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Probability. Introduction. Class 2 Readings & 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 . Introduction to Probability. Probability - a useful tool Inferential statistics

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Probability

Probability

Introduction

statistical processes


Class 2 readings problems
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, and83

statistical processes


Introduction to probability
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


Development of probability theory
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


What is probability
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


Probability most common viewpoint
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


Probability an alternative perspective
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


Bayesian probability what is key
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


Frequentist probability building a foundation

Flip a coin

Heads or tails

Frequentist ProbabilityBuilding a Foundation

  • Experiment

    • Process of obtaining observations

      What are examples?

  • Basic outcome

    • A simple event

      • Elemental outcomes

        What are examples?

statistical processes


Frequentist probability defining terms
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


Visualizing sample space venn diagram

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?

statistical processes


Set terminology
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


Defining probability
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”

statistical processes


Fundamental rules probability
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)

statistical processes


Defining more terms compound events

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}

statistical processes


Visualizing union of sets venn diagrams

B

A

A

A

B

B

Visualizing Union of SetsVenn Diagrams

C = A  B

statistical processes


Defining more terms intersection of sets

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

statistical processes


Intersection of sets dice example

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

statistical processes


Complementarity a useful concept

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

      ~A read as “not A”

      also shown as Ac, A

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

  • p(A) + p(~A) = 1.0

statistical processes


Conditional probability strings attached
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


Conditional probability formula

S

A  B

A

B

Conditional Probability Formula

statistical processes


Conditional probabilities example problem
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)?

statistical processes


Additive rule of probability intuitive result

A  B

S

A

B

Additive Rule of ProbabilityIntuitive Result

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?

statistical processes


Exercise
Exercise

  • Deck of 52 playing cards

    • What is p(picking a heart or a jack)???

statistical processes


Exercise1
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


Multiplicative rule
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


Special case of conditional probability what if the conditions do not matter
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


Confirming independence do not trust intuition
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


Counting rules
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


Permutations representative application
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


Permutations visualizing problem

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

statistical processes


Permutation formula
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

statistical processes


Permutation rule more formal definition
Permutation RuleMore Formal Definition

  • Given SN { ej  j = 1, …, N}

    • Select subset of n members from SN

      • Order is important

statistical processes


Combinations order is not important

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

      • AB, AC, AD, AE

    • Choose B for J1

      • BC, BD, BE

    • Choose C for J1

      • CD, CE

    • Choose D for J1

      • DE

statistical processes


Combinations rule more formal definition
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

statistical processes


Combinations rule different perspective

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?

statistical processes


Interpreting the combinations rule

Original set

One of the subsets

The second subset

Interpreting theCombinations Rule

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

statistical processes


Partitions rule breaking set into k subsets

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

statistical processes


Partitions rule a personal experience
Partitions RuleA Personal Experience

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

statistical processes


Useful excel functions when you work with real data
Useful Excel FunctionsWhen You Work With Real Data

MEAN

MEDIAN

MODE

PERMUT

PERCENTILE

FACT

STDEV

STDEVP

VAR

VARP

DEVSQ

Statistical

Special

Functions

Excel

statistical processes


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