L Berkley Davis Copyright 2009 MER301: Engineering Reliability1 LECTURE 1: Basic Probability Theory.

L Berkley DavisCopyright 2009

MER301: Engineering Reliability 1

MER301: Engineering Reliability

LECTURE 1:

Basic Probability Theory


MER301: Engineering ReliabilityLecture 1

2

Summary-Probability Basic Definitions

Random Experiments, Outcomes, Sample Spaces, Events

Probability Properties Limits and Definitions

The Laws of Chance Classical Probability Relative Frequency Definition Subjective or Bayesian Probability

Probability Rules Addition/Multiplication Conditional Probabiity



3

Probability Probability is used to quantify the likelihood, or

chance, that the outcome of an “event” falls within some specified range of values of a specified random variable Random variable can be discrete or continuous

Probabilities are used in many fields of both everyday and professional life Weather forecasting,insurance, investing, medicine,

genetics, “can I make that green light” Reliability analysis, safety analysis, strength of

materials, quantum mechanics, commercial guarantee policies



4

Probability- Basic Definitions Random Experiment

An experiment that can result in different outcomes when repeated in the same manner

Outcomes of a Random Experiment Outcome-single result of a random experiment, Elementary Outcomes- all possible results of a random

experiment

Sample Space Set or collection of all of the elementary outcomes

Event A collection of outcomes A that share a specified

characteristic Complement of an event A is an event comprising all

outcomes not belonging to A (not A)



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A Random Experiment –Example 1.1

Coin Toss Exercise- flip 3 times and record the results Outcome Sample Space Event Properties of Probability



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Probability –Example 1.1(Solution)

Coin Toss Exercise- flip 3 times and record the results Outcome

A single sequence of Heads/Tails(HTT,etc) Sample Space

The eight possible Outcomes from three coin flips Event

The collection of outcomes with,eg, at least one head Properties of Probability-for three coin flips

ii oallforoP ......8/1)(

18/18/18/18/18/18/18/18/1)( ioP



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Properties of Probability Probability of any particular Elementary Outcome or

Event is greater than or equal to zero and is less than or equal to one

Probability is always non-negative If an outcome/event cannot occur, probability is zero If an outcome/event is certain to occur,its probability is

one Sum of the Probabilities of all the possible Elementary

Outcomes of a Random Experiment is equal to one(All possible Elementary Outcomes by definition equal the Sample Space)

1)(0 ioP

1)()()( 21 noPoPoP



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Properties of Probability The probability that an Elementary Outcome/Event will

occur is one minus the probability that the Elementary Outcome/Event will not occur

Complement of an event A is an event comprising all outcomes not belonging to A (not A).

For the 3 coin toss Example 2.1

)(1)( AnotPAP

4/11)(14/38/18/18/18/18/18/1)(

)()()()()()()(

)(14/18/18/1)()()(

HHxPnotHHxP

or

TTTPTTHPTHTPTHHPHTTPHTHPnotHHxP

notHHxPHHTPHHHPHHxP



9




experiment







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Sample Spaces/Populations A Sample Space or Population is the set of all

possible values of a random variable, called the elementary outcomes, for a given experiment A Sample is a subset of a Sample Space

Definition of a Sample Space depends on what characteristic is to be observed

Types of Sample Spaces Finite Countable Uncountable



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Example 1.2 – Finite Sample Space Consider the random experiment of tossing a

coin three times and recording the results

Two of the possible sample spaces for this experiment are The exact sequence of heads (H) and tails (T) in each

outcome S={TTT, TTH, THT, HTT, HHT, HTH, THH,

HHH} The number of heads (or tails) in each outcome

S={0, 1, 2, 3} Binomial Distribution applies to this case



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Example1.3–Countable Sample Spaces

The random experiment consists of rolling a dice until a 6 is obtained(so a 6 is obtained by definition in each outcome)

Two of the possible sample spaces are The exact values on the dice in each outcome

S={6, 16, 26, 36, 46, 56, 116, 126, 136, 146, 156, 216, …}

If N=1,2,3,4,5 then S={6, N6, NN6,…} The number of throws needed to get a 6 in each outcome

S= {1,2,3,4, …}



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Example1.4 Uncountable Sample Space

The experiment consists of throwing a dart onto a circular dart board marked with three concentric rings. inner ring is worth 3 points middle ring worth 2 points outside ring worth 1 point

Describe two possible Sample Spaces One Sample Space is the exact location of the dart in each

outcome S={(r,)|02, 0rR} The r and theta distributions are continuous

A second is the number of points scored in each outcome S={1,2,3}



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experiment







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Probability- Definitions of an Event

Union of two events A and B is an event comprising all outcomes in A or B or both (A or B)

Intersection of two events A and B is an event comprising outcomes common to both A and B (A and B)

Empty Event (Null Set) is one containing no outcomes

If A and B have no outcomes in common then they are Mutually Exclusive or Disjoint



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Non-Constant Probability-Example 1.5

Consider a group of five potential blood donors – A, B, C, D, and E – of whom only A and B have type O+ blood. Five blood samples, one from each individual, will be typed in random order until an O+ individual is identified. Let X=the number of typings necessary to

identify an O+ individual Determine the probability that an O+ individual

will be identified in three typings



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Summary of Probability Properties

Rule 1 The probability of an elementary outcome/event will be a number

between zero and one Rule 2

If an elementary outcome/event cannot occur, the probability is zero

Rule 3 If an elementary outcome/event is certain, the probability is one

Rule 4 The sum of probabilities of all the elementary outcomes or events

in a sample space is one Rule 5

The probability an elementary outcome/event will not occur is one minus the probability that the outcome/event will occur



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Summary of Probability Properties-Text

Rule 1 The probability on an elementary outcome/event will be a number

between zero and one Rule 2

If an elementary outcome/event cannot occur, the probability is zero

Rule 3 If an elementary outcome/event is certain, the probability is one

Rule 4 The sum of probabilities of all the elementary outcomes or events

in a sample space is one Rule 5

The probability an elementary outcome/event will not occur is one minus the probability that the outcome/event will occur



19

Probability-what is it?

Probability is used to quantify the likelihood, or chance, that the outcome of an “event” falls within some specified range of values of a specified random variable Random variable can be discrete or continuous

Probabilities are used in many fields of both everyday and professional life Weather forecasting,insurance, investing, medicine,

genetics, “can I make that green light” Reliability analysis, safety analysis, strength of

materials, quantum mechanics, commercial guarantee policies



20

Probability:The Laws of Chance….

Objective Definitions Classical Probability assumes the game is “fair” and

all elementary outcomes have the same probability Relative Frequency Probability of a result is

proportional to the number of times the result occurs in repeated experiments

Subjective or Bayesian Definition Bayesian Probability is an assessment of the

likelihood of the truth of each of several competing hypotheses ,given data and some additional assumptions.



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Probability Rules Addition ( A or B)

A and B are mutually exclusive

A and B are not mutually exclusive

Multiplication( A and B) A and B are independent

A and B are not independent/conditional probability

)()()()()( BandAPBPAPBAPBorAP

0)()(

)()()()(

BAPBandAP

BPAPBAPBorAP

)()()( BPAPBandAP

)()()( APABPBPBAPBandAP



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Dice Example 1.6

36 Elementary Outcomes Probability of a specific outcome is 1/36 Probability of the event “sum of dice equals 7” is 6/36

Addition-P(A or B) Events “sum=7” and “sum=11” mutually exclusive P=(6/36+2/36) Events “sum=7” and “dice 1=3” not mutually exclusive P=(11/36)

Multiplication-P(A and B) Events A=(6,6) and B=(6,6 repeated) are independent P=(1/36)2

Event A= (6,6) given B=(n1=n2=even) are not independent P=(1/3)

Dice 11 2 3 4 5 6

1 1,1 2,1 3,1 4,1 5,1 6,12 1,2 2,2 3,2 4,2 5,2 6,2

Dice 2 3 1,3 2,3 3,3 4,3 5,3 6,34 1,4 2,4 3,4 4,4 5,4 6,45 1,5 2,5 3,5 4,5 5,5 6,56 1,6 2,6 3,6 4,6 5,6 6,6



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Dice Example 1.6-Probabilities For the dice experiment, determine the probability for each event (A, B,

C) A = {the sum on the two dice is 6}

B = {both dice show the same number}

C = {at least one of the faces is divisible by 2} The Sample Space and Events are given by

For the Sample Space S, N=36 and the probability of an event is the number of outcomes in that event divided by N A=5/36 B=6/36 C=27/36

6,6.........:1,3:,6,2.....:1,2:,6,1....:3,1:2,1:1,1S 1,5;2,4;3,3;4,2;5,1A 6,6;5,5;4,4;3,3;2,2;1,1B

....6,1;....4,1;2,1;6,6;......1,6;6,4;.....1,4,6,2;5,2;4,2;3,2;2,2;1,2C


The Dice Game of Craps and Probability Rules

Player has two Dice 1st Roll

7 or 11 wins immediately

2,3, or 12 loses immediately

Rolls of 4,5,6,8,9,10 Continue

Subsequent Rolls Continue to roll until get the same number as on 1st roll

(win) or a 7 (lose)

Probability of Winning Overall


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%2236/8)117( orP

%1136/4)1232( ororP

)( GFEDCBAP

Union CollegeMechanical Engineering


21

Probability Rules

Addition ( A or B) A and B are mutually exclusive

A and B are not mutually exclusive

Multiplication( A and B) A and B are independent

A and B are not independent/conditional probability

)()()()()( BandAPBPAPBAPBorAP

0)()(

)()()()(

BAPBandAP

BPAPBAPBorAP

)()()( BPAPBandAP


Dice 11 2 3 4 5 6

1 1,1 2,1 3,1 4,1 5,1 6,12 1,2 2,2 3,2 4,2 5,2 6,2

Dice 2 3 1,3 2,3 3,3 4,3 5,3 6,34 1,4 2,4 3,4 4,4 5,4 6,45 1,5 2,5 3,5 4,5 5,5 6,56 1,6 2,6 3,6 4,6 5,6 6,6


The Dice Game of Craps Define the following probabilities

Let A= probability of 7 or 11 on 1st roll Let B= probability of 4 on 1st roll and then another 4 before a 7 Let C= probability of 5 on 1st roll and then another 5 before a 7 Let D= probability of 6 on 1st roll and then another 6 before a 7 Let E= probability of 8 on 1st roll and then another 8 before a 7 Let F= probability of 9 on 1st roll and then another 9 before a 7 Let G= probability of 10 on 1st roll and then another 10 before a 7

These are mutually exclusive events so the probability of winning is

or

The House always wins…


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4929.0)()()()()()()()( GPFPEPDPCPBPAPwinP

Dice 11 2 3 4 5 6

1 1,1 2,1 3,1 4,1 5,1 6,12 1,2 2,2 3,2 4,2 5,2 6,2

Dice 2 3 1,3 2,3 3,3 4,3 5,3 6,34 1,4 2,4 3,4 4,4 5,4 6,45 1,5 2,5 3,5 4,5 5,5 6,56 1,6 2,6 3,6 4,6 5,6 6,6

)()()()()()()( GPFPEPDPCPBPAP



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Probability Rules- Conditional Probability

The probability of A given that B has already occurred is called a conditional probability

Conditional Probability is calculated from

This can be written as

If A and B are Independent then so that

BAP

)(

)(

)(

)(

BP

APABP

BP

BandAPBAP


)()()( BPAPBandAP

)(APBAP



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Dice Example 1.6

36 Elementary Outcomes Probability of a specific outcome is 1/36 Probability of the event “sum of dice equals 7” is 6/36

Addition-P(A or B) Events “sum=7” and “sum=10” mutually exclusive P=(6/36+3/36) Events “sum=7” and “dice 1=3” not mutually exclusive P=(11/36)

Multiplication-P(A and B) Events A=(6,6) and B=(6,6 repeated) are independent P=(1/36)2

Event A= (6,6) given B=(n1=n2=even) are not independent

Dice 11 2 3 4 5 6

1 1,1 2,1 3,1 4,1 5,1 6,12 1,2 2,2 3,2 4,2 5,2 6,2

Dice 2 3 1,3 2,3 3,3 4,3 5,3 6,34 1,4 2,4 3,4 4,4 5,4 6,45 1,5 2,5 3,5 4,5 5,5 6,56 1,6 2,6 3,6 4,6 5,6 6,6

3/1BAP

3/1BAP



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Medical testing and False positives..

A certain disease affects 1 out of every 1000 people. There is a test that will give a positive result 99% of the time if an individual has the disease. It will also show a positive result 2% of the time for individuals who do not have the disease.

If you test positive, what is the probability that you have the disease?



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Medical testing and False positives….What do we know? What do we want to know?

Define the events as A : person has the disease B : person tests positive

The known information can be written as 1/1000 has the disease

Probability of a positive result for a person with the disease

Probability of a false positive for a person without the disease

We want to know

001.0)( AP

99.0ABP

02.0AnotBP

?whatBAP



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Medical testing and False positives….

Sample space is four mutually exclusive events…

A not AB A and B not A and B

not B A and not B not A and not B



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A not A SumB A and B not A and B P(B)

not B A and not B not A and not B P(not B)P(A) P(not A) 1

Sample space is four mutually exclusive events..the known quantities are

001.0)( AP

999.0)( notAP

00099.0001.099.0)()( APABPBandAP

01998.0999.002.0)()( notAPnotABPBandnotAP



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A not A SumB 0.00099 0.01998 0.02097

not B A and not B not A and not B P(not B)0.001 0.999 1

The rows and columns must add up so

979034.097902.000001.0)( notBP

00001.000099.0001.0)( notBandAP

97902.001998.0999.0)( notBandnotAP



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The probability of actually having the disease given a positive test is then

A not A SumB 0.00099 0.01998 0.02097

not B 0.00001 0.97902 0.979030.001 0.999 1

0472.002097.0

00099.0

)(

)(

BP

BandAPBAP



34


A not A SumB 1 20 21

not B 0 979 9791 999 1000

Even though the test is accurate,less than 5% of those who test positive actually have the disease. This “False Positive Paradox” is one reason repeat or alternative medical tests are often required to establish if a person really has a particular disease.



35

Summary-Probability Basic Definitions

Random Experiments,Outcomes,Sample Spaces ,Events

Probability Properties

The Laws of Chance Classical Probability Relative Frequency Definition Subjective or Bayesian Probability

Probability Rules Addition/Multiplication Conditional Probabiity

L Berkley Davis Copyright 2009 MER301: Engineering Reliability1 LECTURE 1: Basic Probability Theory.

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