Probability

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# Probability - PowerPoint PPT Presentation

Probability. Introduction. Class 2 Readings &amp; Problems. Reading assignment M &amp; S Chapter 3 - Sections 3.1 - 3.10 (Probability) Recommended Problems M &amp; S Chapter 3 1, 20, 25, 29, 33, 57, 75, and 83 . Introduction to Probability. Probability - a useful tool Inferential statistics

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### Probability

Introduction

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• M & S
• Chapter 3 - Sections 3.1 - 3.10 (Probability)
• Recommended Problems
• M & S Chapter 3
• 1, 20, 25, 29, 33, 57, 75, and83

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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 …

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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!

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ProbabilityMost Common Viewpoint
• Frequentist view
• Probability is relative frequency of occurrence
• Most often associated with probability
• Probability inherent to physical process
• Property of large number () of trials
• Examples of applications??

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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

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Flip a coin

Frequentist ProbabilityBuilding a Foundation
• Experiment
• Process of obtaining observations

What are examples?

• Basic outcome
• A simple event
• Elemental outcomes

What are examples?

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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?

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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 Diagram

Venn 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

S0S1

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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Event

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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B

A

A

A

B

B

Visualizing Union of SetsVenn Diagrams

C = A  B

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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 Sets

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A

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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S

~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

also shown as Ac, A

Ac and A read as “the complement of A”

• 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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S

A  B

A

B

Conditional Probability Formula

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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(AC)?

p(C)?

p(AC)?

p(BC)?

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A  B

S

A

B

Additive Rule for Mutually Exclusive Events

1) p(AB)=0

2) p(AB) = p(A) + p(B)

What if A & B are mutually exclusive?

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Exercise
• Deck of 52 playing cards
• What is p(picking a heart or a jack)???

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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)?
• 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
• 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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B

C

A

J2

D

E

B

J1

C

D

E

PermutationsVisualizing Problem

And 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

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A total of 10

combinations!

CombinationsOrder Is Not Important
• Suppose J1 and J2 were the same
• Order not important
• How would you enumerate combinations?
• Choose A for J1
• Choose B for J1
• BC, BD, BE
• Choose C for J1
• CD, CE
• Choose D for J1
• DE

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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

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Subset with

n members

SN

Set with

N members

Subset with

(N-n) members

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?

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Original set

One of the subsets

The second subset

Interpreting theCombinations Rule

Can you generalize breaking up into > 2 subsets???

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Note special

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

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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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