ch08 probability

8/8/2019 Ch08 Probability

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Review Homework

Chapter 6: 1, 2, 3, 4, 13

Chapter 7 - 2, 5, 11 Probability

Control charts forattributes

Week 13 Assignment Read Chapter 10:

Reliability

Homework

Chapter 8: 5, 9,10,

20, 26, 33, 34

Chapter 9: 9, 23

Week 12AgendaAgenda


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Probability

ProbabilityProbability

Chapter Eight


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Probability

Probability theoremsProbability theorems

For mutually exclusive events, theprobability that either event A or event B

will occur is the the sum of theirrespective probabilities.

When events A and B are not mutually

exclusive events, the probability thateither event A or event B will occur is

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


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Probability

Probability theoremsProbability theorems

If A and B are dependent events, theprobability that both A and B will occur is

P(A and B) = P(A) x P(B|A) If A and B are independent events, then

the probability that both A and B will

occur is P(A and B) = P(A) x P(B)


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Probability

PermutationsandPermutationsand

combinationscombinationsA permutation is the number of

arrangements that n objects can have

when r of them are used. When the order in which the items are

used is not important, the number of

possibilities can be calculated by usingthe formula for a combination.


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Probability

Discrete probabilityDiscrete probability

distributionsdistributions Hypergeometric - random samples from

small lot sizes.

Population must be finite samples must be taken randomly without

replacement

Binomial - categorizes success andfailure trials

Poisson - quantifies the count of discrete

events.


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Probability

Continuous probabilityContinuous probability

distributionsdistributions Normal

Uniform

Exponential Chi Square

F

student t


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Probability

Fundamental conceptsFundamental concepts

Probability = occurrences/trials

0 < P < 1

The sum of the simple probabilities for allpossible outcomes must equal 1

Complementary rule - P(A) + P(A) = 1


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Probability

Addition ruleAddition rule

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

If mutually exclusive; just P(A) + P(B)

P(A) P(B)

P(AandB)


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Probability

Addition ruleexampleAddition ruleexample

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

Roll one die

Probability of even and divisible by 1.5? Sample space {1,2,3,4,5,6}

Event A - Even {2,4,6}

Event B - Divisible by 1.5 {3,6} Event A and B ?

Solution?


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Probability

Conditional probability ruleConditional probability rule

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

A die is thrown and the result is known to

be an even number. What is theprobability that this number is divisible by1.5?

P(/1.5|Even)=P(/1.5 and even)/P(even) 1/6 / 3/6 = 1/3


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Probability

Compoundor jointCompoundor joint

probabilityprobability The probability of the simultaneous

occurrence of two or more events is

called the compound probability or,synonymously, the joint probability.

Mutually exclusive events cannot be

independent unless one of them is zero.


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Probability

Multiplication forMultiplication for

independenteventsindependentevents P(A and B) = P(A) x P(B)

P(ace and heart) = P(ace) x P(heart)

1/13 x 1/4 = 1/52


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Probability

Computing conditionalComputing conditional

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

P(Own and Less than 2 years)?

Number of credit applicants by category

On present job 2

years or less

On present job

more than 2 years

Own Home 20 40

Rent Home 80 60

Total 100 100


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Probability

P(A) P(B)

P(AandB)

Computing conditionalComputing conditional

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


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Probability

Joi t robabilit

tabl

On r nt jr r l

On r nt jr t n

r

M rginalr abilit

Own Home . . .3

Rent Home . .3 .7

Marginal probability .5 .5 .


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Probability

Conditional probabilityConditional probability

Satisfied Not Satisfied Totals

New 300 100 400

Used 450 150 600

Total 750 250 1000

S=satisfied N= bought new car

P(N|S) = ?


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Probability

Just for funJust for fun

60 business studentsfrom a large universityare surveyed with the

following results: 19 read Business Week

18 read WSJ

50 read Fortune

13 read BW and WSJ 11 read WSJ and Fortune

13 read BW and Fortune

9 read all three

How many read none?

How many read onlyFortune?

How many read BW, theWSJ, but not Fortune?

Hint: Try a Venn

diagram.


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Probability

ProbabilityProbabilityDistributionsDistributions


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Probability

Learning objectivesLearning objectives

Know the difference between discreteand continuous random variables.

Provide examples of discrete andcontinuous probability distributions.

Calculate expected values and

variances. Use the normal distribution table.


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Probability

RandomvariablesRandomvariables

A random variable is a numericalquantitywhose value is determined by chance.

A random variable assigns a number toevery possible outcome or event in anexperiment.

For non-numerical outcomes such as a coin

flip you must assign a random variable thatassociates each outcome with a unique realnumber.


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Probability

Randomvariable typesRandomvariable types

Discrete random variable - assumes alimited set of values; non-continuous,

generally countable number of Mark McGwire homeruns in a

season

number of auto parts passing assembly-lineinspection

GRE exam scores


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Probability

Randomvariable typesRandomvariable types

Continuous random variable - randomvariable with an infinite set of values.

Can occur anywhere on a continuous number scale

0.000 1.000Baseball players batting average


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Probability

RandomvariablesandRandomvariablesand

probabilitydistributionsprobabilitydistributions The relationship between a random

variables values and their probabilities is

summarized by itsprobability distribution.


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Probability

ProbabilitydistributionProbabilitydistribution

Whether continuous or discrete, theprobability distribution provides a

probability for each possible value of arandom variable, and follows these rules:

The events are mutually exclusive

The individual probability values arebetween 0 and 1.

The total value of the probability values sumto 1


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Probability

Probabilitydistribution forProbabilitydistribution for

rates of returnrates of return Possible rate of

return

10%

11%

12%

13%

14% 15%

16%

17%

Probability

.05

.15

.20

.35

.10

.10 .03

.02

Total = .


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Probability

DescribingdistributionsDescribingdistributions

Measures ofcentral tendency

expected value (weighted

average)

Measures ofvariability

variance standard deviation


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Probability

Expectedvalue ofadiscreteExpectedvalue ofadiscrete

randomvariablerandomvariable For discrete random variables, the

expected value is the sum of all the

possible outcomes times the probabilitythat they occur.

E(X) =7 {xi * P(xi)}


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Probability

Example:A fairdieExample:A fairdie

Roll 1 die: x P(x) x*P(x) E(x)=?

1 1/6 1/6

2 1/6 2/63 1/6 3/6

4 1/6 4/6

5 1/6 5/6

6 1/6 6/6

Can you sketch

the distribution?


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Probability

Fairdie illustratesadiscreteFairdie illustratesadiscrete

uniformdistributionuniformdistribution The random variable, x, has n possible

outcomes and each outcome is equally

likely. Thus, x is distributed uniform.


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Probability

x

P(x)

1/6

1 2 3 4 5 6

ProbabilitydistributionProbabilitydistribution


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Probability

Example:AnunfairdieExample:Anunfairdie

Roll 1 die: x P(x) x*P(x) E(x)=?

1 1/12 1/12

2 2/12 4/123 2/12 6/12

4 2/12 8/12

5 2/12 10/12

6 3/12 18/12

Can you sketch

the distribution?


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Probability

Expectedvalue ofa betExpectedvalue ofa bet

Suppose I offer you the following wager:You roll 1 die. If the result is even, I pay

you $2.00. Otherwise you pay me$1.00.

E(your winnings)=.5 ($2.00) + .5 (-1.00)

= 1.00 - .50 = $0.50


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Probability

ExpectedValue ofa BetExpectedValue ofa Bet

Suppose I offer you the following wager:You roll 1 die. If the result is 5 or 6 I pay

you $3.00. Otherwise you pay me $2.00. What is your expected value?


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Probability

Variance ofadiscreteVariance ofadiscrete

randomvariablerandomvariableThe variance of a random variable is ameasure of dispersion calculated by

squaring the differences between theexpected value and each randomvariable and multiplying by its associatedprobability.

7{(xi-E(x))2 * P(xi)}


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Probability

Roll 1 die: [x- E(X)] 2 P(x) *P(x)

1 - 21/6 6.25 1/6 1.04

2 - 21/6 2.25 1/6 .375

3 - 21/6 .25 1/6 .04

4 - 21/6 .25 1/6 .045 - 21/6 2.25 1/6 .375

6 - 21/6 6.25 1/6 1.042.91

Example:A fairdieExample:A fairdie


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Probability

Probabilitydistributions forProbabilitydistributions for

continuous randomvariablescontinuous randomvariablesA continuous mathematical function

describes the probability distribution.

Its called the probability density functionand designated (x)

Some well know continuous probability

density functions: Normal Beta

Exponential Student t


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Probability

Continuous probabilityContinuous probability

density functiondensity function -- UniformUniformIf a random variable, x, is distributed

uniform over the interval [a,b], then its

pdf is given byf x

b a( ) !

1

a b

1

b-a


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Probability

UniformUniform

a b

1

b-a

What is the probability of x?

x


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Probability

UniformUniform

a b

1

b-a

Area under the rectangle = base*height

= (b-a)* 1 = 1

b-a


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Probability

UniformUniform

a b

1

b-a

c

P(c


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Probability

UniformdistributionUniformdistribution

If a random variable, x, is distributeduniform over the interval [a,b], then its pdfis given by

f xb a

( ) !1

And, the mean and variance are

(a+b) ( b-a )2E(x) = ------- Var(x)=---------

2 12


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Probability

UniformUniform

3 8

Mean? Variance?


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Probability

f x( ) !

1

5

And, the mean and variance are

(a+b) ( b-a )2 25

E(x) = ------ = 5.5 V(x)=--------- = ----- = 2.08

2 12 12

So, if a = 3 and b = 8

Calculateuniformmean,Calculateuniformmean,

variancevariance


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Probability

Continuous pdfContinuous pdf -- NormalNormal

f x e

x

( )

( )

!

1

2 22

2

2

T W

Q

W

If x is a normally distributed variable, then

is the pdf for x. The expected value is Q andthe variance is W2.


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Probability

OnestandarddeviationOnestandarddeviation

68.3%

W W


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Probability

Two standarddeviationsTwo standarddeviations

95.5%

2W 2W


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Probability

ThreestandarddeviationsThreestandarddeviations

99.73%

3W 3W


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P b bili

Continuous PDFContinuous PDF-- StandardStandardNormalNormal

f z ez

( ) !

1

2

2

2

T

If z is distributed standard normal,

then Q!and W!

f x e

x

( )

( )

!

1

22

2

2

2

T W

Q

W

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