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statistical processes 1
Probability
Introduction
statistical processes 2
Class 2Readings & Problems
Reading assignment
M & S Chapter 3 - Sections 3.1 - 3.10
(Probability)
Recommended Problems
M & S Chapter 3 1, 20, 25, 29, 33, 57, 75, and 83
statistical processes 3
Introduction to Probability
Probability - a useful tool
Inferential statistics Infer population parameters probabilistically
Stochastic modeling (engineering applications) Decision analysis Simulation Reliability Statistical process control Others …
statistical processes 4
Development of Probability Theory
Chapter 3 - Introduction to probability
Basic concepts
Chapter 4 - Discrete random variables
What is a random variable???
What is a discrete random variable???
Chapter 5 - Continuous random variable
What is a continuous random variable???
Do not be afraid of random variables!!
statistical processes 5
What Is Probability?
Deterministic models
All parameters known with certainty
Stochastic models
One or more parameters are uncertain May be unknown Known but may take on more than 1
value
Measure of uncertainty probability Probability quantifies uncertainty!
statistical processes 6
ProbabilityMost Common Viewpoint
Frequentist view
Probability is relative frequency of occurrence
Most often associated with probability Adopted in textbook
Probability inherent to physical process Property of large number () of trials
Examples of applications??
statistical processes 7
ProbabilityAn Alternative Perspective
Bayesian view (aka personalist or subjective)
Many real world applications not amenable to frequentist viewpoint
What is probability of permanent lunar colony by 2015?
What if asked in 1970?
What if asked in 1998?
What if asked in 2004??! Is probability here a property inherent to
physical process?
statistical processes 8
Bayesian ProbabilityWhat is key?
What is probability RPI beat Cornell in hockey February 1971?
What is probability RPI beat Cornell in hockey February 1971?
RPI was ECAC champ that year
What is probability RPI beat Cornell in hockey February 1971?
The score was RPI 3, Cornell 1
State of knowledge defines probability
statistical processes 9
Frequentist ProbabilityBuilding a Foundation
Experiment
Process of obtaining observations
What are examples?
Basic outcome
A simple event Elemental outcomes
What are examples?
Flip a coin Heads or tails
statistical processes 10
Frequentist ProbabilityDefining Terms
Sample space
Collection of all simple events of experiment Could be population or sample
Set notation
S = { e1, e2, …, en}
where, S sample space
ei possible simple event
(outcome)
What is sample space for rolling 1 die?
What is sample space for rolling 2 dice?
statistical processes 11
Visualizing Sample SpaceVenn Diagram
Venn diagram represents all simple events in sample space
Is S0 part of a larger sample space?
S0S0 all men in VA
S1
S1 all >6’ men in VA
S2 S2 all men >50 in VA
statistical processes 12
Set Terminology
Subsets
S0 S1
S1 is a subset of S0 (S0 is a superset of S1)
Every point in S1 is in S0
NOTE: S1 could be the same as S0
S0 S1
S1 is a strict subset of S0
Every point in S1 is in S0 and S0 S1
statistical processes 13
Defining Probability
p(ei) probability of ei
Likelihood of ei occurring if perform experiment
Proportion of times you observe ei
S
eep i
i space sample of size""
event ofsize"")(
Recall frequentist viewpoint in word “size”
statistical processes 14
Fundamental RulesProbability
If p(ei) = 0 ei will never occur
If p(ei) = 1.0 ei will occur with certainty
Let, E = {ei, …, ej}
then, p(E) = p(ei) + … + p(ej)
Have 2 dice, find p(toss a 7), p(toss an 11)
n
ii
n
Seep
eeeS
1ii
21
0.1)p(e )2
0.1)(0 )1
then,
},...,,{Let
statistical processes 15
Defining More TermsCompound Events
Let A event, B event
A B is the union of A and B (either A or B or both occur)
If C = A B
then A C, and B C
If A event you toss 7, B event you toss 11, and C = A B What is C
Recall E = {ei, …, ej}
Event Simple events
statistical processes 16
Visualizing Union of SetsVenn Diagrams
AB
C = A B
AB
A
B
statistical processes 17
Defining More TermsIntersection of Sets
S0S0 all men in VA
S1S1 all >6’ men in VA
S2 S2 all men >50 in VA
Let C = S1 S2
What does C represent??
statistical processes 18
Intersection of SetsDice Example
Consider toss of 2 dice, let
A = event you toss a 7
B = event you toss an 11
C = A B
Draw Venn Diagram showing C
AB
A and B are mutually exclusive A B = (the null set)
statistical processes 19
ComplementarityA Useful Concept
Let A be an event
then ~A is event that A does not occur
~A is the complement of A
~A read as “not A”
also shown as Ac, AAc and A read as “the
complement of A”
p(A) + p(~A) = 1.0S
A ~A
statistical processes 20
Conditional ProbabilityStrings Attached
Are these likely the same?
p(person in VA > 6’ tall)
p(person in VA > 6’ tall given person is a man)
Former is an unconditional probability
Latter is a conditional probability
Probability of one event given another event has occurred
Formal nomenclature
p(A B)
statistical processes 21
Conditional Probability Formula
)(
)()(
Bp
BApBAp
S
BA A B
)(
)(S of size""
B of size""S of size""
BA of size""B of size""
BA of size"")(
Bp
BAp
BAp
statistical processes 22
Conditional ProbabilitiesExample Problem
Study of SPC success at plants
A = plant reports success; B = plant reports failure
C = plant has formal SPC; D = plant has no formal SPC
FormalC
No FormalD
MarginalProbability
SuccessA
0.4 0.3 0.7
FailureB
0.1 0.2 0.3
MarginalProbability
0.5 0.5 1.0
What are:p(AC)?p(C)?p(AC)?p(BC)?
statistical processes 23
Additive Rule of ProbabilityIntuitive Result
Additive Rule for Mutually Exclusive Events1) p(AB)=02) p(AB) = p(A) + p(B)
)()()()( BApBpApBAp
What if A & B are mutually exclusive?
SBA A B
statistical processes 24
Exercise
Deck of 52 playing cards
What is p(picking a heart or a jack)???
5216)(
521)( ;52
4)( ;5213)(
)()()()(
)( find Want to
jack is card B heart; is cardALet
BAp
BApBpAp
BApBpApBAp
BAp
statistical processes 25
Exercise
Same deck of 52 cards
What is p(jack card is a heart)?
What is p(heart card is a jack)?
Your results should make sense
statistical processes 26
Multiplicative Rule
Recall, conditional probability formula
p(A B) = p(A B) / p(B)
Multiplicative Rule
p(A B) = p(B) p(A B)
= p(A) p(B A)
Remember:
Additive rule applies to p(A B)
Multiplicative rule applies to p(A B)
statistical processes 27
Special Case of Conditional Probability:What if the Conditions Do Not Matter?
What is p(toss head previous toss was tail)?
p(toss head previous toss was tail) = p(toss head)
Independent events defined as
p(A B) = p(A)
p(B A) = p(B)
Multiplicative rule for independent events
p(A B) = p(B) p(A)
= p(A) p(B)
statistical processes 28
Confirming IndependenceDo Not Trust Intuition
Can Venn Diagrams illustrate independence?
No!
Unlike mutually exclusive events
How to demonstrate A & B are independent?
See if p(A B) = p(B) p(A) See Examples 3.16 & 3.17, assigned
problem 3.24
Not through Venn Diagram
Are mutually exclusive events independent?
No! p(A B) = 0 p(B) p(A)
statistical processes 29
Counting Rules
Counting rules
Finding number of simple events in experiment
aka Combinatorial Analysis
Why would this be important?
Most important rules
Permutations
Combinations
statistical processes 30
PermutationsRepresentative Application
You are employer
2 open positions, J1 and J2
5 applicants {A, B, C, D, E} for either job
How many ways to fill positions??
statistical processes 31
PermutationsVisualizing Problem
Decisions to fill open jobs
And so forth.Total of 20 possibilities.
Decision tree representationTool for sequential combinatorialanalysis
J1
A
B
C
D
E
B
J2
CDE
statistical processes 32
Permutation Formula
Is A getting J1 same as A getting J2?
Order important
Basic distinction of permutation problems
Permutation formula
)!(
!
nN
NPN
n
N! said as “N factorial”
N! = (N)(N-1) … (1)
0! = 1
Multiplicative Rule:
Basis of permutation formula
knnnn 321
statistical processes 33
Permutation RuleMore Formal Definition
Given SN { e { ejj j = 1, …, N} j = 1, …, N}
Select subset of n members from SSNN
Order is important
)!(
! sets unique of #
nN
NPP nN
Nn
20!3
!5
)!25(
!5
example job Previous
52
P
statistical processes 34
CombinationsOrder Is Not Important
Suppose J1 and J2 were the same
Order not important
How would you enumerate combinations?
Choose A for J1
AB, AC, AD, AE
Choose B for J1
BC, BD, BE
Choose C for J1
CD, CE
Choose D for J1
DE
A total of 10 combinations!
statistical processes 35
Combinations Rule More Formal Definition
Given SN { e { ejj j = 1, …, N} j = 1, …, N}
Select subset of n members from SN
Order is not important
Effectively a sample from SN
)!(!
!CC sets unique of #
nNn
NnN
Nn
102!3!
5!iespossibilit of #
:example Job
52 C
statistical processes 36
Combinations RuleDifferent Perspective
How many ways can you break up set SN into two subsets: one with
n and the other with (N-n) members?
SN
Set withN members
SN
Set withN members
Subset withn members
Subset withn members
Subset with(N-n) members
Subset with(N-n) members
statistical processes 37
Interpreting theCombinations Rule
)!(!
!C sets unique of #
nNn
NNn
Original set
One of the subsets The second subset
Can you generalize breaking up into > 2 subsets???
statistical processes 38
Partitions RuleBreaking Set into k Subsets
Given SN { e { ejj j = 1, …, N} j = 1, …, N}
Select k subsets from SN
Each subset has n1, n2, … , nk
members Order is not important
ii
21
n
where,
!!!
! sets unique of #
N
nnn
N
kNote specialcase when k=2
statistical processes 39
Partitions RuleA Personal Experience
Have 55 kids, how many different teams of 11 players each?
E3525.1!11
!55!!!
! sets unique of #
5
21
knnn
N
statistical processes 40
Useful Excel FunctionsWhen You Work With Real Data
StatisticalSpecial
Functions
StatisticalSpecial
FunctionsExcel
MEANMEDIANMODEPERMUTPERCENTILEFACTSTDEVSTDEVPVARVARPDEVSQ