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Class 2Readings & Problems

- Reading assignment
- M & S
- Chapter 3 - Sections 3.1 - 3.10 (Probability)

- M & S
- Recommended Problems
- M & S Chapter 3
- 1, 20, 25, 29, 33, 57, 75, and83

- M & S Chapter 3

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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 …

- Inferential statistics

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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!!

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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!

- One or more parameters are uncertain

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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??

- Probability is relative frequency of occurrence

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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?

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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

- The score was RPI 3, Cornell 1

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Heads or tails

Frequentist ProbabilityBuilding a Foundation- Experiment
- Process of obtaining observations
What are examples?

- Process of obtaining observations
- Basic outcome
- A simple event
- Elemental outcomes
What are examples?

- Elemental outcomes

- A simple event

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Frequentist ProbabilityDefining Terms

- Sample space
- Collection of all simple events of experiment
- Could be population or sample

- Collection of all simple events of experiment
- 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?

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S0 all men in VA

S0

S1

S1 all >6’ men in VA

S2

S2 all men >50 in VA

Visualizing Sample SpaceVenn DiagramVenn diagram represents all simple events in sample space

Is S0 part of a larger sample space?

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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

S0S1

S1 is a strict subset of S0

Every point in S1 is in S0 and S0 S1

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Defining Probability

- p(ei) probability of ei
- Likelihood of ei occurring if perform experiment
- Proportion of times you observe ei

Recall frequentist viewpoint in word “size”

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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)

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Simple events

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 BWhat is C

Recall E = {ei, …, ej}

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S0 all men in VA

S0

S1

S1 all >6’ men in VA

S2

S2 all men >50 in VA

Let C = S1 S2

What does C represent??

Defining More TermsIntersection of Setsstatistical processes

B

A and B are mutually exclusive

A B = (the null set)

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

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~A

A

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, A

Ac and A read as “the complement of A”

- then ~A is event that A does not occur
- p(A) + p(~A) = 1.0

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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)

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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

What are:

p(AC)?

p(C)?

p(AC)?

p(BC)?

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A B

S

A

B

Additive Rule of ProbabilityIntuitive ResultAdditive Rule for Mutually Exclusive Events

1) p(AB)=0

2) p(AB) = p(A) + p(B)

What if A & B are mutually exclusive?

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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

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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)

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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)

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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

- See if p(A B) = p(B) p(A)
- Are mutually exclusive events independent?
- No! p(A B) = 0 p(B) p(A)

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Counting Rules

- Counting rules
- Finding number of simple events in experiment
- aka Combinatorial Analysis
- Why would this be important?

- Most important rules
- Permutations
- Combinations

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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??

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C

A

J2

D

E

B

J1

C

D

E

PermutationsVisualizing ProblemAnd so forth.

Total of 20 possibilities.

Decision tree representation

Tool for sequential combinatorial

analysis

Decisions to fill open jobs

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Permutation Formula

- Is A getting J1 same as A getting J2?
- Order important
- Basic distinction of permutation problems

- Permutation formula

- N! said as “N factorial”
- N! = (N)(N-1) … (1)
- 0! = 1

- Multiplicative Rule:
Basis of permutation formula

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Permutation RuleMore Formal Definition

- Given SN { ej j = 1, …, N}
- Select subset of n members from SN
- Order is important

- Select subset of n members from SN

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combinations!

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

- Choose A for J1

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Combinations Rule More Formal Definition

- Given SN { ej j = 1, …, N}
- Select subset of n members from SN
- Order is not important
- Effectively a sample from SN

- Select subset of n members from SN

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n members

SN

Set with

N members

Subset with

(N-n) members

Combinations RuleDifferent PerspectiveHow many ways can you break up set SN into two subsets: one with

n and the other with (N-n) members?

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One of the subsets

The second subset

Interpreting theCombinations RuleCan you generalize breaking up into > 2 subsets???

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case when k=2

Partitions RuleBreaking Set into k Subsets- Given SN { ej j = 1, …, N}
- Select k subsets from SN
- Each subset has n1, n2, … , nk members
- Order is not important

- Select k subsets from SN

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Partitions RuleA Personal Experience

- Have 55 kids, how many different teams of 11 players each?

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Useful Excel FunctionsWhen You Work With Real Data

MEAN

MEDIAN

MODE

PERMUT

PERCENTILE

FACT

STDEV

STDEVP

VAR

VARP

DEVSQ

Statistical

Special

Functions

Excel

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