Chapter 4Chapter 4

ProbabilityProbability

©

Sample SpaceSample Space

The possible outcomes of a random The possible outcomes of a random experiment are called the experiment are called the basic outcomesbasic outcomes, , and the set of all basic outcomes is called and the set of all basic outcomes is called the the sample space.sample space. The symbol The symbol SS will be will be used to denote the sample space.used to denote the sample space.

Sample SpaceSample Space- An Example -- An Example -

What is the sample space for a roll of What is the sample space for a roll of a single six-sided die?a single six-sided die?

S = [1, 2, 3, 4, 5, 6]

Mutually ExclusiveMutually Exclusive

If the events A and B have no common basic If the events A and B have no common basic outcomes, they are outcomes, they are mutually exclusivemutually exclusive and and their intersection A their intersection A B is said to be the empty B is said to be the empty set indicating that A set indicating that A B cannot occur. B cannot occur.

More generally, the K events EMore generally, the K events E11, E, E22, . . . , E, . . . , EKK are said to be mutually exclusive if every pair are said to be mutually exclusive if every pair of them is a pair of mutually exclusive events.of them is a pair of mutually exclusive events.

Venn DiagramsVenn Diagrams

Venn DiagramsVenn Diagrams are drawings, usually are drawings, usually using geometric shapes, used to depict using geometric shapes, used to depict basic concepts in set theory and the basic concepts in set theory and the outcomes of random experiments.outcomes of random experiments.

Intersection of Events A and BIntersection of Events A and B

A B A BAB

(a) AB is the striped area

S S

(b) A and B are Mutually Exclusive

Collectively ExhaustiveCollectively Exhaustive

Given the K events EGiven the K events E11, E, E22, . . ., E, . . ., EKK in the in the

sample space S. If Esample space S. If E11 E E2 2 . . . . . . EEKK = S, = S,

these events are said to be these events are said to be collectively collectively exhaustiveexhaustive..

ComplementComplement

Let A be an event in the sample space S. Let A be an event in the sample space S. The set of basic outcomes of a random The set of basic outcomes of a random experiment belonging to S but not to A is experiment belonging to S but not to A is called the called the complement complement of A and is denoted of A and is denoted by A.by A.

Venn Diagram for the Complement Venn Diagram for the Complement of Event Aof Event A

AA

S

Unions, Intersections, and Unions, Intersections, and ComplementsComplements

A die is rolled. Let A be the event “Number rolled is even” and B be the event “Number rolled is at least 4.” Then

A = [2, 4, 6] and B = [4, 5, 6]

3] 2, [1, B and 5] 3, [1, A 6] [4, BA

6] 5, 4, [2, BA S 6] 5, 4, 3, 2, [1, AA

Classical ProbabilityClassical Probability

The The classical definition of probabilityclassical definition of probability is the is the proportion of times that an event will occur, proportion of times that an event will occur, assuming that all outcomes in a sample space assuming that all outcomes in a sample space are equally likely to occur. The probability of are equally likely to occur. The probability of an event is determined by counting the number an event is determined by counting the number of outcomes in the sample space that satisfy of outcomes in the sample space that satisfy the event and dividing by the number of the event and dividing by the number of outcomes in the sample space.outcomes in the sample space.

Classical ProbabilityClassical Probability

The probability of an event A isThe probability of an event A is

where Nwhere NAA is the number of outcomes that satisfy the is the number of outcomes that satisfy the

condition of event A and N is the total number of condition of event A and N is the total number of outcomes in the sample space. The important idea outcomes in the sample space. The important idea here is that one can develop a probability from here is that one can develop a probability from fundamental reasoning about the process.fundamental reasoning about the process.

N

N P(A) A

CombinationsCombinations

The counting process can be generalized The counting process can be generalized by using the following equation to by using the following equation to compare the number of combinations of n compare the number of combinations of n things taken k at a time.things taken k at a time.

1!0)!(!

!

knk

n C n

k

Relative FrequencyRelative Frequency

The The relative frequency definition of probabilityrelative frequency definition of probability is the limit of the proportion of times that an is the limit of the proportion of times that an event A occurs in a large number of trials, event A occurs in a large number of trials, nn, ,

where nwhere nAA is the number of A outcomes and n is the number of A outcomes and n

is the total number of trials or outcomes in is the total number of trials or outcomes in the population. The probability is the limit the population. The probability is the limit as n becomes large.as n becomes large.

n

n P(A) A

Subjective ProbabilitySubjective Probability

The The subjective definition of probabilitysubjective definition of probability expresses an individual’s degree of belief expresses an individual’s degree of belief about the chance that an event will occur. about the chance that an event will occur. These subjective probabilities are used in These subjective probabilities are used in certain management decision procedures.certain management decision procedures.

Probability PostulatesProbability PostulatesLet S denote the sample space of a random experiment, Oi, the basic outcomes, and A, an event. For each event A of the sample space S, we assume that a number P(A) is defined and we have the postulates

1. If A is any event in the sample space S

2. Let A be an event in S, and let Oi denote the basic outcomes. Then

where the notation implies that the summation extends over all the basic outcomes in A.

3. P(S) = 1

1)(0 AP

)()( A

iOPAP

Probability RulesProbability Rules

Let A be an event and A its complement. Let A be an event and A its complement. The the The the complement rule iscomplement rule is::

)(1)( APAP

Probability RulesProbability Rules

The Addition Rule of ProbabilitiesThe Addition Rule of Probabilities::

Let A and B be two events. The probability Let A and B be two events. The probability of their union isof their union is

)()()()( BAPBPAPBAP

Probability RulesProbability Rules

Conditional ProbabilityConditional Probability::

Let A and B be two events. The Let A and B be two events. The conditional probabilityconditional probability of event A, given that event B has occurred, is of event A, given that event B has occurred, is denoted by the symbol P(A|B) and is found to be:denoted by the symbol P(A|B) and is found to be:

provided that P(B > 0).provided that P(B > 0).

)(

)()|(

BP

BAPBAP

Probability RulesProbability Rules

Conditional ProbabilityConditional Probability::

Let A and B be two events. The Let A and B be two events. The conditional probabilityconditional probability of event B, given that event A has occurred, is of event B, given that event A has occurred, is denoted by the symbol P(B|A) and is found to be:denoted by the symbol P(B|A) and is found to be:

provided that P(A > 0).provided that P(A > 0).

)(

)()|(

AP

BAPABP

Probability RulesProbability Rules

The Multiplication Rule of ProbabilitiesThe Multiplication Rule of Probabilities::

Let A and B be two events. The probability Let A and B be two events. The probability of their intersection can be derived from of their intersection can be derived from the conditional probability asthe conditional probability as

Also,Also,)()|()( BPBAPBAP

)()|()( APABPBAP

Statistical IndependenceStatistical Independence

Let A and B be two events. These events are said to Let A and B be two events. These events are said to be statistically be statistically independentindependent if and only if if and only if

From the From the multiplication rulemultiplication rule it also follows that it also follows that

More generally, the events EMore generally, the events E11, E, E22, . . ., E, . . ., Ekk are mutually are mutually statistically independent if and only ifstatistically independent if and only if

)()()( BPAPBAP

0)P(B) if(P(A)B)|P(A 0)P(A) if(P(B)A)|P(B

)P(E)P(E )P(E)EEP(E K21K21

Bivariate ProbabilitiesBivariate ProbabilitiesB1 B2 . . . Bk

A1P(A1B1) P(A1B2) . . . P(A1Bk)

A2P(A2B1) P(A2B2) . . . P(A2Bk)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

AhP(AhB1) P(AhB2) . . . P(AhBk)

Figure 4.1 Outcomes for Bivariate Events

Joint and Marginal Joint and Marginal ProbabilitiesProbabilities

In the context of bivariate probabilities, the In the context of bivariate probabilities, the intersection probabilities P(Aintersection probabilities P(Ai i B Bjj) are called) are called joint joint

probabilities.probabilities. The probabilities for individual The probabilities for individual events P(Aevents P(Aii) and P(B) and P(Bjj) are called ) are called marginal marginal

probabilitiesprobabilities.. Marginal probabilities are at the Marginal probabilities are at the margin of a bivariate table and can be computed margin of a bivariate table and can be computed by summing the corresponding row or column.by summing the corresponding row or column.

Probabilities for the Television Probabilities for the Television Viewing and Income ExampleViewing and Income Example

Viewing Frequenc

y

High Income

Middle

Income

Low Income

Totals

Regular 0.04 0.13 0.04 0.21

Occasional 0.10 0.11 0.06 0.27

Never 0.13 0.17 0.22 0.52

Totals 0.27 0.41 0.32 1.00

Tree DiagramsTree Diagrams

P(A3 )

= .52

P(A1 B1) = .04

P(A2) = .27

P(A 1

) = .2

1 P(A1 B2) = .13

P(A1 B3) = .04

P(A2 B1) = .10

P(A2 B2) = .11

P(A2 B3) = .06

P(A3 B1) = .13

P(A3 B2) = .17

P(A3 B3) = .22

P(S) = 1

Probability RulesProbability Rules

Rule for Determining the Independence of AttributesRule for Determining the Independence of Attributes

Let A and B be a pair of attributes, each broken into Let A and B be a pair of attributes, each broken into mutually exclusive and collectively exhaustive mutually exclusive and collectively exhaustive event categories denoted by labels Aevent categories denoted by labels A11, A, A22, . . ., A, . . ., Ahh

and Band B11, B, B22, . . ., B, . . ., Bkk. If every A. If every Aii is is statistically statistically

independentindependent of every event B of every event Bjj, then the attributes A , then the attributes A

and B are independent.and B are independent.

Odds RatioOdds Ratio

The The odds in favorodds in favor of a particular event are of a particular event are given by the ratio of the probability of the given by the ratio of the probability of the event divided by the probability of its event divided by the probability of its complement. The odds in favor of A arecomplement. The odds in favor of A are

)AP(

P(A)

P(A)-1

P(A) odds

Overinvolvement RatioOverinvolvement RatioThe probability of event AThe probability of event A11 conditional on event conditional on event

BB11divided by the probability of Adivided by the probability of A11 conditional on conditional on

activity Bactivity B22 is defined as the is defined as the overinvolvement ratiooverinvolvement ratio::

An overinvolvement ratio greater than 1,An overinvolvement ratio greater than 1,

Implies that event AImplies that event A11 increases the conditional odds increases the conditional odds

ration in favor of Bration in favor of B11::

)B|P(A

)B|P(A

21

11

0.1)B|P(A

)B|P(A

21

11

)P(B

)P(B

)A|P(B

)A|P(B

2

1

12

11

Bayes’ TheoremBayes’ Theorem

Let A and B be two events. Then Let A and B be two events. Then Bayes’ TheoremBayes’ Theorem states that:states that:

andand

P(A)

B)P(B)|P(A)|( BAP

P(B)

A)P(A)|P(B)|( BAP

Bayes’ TheoremBayes’ Theorem(Alternative Statement)(Alternative Statement)

Let ELet E11, E, E22, . . . , E, . . . , Ekk be mutually exclusive and be mutually exclusive and

collectively exhaustive events and let A be some collectively exhaustive events and let A be some other event. The conditional probability of Eother event. The conditional probability of Eii

given A can be expressed as given A can be expressed as Bayes’ TheoremBayes’ Theorem::

))P(EE|P(A))P(EE|P(A))P(EE|P(A

))P(EE|P(AA)|P(E

KK2211

iii

))P(EE|P(A))P(EE|P(A))P(EE|P(A

))P(EE|P(AA)|P(E

KK2211

iii

Bayes’ TheoremBayes’ Theorem- Solution Steps -- Solution Steps -

1. Define the subset events from the problem.

2. Define the probabilities for the events defined in step 1.

3. Compute the complements of the probabilities.

4. Apply Bayes’ theorem to compute the probability for the problem solution.