statistical processes 1 probability introduction

statistical processes 1

Probability

Introduction


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


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 …


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!!


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!


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??


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?


Bayesian ProbabilityWhat is key?

What is probability RPI beat Cornell in hockey February 1971?


RPI was ECAC champ that year


The score was RPI 3, Cornell 1

State of knowledge defines probability


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


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?


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


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


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”


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


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


Visualizing Union of SetsVenn Diagrams

AB

C = A B

AB

A

B


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??


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)


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


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)


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


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)?


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


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


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


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)


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)


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)


Counting Rules

Counting rules

Finding number of simple events in experiment

aka Combinatorial Analysis

Why would this be important?

Most important rules

Permutations

Combinations


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??


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


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


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


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!


Combinations Rule More Formal Definition


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


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


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???


Partitions RuleBreaking Set into k Subsets


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


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


Useful Excel FunctionsWhen You Work With Real Data

StatisticalSpecial

Functions

StatisticalSpecial

FunctionsExcel

MEANMEDIANMODEPERMUTPERCENTILEFACTSTDEVSTDEVPVARVARPDEVSQ

statistical processes 1 probability introduction

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