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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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