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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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Two standarddeviationsTwo standarddeviations
95.5%
2W 2W
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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