Chapter 4 Using Probability and Discrete Probability Distributions ©

Chapter 4Chapter 4

Using Probability and Using Probability and Discrete Probability Discrete Probability

DistributionsDistributions

©

Chapter 4 - Chapter 4 - Chapter Chapter OutcomesOutcomes

After studying the material in this chapter, you should be able to:

• Understand the three approaches to assessing probabilities.

• Apply the common rules of probability.

• Identify the types of processes that are represented by discrete probability distributions.

Chapter 4 - Chapter 4 - Chapter Chapter OutcomesOutcomes

(continued)(continued)

After studying the material in this chapter, you should be able to:

• Know how to determine probabilities associated with binomial and Poisson distribution applications.

ProbabilityProbability

ProbabilityProbability refers to the chance refers to the chance that a particular event will that a particular event will

occur.occur.•The probability of an event will be a value in the range 0.00 to 1.00. A value of 0.00 means the event will not occur. A probability of 1.00 means the event will occur. Anything between 0.00 and 1.00 reflects the uncertainty of the event occurring.

Events and Sample SpaceEvents and Sample Space

An experimentexperiment is a process that produces a single outcome whose result cannot be predicted with certainty.


Elementary eventsElementary events are the most rudimentary outcomes resulting from a simple experiment.


The sample spacesample space is the collection of all elementary outcomes that can result from a selection or decision.


An eventevent is a collection of elementary events.

Elementary Event Audit 1 Audit 2e1 early earlye2 early on timee3 early latee4 on time earlye5 on time on timee6 on time latee7 late earlye8 late on timee9 late late

Events and Sample SpaceEvents and Sample Space(Able Accounting Example)(Able Accounting Example)

Sample Space = SS = (e1, e2, e3, e4, e5, e6, e7, e8, e9)

Mutually Exclusive EventsMutually Exclusive Events

Two events are mutually mutually exclusiveexclusive if the occurrence of one event precludes the occurrence of a second event.

Mutually Exclusive EventsMutually Exclusive Events(Able Accounting Example)(Able Accounting Example)

The event in which at least one of the two audits is late:

E1 = {e3, e6, e7, e8, e9}

The event that neither audit is late:

E2 = {e1, e2, e4, e5}

EE11 and E and E22 are mutually are mutually exclusive!exclusive!

Independent and Independent and Dependent EventsDependent Events

Two events are independent independent if the occurrence of one event in no way influences the probability of the occurrence of the other event.

Independent and Independent and Dependent EventsDependent Events

Two events are dependent dependent if the occurrence of one event impacts the probability of the other event occurring.

Classical Probability Classical Probability AssessmentAssessment

Classical Probability AssessmentClassical Probability Assessment refers to the method of determining probability based on the ratio of the number of ways the event of interest can occur to the total number of ways any event can occur when the individual elementary events are equally likely.

CLASSICAL PROBABILITY MEASUREMENTCLASSICAL PROBABILITY MEASUREMENT

Classical Probability Classical Probability AssessmentAssessment

events elementaryof number Totaloccur can E of ways Number

P(E ii )

Relative Frequency of Relative Frequency of OccurrenceOccurrence

Relative Frequency of OccurrenceRelative Frequency of Occurrence refers to a method that defines probability as the number of times an event occurs, divided by the total number of times an experiment is performed in a large number of trials.


RELATIVE FREQUENCY OF OCCURRENCERELATIVE FREQUENCY OF OCCURRENCE

where:

Ei = the event of interest

RF(Ei) = the relative frequency of Ei occurring

n = number of trials

noccurs E timesof Number

RF(E ii )


(Example 4-6)(Example 4-6)

Commercial Residential TotalHeating Systems 55 145 200Air-ConditioningSystems 45 255 300Total 100 400 500

80.0500

400)(Re)(Re sidentialRFsidentialP

40.0500

200)()( HeatingRFHeatingP

Subjective Probability Subjective Probability AssessmentAssessment

Subjective Probability Subjective Probability AssessmentAssessment refers to the method that defines probability of an event as reflecting a decision maker’s state of mind regarding the chances that the particular event will occur.

The Rules of ProbabilityThe Rules of Probability

PROBABILITY RULE 1PROBABILITY RULE 1

For any event Ei

0.0 P(Ei) 1.0 for all i



where:k = Number of elementary

events in the sample space

ei = ith elementary event

0.1)(1

k

iieP


PROBABILITY RULE 3PROBABILITY RULE 3The probability of an event Ei is equal to the sum of the probabilities of the elementary events forming Ei. That is, if:

Ei = {e1, e2, e3}

then:

P(Ei) = P(e1) + P(e2) + P(e3)

ComplementsComplements

The complement complement of an event E is the collection of all possible elementary events not contained in event E. The complement of event E is represented by E.


COMPLEMENT RULECOMPLEMENT RULE

)(1)( EPEP



Addition rule for any two events E1 and E2:

P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2)



Addition rule for mutually exclusive events E1 and E2:

P(E1 or E2) = P(E1) + P(E2)

Conditional ProbabilityConditional Probability

Conditional probabilityConditional probability refers to the probability that an event will occur given that some other event has already happened.



Conditional probability for any two events E1 , E2:

0)(

)(

)()|(

2

2

2121

EP

EP

EandEPEEP

Tree DiagramsTree Diagrams

Another way of organizing events of an experiment that aids in the calculation of probabilities is the tree tree diagramdiagram.

Tree DiagramsTree Diagrams(Figure 4-1)(Figure 4-1)

Female P(E4) =

0.34

Male P(E5) =

0.66

Tree DiagramsTree Diagrams (Figure 4-1)(Figure 4-1)

Female P(E4) =

0.34

Male P(E5) =

0.66

P(E1) = 0.38 P(E2) =

0.44

P(E3) = 0.18

P(E1) = 0.38 P(E2) =

0.44

P(E3) = 0.18

Tree DiagramsTree Diagrams (Figure 4-1)(Figure 4-1)

Female P(E4) =

0.34

Male P(E5) =

0.66

P(E1) = 0.38 P(E2) =

0.44

P(E3) = 0.18

P(E1) = 0.38 P(E2) =

0.44

P(E3) = 0.18

P(E1 and E5) = 0.38 x 0.66 = 0.20

P(E2 and E5) = 0.44 x 0.66 = 0.32

P(E3 and E5) = 0.18 x 0.66 = 0.14

P(E1 and E4) = 0.38 x 0.34 = 0.18 P(E2 and E4) = 0.44 x 0.34 = 0.12

P(E3 and E4) = 0.18 x 0.34 = 0.04



Conditional probability for independent events E1 , E2:

0)();()|(

0)();()|(

1212

2121

EPEPEEP

and

EPEPEEP



Multiplication rule two events E1 and E2:

)|()()(

)|()()(

21212

12121

EEPEPEandEP

and

EEPEPEandEP



Multiplication rule independent events E1 , E2:

)()()( 2121 EPEPEandEP

Bayes’ TheoremBayes’ Theorem

BAYES’ THEOREMBAYES’ THEOREM

where:Ei = ith event of interest of the k

possible eventsB = new event that might impact

P(Ei)

)|()()|()()|()(

)|()()|(

2211 kk

iii EBPEPEBPEPEBPEP

EBPEPBEP

Discrete Probability Discrete Probability DistributionsDistributions

A random variablerandom variable is a variable that assigns a numerical value to each outcome of a random experiment or trial.


A discrete random discrete random variablevariable is a variable that can only assume a countable number of values.


A continuous random variablecontinuous random variable is a variable that can assume any value on a continuum. Alternatively, they are random variables that can assume an uncountable number of values.

Discrete DistributionsDiscrete Distributions(Example 4-19)(Example 4-19)

Service Calls = x Frequency P(x)0 3 0.0751 4 0.1002 10 0.2503 8 0.2004 7 0.1755 6 0.1506 2 0.050

1 0 00.


Discrete Probability Distribution

x = Number of service calls

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 1 2 3 4 5 6


The uniform probability uniform probability distributiondistribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable


Uniform Probability Distribution

Delivery Lead Time

0

0.05

0.1

0.15

0.2

0.25

0.3

1 week 2 weeks 3 weeks 4 weeks

Mean and Standard Mean and Standard Deviation of Discrete Deviation of Discrete


EXPECTED VALUE FOR A DISCRETE EXPECTED VALUE FOR A DISCRETE DISTRIBUTIONDISTRIBUTION

where:E(x) = Expected value of the random

variable x = Values of the random variableP(x) = Probability of the random variable

taking on the value of x

)()( xxPxE

Mean and Standard Mean and Standard Deviation of Discrete Deviation of Discrete


STANDARD DEVIATION FOR A DISCRETE STANDARD DEVIATION FOR A DISCRETE DISTRIBUTIONDISTRIBUTION

where:E(x) = Expected value of the random

variable x = Values of the random variableP(x) = Probability of the random variable

having the value of x

)()}({ xPxEx 2x

Binomial Probability Binomial Probability DistributionDistribution

• A manufacturing plant labels items as either defective or acceptable.

• A firm bidding for a contract will either get the contract or not.

• A marketing research firm receives survey responses of “Yes, I will buy,” or “No, I will not.”

• New job applicants either accept the offer or reject it.


Characteristics of the Binomial Characteristics of the Binomial Probability Distribution:Probability Distribution:

• A trial has only two possible outcomes – a success or a failure.

• There is a fixed number, n, of identical trials.• The trials of the experiment are independent

of each other and randomly generated.• The probability of a success, p, remains

constant from trial to trial.• If p represents the probability of a success,

then (1-p) = q is the probability of a failure.

CombinationsCombinations

A combinationcombination is an outcome of an experiment where x objects are selected from a group of n objects.

CombinationsCombinations

COUNTING RULE FOR COMBINATIONSCOUNTING RULE FOR COMBINATIONS

where:n! =n(n - 1)(n - 2) . . . (2)(1)

x! = x(x - 1)(x - 2) . . . (2)(1) 0! = 1

)!(!

!

xnx

nC nx


BINOMIAL FORMULABINOMIAL FORMULA

where:

n = sample size

x = number of successes n - x = number of failures

p = probability of a successq = 1 - p = probability of a failuren! =n(n - 1)(n - 2) . . . (2)(1)

x! = x(x - 1)(x - 2) . . . (2)(1) 0! = 1

xnxqpxnx

nxP

)!(!

!)(

Binomial Probability Binomial Probability Distribution TableDistribution Table


MEAN OF THE BINOMIAL MEAN OF THE BINOMIAL DISTRIBUTIONDISTRIBUTION

where:n = Sample sizep = Probability of a

success

npxEx )(


STANDARD DEVIATION FOR THE STANDARD DEVIATION FOR THE BINOMIAL DISTRIBUTIONBINOMIAL DISTRIBUTION

where:n = Sample sizep = Probability of a successq = (1 - p) = Probability of a

failure

npq

Poisson Probability Poisson Probability DistributionDistribution

Characteristics of the Poisson Characteristics of the Poisson Probability Distribution:Probability Distribution:

• The outcomes of interest are rare relative to the possible outcomes.

• The average number of outcomes of interest per segment is ..

• The number of outcomes of interest are random, and the occurrence of one outcome does not influence the chances of another outcome of interest.

• The probability of that an outcome of interest occurs in a given segment is the same for all segments.

Poisson Probability Poisson Probability DistributionDistribution

POISSON PROBABILITY DISTRIBUTIONPOISSON PROBABILITY DISTRIBUTION

where:

x = number of successes in segment t

t = expected number of successes in segment te =base of the natural number system (2.71828)

!

)()(

x

etxP

tx

Poisson Probability Poisson Probability Distribution TableDistribution Table

Mean and Standard Mean and Standard Deviation for the Poisson Deviation for the Poisson Probability DistributionProbability Distribution

MEAN OF THE POISSON MEAN OF THE POISSON DISTRIBUTIONDISTRIBUTION

STANDARD DEVIATION FOR THE STANDARD DEVIATION FOR THE POISSON DISTRIBUTIONPOISSON DISTRIBUTION

t

t

Key TermsKey Terms

• Binomial Probability Distribution

• Classical Probability• Conditional

Probability• Continuous Random

Variable• Dependent Events• Discrete Random

Variable

• Elementary Events• Event• Independent Events• Mutually Exclusive

Events• Poisson Probability

Distribution• Probability• Random Variable

Key TermsKey Terms(continued)(continued)

• Relative Frequency of Occurrence

• Sample Space• Subjective Probability

Assessment• Uniform Probability

Distribution

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Chapter 4 Using Probability and Discrete Probability Distributions ©