Chapter 4Chapter 4

ProbabilityProbability

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Sample SpaceSample Space

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

Sample SpaceSample Space- An Example -- An Example -

What is the sample space for a roll of 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 outcomes, they are mutually mutually exclusive exclusive and their intersection A B is said to be the empty set indicating that A B cannot occur.

More generally, the K events E1, E2, . . . , EK are said to be mutually exclusive if every pair of them is a pair of mutually exclusive events.

Venn DiagramsVenn Diagrams

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

Intersection of Events A and Intersection of Events A and BB

(Figure 4.1)(Figure 4.1)

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 E1, E2, . . ., EK in the sample space S. If E1 E2

. . . EK = S, these events are said to be collectively collectively exhaustiveexhaustive.

ComplementComplement

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

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

(Figure 4.3)(Figure 4.3)

AA

S

Unions, Intersections, and Unions, Intersections, and ComplementsComplements

(Example 4.3)(Example 4.3)

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

Classical ProbabilityClassical Probability

The probability of an event A is

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

N

N P(A) A

CombinationsCombinations

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

1!0)!(!

!

knk

n C n

k

Relative FrequencyRelative Frequency

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

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

n

n P(A) A

Subjective ProbabilitySubjective Probability

The subjective definition of subjective definition of probabilityprobability expresses an individual’s degree of belief about the chance that an event will occur. These subjective probabilities are used in 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. The the complement complement rule isrule is:

)(1)( APAP

Probability RulesProbability Rules

The Addition Rule of ProbabilitiesThe Addition Rule of Probabilities:Let A and B be two events. The probability of their union is

)()()()( BAPBPAPBAP

Probability RulesProbability RulesVenn Diagram for Addition Venn Diagram for Addition

RuleRule(Figure 4.8)(Figure 4.8)

)()()()( BAPBPAPBAP P(AB)

A B

P(A)

A B

P(B)

A B

P(AB)

A B+ -

=

Probability RulesProbability Rules

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

provided that P(B > 0).

)(

)()|(

BP

BAPBAP

Probability RulesProbability Rules

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

provided that P(A > 0).

)(

)()|(

AP

BAPABP

Probability RulesProbability Rules

The Multiplication Rule of The Multiplication Rule of ProbabilitiesProbabilities:Let A and B be two events. The probability of their intersection can be derived from the conditional probability as

Also,

)()|()( BPBAPBAP

)()|()( APABPBAP

Statistical IndependenceStatistical Independence

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

From the multiplication rule it also follows that

More generally, the events E1, E2, . . ., Ek are mutually statistically 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 Probabilities

B1 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 intersection probabilities P(Ai Bj) are called joint probabilities.joint probabilities. The probabilities for individual events P(Ai) and P(Bj) are called marginal probabilitiesmarginal probabilities. Marginal probabilities are at the margin of a bivariate table and can be computed by summing the corresponding row or column.

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

(Table 4.2)(Table 4.2)

Viewing Frequen

cy

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 (Figure 4-10)(Figure 4-10)

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 Rule for Determining the Independence of AttributesAttributesLet A and B be a pair of attributes, each broken into mutually exclusive and collectively exhaustive event categories denoted by labels A1, A2, . . ., Ah and

B1, B2, . . ., Bk. If every Ai is statistically statistically independentindependent of every event Bj, then the attributes A and B are independent.

Odds RatioOdds Ratio

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

)AP(

P(A)

P(A)-1

P(A) odds

Overinvolvement RatioOverinvolvement Ratio

The probability of event A1 conditional on event B1divided by the probability of A1 conditional on activity B2 is defined as the overinvolvement ratiooverinvolvement ratio:

An overinvolvement ratio greater than 1,

Implies that event A1 increases the conditional odds ration in favor of B1:

)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 Bayes’ Bayes’ TheoremTheorem states that:

and

P(A)

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

P(B)

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

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

Let E1, E2, . . . , Ek be mutually exclusive and collectively exhaustive events and let A be some other event. The conditional probability of Ei given A can be expressed as Bayes’ Bayes’ TheoremTheorem:

))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.

Key WordsKey Words

Addition Rule of Probabilities

Bayes’ Theorem Bayes’ Theorem

(Alternative Statement) Classical Probability Collectively Exhaustive Complement Complement Rule Conditional Probability

Event Independence for

Attributes Intersection Joint Probabilities Marginal Probabilities Multiplication Rule of

Probability Mutually Exhaustive Number of Combinations Odds Ratio

Key WordsKey Words(continued)(continued)

Overinvolvement Ratios Probability Postulates Random Experiment Relative Frequency

Probability Sample Space Solution Steps: Bayes’

Theorem Statistical

Independence Subjective Probability Union