Intro Proba

7/31/2019 Intro Proba

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Probability Concepts

FOUR

Define the terms: conditional probability and jointprobability.

FIVECalculate probabilities applying the rules of addition

and the rules of multiplication.

Goals


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Movie


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Definitions continued

There are three definitions of probability: classical,empirical, and subjective.

A PrioriClassical

probabilityapplies

when thereare n

equally likelyoutcomes.

Empirical

Classicalprobabilityapplies whenthe number of

times the eventhappens is

divided by thenumber of

observations.

Subjectiveprobability is

based onwhateverinformation is

available.


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Movie


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Definitions continued

An Event isthe collection

o f o n e o rm o r eoutcomes ofa nex er iment .

An Outcome is

the part icularr e s u l t o f a ne x p e r i m e n t .

Experiment: A fair die is cast.

Possible outcomes: Thenumbers 1, 2, 3, 4, 5, 6

One possible event: The

occurrence of an evennumber. That is, wecollect the outcomes 2,4 , a n d 6 .


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Collectively ExhaustiveEvents

E v e n t s a r e C o l l e c t i v e l y

Exhaustive if at least one of theevents must occur when ane x p e r i m e n t i s c o n d u c t e d .


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Example 2

155.01200186)( AP

Throughout herteaching careerProfessor Joneshas awarded 186As out of 1,200

students. Whatis the probabilitythat a student inher section this

s e m e s t e r w i l lr e ce i ve a n A?

This is an example of theempirical definition ofprobability.

To find the probability aselected student earned

an A:


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Subjective Probability

Examples of subjective probabilities are:

Estimating the probability theWashington Redskins will winthe Super Bowl this year.

Estimating the probabilitymortgage rates for home

loans wil l top 8 percent.


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Basic Rules of ProbabilityP(A or B) = P(A) + P(B)

If two eventsA and B are mutually

exclusive, the

Special Rule ofAddition states that

theProbability of A or B

occurring equals the sumof their respective

probabilities.


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Example 3

Arrival Frequency

Early 100On Time 800

Late 75

Canceled 25

Total 1000

New England Commuter Airways recentlysupplied the following information on their

commuter flights from Boston to New York:


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Example 3 continued

The probability that a flight is either early orlate is:

P(A or B) = P(A) + P(B) = .10 + .075 =.175.

If A is the event thata flight arrivesearly, then P(A) =100/1000 = .10.

If Bis the event that a

flight arrives late, thenP(B) = 75/1000 = .075.


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The Complement Rule

If P(A) is the probability of event A andP(~A) is the complement of A,

P(A) + P(~A) = 1 or P(A) = 1 - P(~A).

The Complement Rule is used to determinethe probability of an event occurring by subtracting

the probability of the event notoccurring from 1.


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The Complement Rule continued

A~A

A Venn Diagram illustrating the complementrule would appear as:


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Example 4

If Dis the event that

a flight is canceled,then

P(D) = 25/1000 =.025.

Recall example 3. Use thecomplement rule to find the

probability of an early (A) or a late(B) flight

If Cis the event that a

flight arrives on time,then P(C) = 800/1000 =.8.


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Example 4 continued

C.8

D.025

~(C or D) = (A or B).175

P(A or B) = 1 - P(Cor D)= 1 - [.8 +.025]

=.175


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The General Rule ofAddition

If A and Bare two

events that are notmutually exclusive,then P(A or B) isgiven by the

following formula:

P(A or B) = P(A) + P(B) - P(A and B)


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The General Rule of

Addition

A and B

A

B

The Venn Diagram illustrates this rule:


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EXAMPLE 5

Stereo

220

Both100

TV75

In a sample of 500 students, 320 said they had

a stereo, 175 said they had a TV, and 100 saidthey had both. 5 said they had neither.

If d i l d


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Example 5 continued

P(S or TV) = P(S) + P(TV) - P(S and TV)

= 320/500 + 175/500

100/500= .79.

P(S and TV) = 100/500

= .20

If a student is selectedat random, what is theprobab i l i ty tha t the

s t u d e n t h a s o n l y astereo or TV? What isthe probability that thes tudent has bo th a

s t e r e o a n d T V ?


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Joint Probability

A Joint Probability measures thelikelihood that two or more events will

happen concurrently.

An example would

be the event that astudent has botha stereo and TV inhis or her dorm

r o o m .


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Special Rule of Multiplication

Two events A and Bare independentif the

occurrence of one has no effect on theprobability of the occurrence of the other.

This rule is written: P(A and B) = P(A)P(B)

The Special Rule of

Multiplication requires that twoevents A and Bare independent.


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Example 6

5-year stock prices

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5

Year

Stockp

rice

$

IBM

GE

P(IBMand GE) = (.5)(.7) = .35

Chris owns two stocks,IBM and General Electric(GE). The probability that

IBM stock will increase invalue next year is .5 andthe probability that GEstock will increase in

value next year is .7.Assume the two stocksare independent. What isthe probability that bothstocks will increase invalue next year?


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Example 6

continued

What

is theprobability

that at least

one of these stocks

increases in value in

the next year?

This means that

either one can

increase or

both.

P(at least one)

= P(IBM but not GE)

+ P(GE but not IBM)

+ P(IBM and GE)

(.5)(1-.7)+ (.7)(1-.5)

+ (.7)(.5)= .85


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General Multiplication Rule

It states that for two

events A and B, thejoint probability thatboth events willhappen is found by

multiplying theprobability that eventA will happen by theconditional

probability of Bgiventhat A has occurred.

The General

Rule ofMultiplicationis used to find the

joint probability thattwo events willoccur.


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General Multiplication Rule

The joint probability,P(A and B), is given bythe following formula:

P(A and B) = P(A)P(B/A)or

P(A and B) = P(B)P(A/B)


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Example 7

Major Male Female Total

Accounting 170 110 280

Finance 120 100 220

Marketing 160 70 230

Management 150 120 270

Total 600 400 1000

The Dean of the School of Business at OwensUniversity collected the following information

about undergraduate students in her college:


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Example 7

continued

P(A|F) = P(A and F)/P(F)

=[110/1000]/[400/1000] =.275

If a student is selected at random, whatis the probability that the student is afemale (F)accounting major (A)?

P(A and F) =110/1000.

Given that the student is afemale, what is theprobability that she is anaccounting major?

Intro Proba

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