical Probability as a Numerical MeasureMeasure

of the Likelihood of of the Likelihood of OccurrenceOccurrence

00 11..55

Increasing Likelihood of OccurrenceIncreasing Likelihood of Occurrence

Probability:Probability:

The eventThe eventis veryis very

unlikelyunlikelyto occur.to occur.

The occurrenceThe occurrenceof the event isof the event is

just as likely asjust as likely asit is unlikely.it is unlikely.

The eventThe eventis almostis almostcertaincertain

to occur.to occur.

4.1 An Experiment and Its 4.1 An Experiment and Its Sample SpaceSample Space

An An experimentexperiment is any process that generates is any process that generates well-defined outcomes.well-defined outcomes. An An experimentexperiment is any process that generates is any process that generates well-defined outcomes.well-defined outcomes.

The The sample spacesample space for an experiment is the set of for an experiment is the set of all experimental outcomes.all experimental outcomes. The The sample spacesample space for an experiment is the set of for an experiment is the set of all experimental outcomes.all experimental outcomes.

An experimental outcome is also called a An experimental outcome is also called a samplesample pointpoint.. An experimental outcome is also called a An experimental outcome is also called a samplesample pointpoint..

Example: Bradley InvestmentsExample: Bradley InvestmentsBradley has invested in two stocks, Markley Bradley has invested in two stocks, Markley Oil and Collins Mining. Bradley has Oil and Collins Mining. Bradley has determined that the possible outcomes of determined that the possible outcomes of these investments three months from now are these investments three months from now are as follows.as follows.

Investment Gain or LossInvestment Gain or Loss in 3 Months (in $000)in 3 Months (in $000)

Markley OilMarkley Oil Collins MiningCollins Mining 1010 55 00-20-20

88-2-2

A Counting Rule for A Counting Rule for Multiple-Step ExperimentsMultiple-Step Experiments

If an experiment consists of a sequence of If an experiment consists of a sequence of kk steps steps in which there are in which there are nn11 possible results for the first step, possible results for the first step,

nn22 possible results for the second step, and so on, possible results for the second step, and so on,

then the total number of experimental outcomes isthen the total number of experimental outcomes is given by (given by (nn11 )( )( nn22). . . (). . . (nnkk).).

A helpful graphical representation of a multiple-stepA helpful graphical representation of a multiple-step experiment is a experiment is a tree diagramtree diagram..

Bradley Investments can be viewed as aBradley Investments can be viewed as atwo-step experiment. It involves two stocks, two-step experiment. It involves two stocks, each with a set of experimental outcomes.each with a set of experimental outcomes.Markley Oil:Markley Oil: nn11 = 4 = 4

Collins Mining:Collins Mining: nn22 = 2 = 2Total Number of Total Number of Experimental Outcomes:Experimental Outcomes: nn11nn22 = (4)(2) = 8 = (4)(2) = 8

Example: A Counting Rule for Example: A Counting Rule for

Multiple-Step ExperimentsMultiple-Step Experiments

Tree DiagramTree Diagram

Gain 5Gain 5

Gain 8Gain 8

Gain 8Gain 8

Gain 10Gain 10

Gain 8Gain 8

Gain 8Gain 8

Lose 20Lose 20

Lose 2Lose 2

Lose 2Lose 2

Lose 2Lose 2

Lose 2Lose 2

EvenEven

Markley OilMarkley Oil(Stage 1)(Stage 1)

Collins MiningCollins Mining(Stage 2)(Stage 2)

ExperimentalExperimentalOutcomesOutcomes

(10, 8) (10, 8) Gain $18,000 Gain $18,000

(10, -2) (10, -2) Gain $8,000 Gain $8,000

(5, 8) (5, 8) Gain $13,000 Gain $13,000

(5, -2) (5, -2) Gain $3,000 Gain $3,000

(0, 8) (0, 8) Gain $8,000 Gain $8,000

(0, -2) (0, -2) Lose Lose $2,000$2,000

(-20, 8) (-20, 8) Lose Lose $12,000$12,000

(-20, -2)(-20, -2) Lose Lose $22,000$22,000

A second useful counting rule enables us to A second useful counting rule enables us to count the number of experimental outcomes count the number of experimental outcomes when when nn objects are to be selected from a set of objects are to be selected from a set of NN objects. objects.

Counting Rule for CombinationsCounting Rule for Combinations

CN

nN

n N nnN

!

!( )!C

N

nN

n N nnN

!

!( )!

Number of Number of CombinationsCombinations of of NN Objects Taken Objects Taken nn at a Time at a Time

where: where: NN! = ! = NN((NN - 1)( - 1)(NN - 2) . . . (2)(1) - 2) . . . (2)(1) nn! = ! = nn((nn - 1)( - 1)(nn - 2) . . . (2)(1) - 2) . . . (2)(1) 0! = 10! = 1

Number of Number of PermutationsPermutations of of NN Objects Taken Objects Taken nn at a Time at a Time

where: where: NN! = ! = NN((NN - 1)( - 1)(NN - 2) . . . (2)(1) - 2) . . . (2)(1) nn! = ! = nn((nn - 1)( - 1)(nn - 2) . . . (2)(1) - 2) . . . (2)(1) 0! = 10! = 1

P nN

nN

N nnN

!!

( )!P n

N

nN

N nnN

!!

( )!

Counting Rule for PermutationsCounting Rule for Permutations

A third useful counting rule enables us to count theA third useful counting rule enables us to count thenumber of experimental outcomes when number of experimental outcomes when nn objects are to objects are tobe selected from a set of be selected from a set of NN objects, where the order of objects, where the order ofselection is important.selection is important.

Assigning ProbabilitiesAssigning Probabilities

Classical MethodClassical Method

Relative Frequency MethodRelative Frequency Method

Subjective MethodSubjective Method

Assigning probabilities based on the assumptionAssigning probabilities based on the assumption of of equally likely outcomesequally likely outcomes

Assigning probabilities based on Assigning probabilities based on experimentationexperimentation or historical dataor historical data

Assigning probabilities based on Assigning probabilities based on judgmentjudgment