Transcript Lecture1

MER301: Engineering
Reliability
LECTURE 1:
Basic Probability Theory
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
1
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


L Berkley Davis
Copyright 2009
Addition/Multiplication
Conditional Probabiity
MER301: Engineering Reliability
Lecture 1
2
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
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
3
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

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)
 Event

L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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A Random Experiment –
Example 1.1
 Coin Toss Exercise- flip 3 times
and record the results




L Berkley Davis
Copyright 2009
Outcome
Sample Space
Event
Properties of Probability
MER301: Engineering Reliability
Lecture 1
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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
P(oi )  1 / 8.. for..all ..oi
 P (o )  1 / 8  1 / 8  1 / 8  1 / 8  1 / 8  1 / 8  1 / 8  1 / 8  1
i
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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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
0  P(oi )  1



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)
P(o1 )  P(o2 )       P(on )  1
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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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
P( A)  1  P(not  A)


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
P( HHx )  P( HHH )  P( HHT )  1 / 8  1 / 8  1 / 4  1  P(notHHx )
P(notHHx )  P( HTH )  P( HTT )  P(THH )  P(THT )  P(TTH )  P(TTT )
or
P(notHHx )  1 / 8  1 / 8  1 / 8  1 / 8  1 / 8  1 / 8  3 / 4  1  P( HHx )  1  1 / 4
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
8
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

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)
 Event

L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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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



L Berkley Davis
Copyright 2009
Finite
Countable
Uncountable
MER301: Engineering Reliability
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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
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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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, …}
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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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,)|02, 0rR}
 The r and theta distributions are continuous
A second is the number of points scored in each
outcome
 S={1,2,3}
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
13
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

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)
 Event

L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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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
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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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
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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Summary of Probability Properties

Rule 1


Rule 2


If an elementary outcome/event is certain, the probability is
one
Rule 4


If an elementary outcome/event cannot occur, the probability
is zero
Rule 3


The probability of an elementary outcome/event will be a
number between zero and one
The sum of probabilities of all the elementary outcomes or
events in a sample space is one
Rule 5

L Berkley Davis
Copyright 2009
The probability an elementary outcome/event will not occur is
one minus the probability that the outcome/event will occur
MER301: Engineering Reliability
Lecture 1
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Summary of Probability Properties-Text

Rule 1


Rule 2


If an elementary outcome/event is certain, the probability is
one
Rule 4


If an elementary outcome/event cannot occur, the probability
is zero
Rule 3


The probability on an elementary outcome/event will be a
number between zero and one
The sum of probabilities of all the elementary outcomes or
events in a sample space is one
Rule 5

L Berkley Davis
Copyright 2009
The probability an elementary outcome/event will not occur is
one minus the probability that the outcome/event will occur
MER301: Engineering Reliability
Lecture 1
18
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
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
19
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.
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
20
Probability Rules
 Addition ( A or B)

A and B are mutually exclusive

A and B are not mutually exclusive
P( A  or  B)  P( A  B)  P( A)  P( B)
P( A  and  B)  P( A  B)  0
P( A  or  B)  P( A  B)  P( A)  P( B)  P( A  and  B)
 Multiplication( A and B)

A and B are independent
P( A  and  B)  P( A)  P( B)

A and B are not independent/conditional probability
P( A  and  B)  P A B  P( B)  P B A  P( A)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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Dice Example 1.6
Dice 1
3
3,1
3,2
3,3
3,4
3,5
3,6
1
2
4
5
6
1
1,1
2,1
4,1
5,1
6,1
2
1,2
2,2
4,2
5,2
6,2
Dice 2
3
1,3
2,3
4,3
5,3
6,3
4
1,4
2,4
4,4
5,4
6,4
5
1,5
2,5
4,5
5,5
6,5
6
1,6
2,6
4,6
5,6
6,6
 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)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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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
S  1,1 : 1,2 : 1,3 : ....1,6 :, 2,1 : .....2,6 :, 3,1 : .........6,6
A  1,5;2,4;3,3;4,2; 5,1
B  1,1;2,2;3,3;4,4;5,5; 6,6

C  2,1;2,2;2,3;2,4;2,5;2,6,4,1;.....4,6;6,1;......6,6;1,2;1,4;....1,6....
For the Sample Space S, N=36 and the probability of an event is
the number of outcomes in that event divided by N



L Berkley Davis
Copyright 2009
A=5/36
B=6/36
C=27/36
MER301: Engineering Reliability
Lecture 1
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Probability Rules
The Dice Game of Craps
and Probability Rules
 Addition ( A or B)
 Player has two Dice
 1st Roll
A and B are mutually exclusive

A and B are not mutually exclusive
P( A  or  B)  P( A  B)  P( A)  P( B)
P( A  and  B)  P( A  B)  0
P( A  or  B)  P( A  B)  P( A)  P( B)  P( A  and  B)
 Multiplication( A and B)
 7 or 11 wins immediately

A and B are independent

A and B are not independent/conditional probability
P( A  and  B)  P( A)  P( B)
P(7  or 11)  8 / 36  22%
 2,3, or 12 loses immediately

P( A  and  B)  P A B  P( B)  P B A  P( A)
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 1
P(2  or  3  or 12)  4 / 36  11%
 Rolls of 4,5,6,8,9,10 Continue
 Subsequent Rolls
Dice 2
1
2
3
4
5
6
1
1,1
1,2
1,3
1,4
1,5
1,6
2
2,1
2,2
2,3
2,4
2,5
2,6
Dice 1
3
3,1
3,2
3,3
3,4
3,5
3,6
4
4,1
4,2
4,3
4,4
4,5
4,6
5
5,1
5,2
5,3
5,4
5,5
5,6
 Continue to roll until get the same number as on
1st roll (win) or a 7 (lose)
 Probability of Winning
 Overall
L Berkley Davis
Copyright 2009
P( A  B  C  D  E  F  G )
MER301: Engineering Reliability
Lecture 1
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6
6,1
6,2
6,3
6,4
6,5
6,6
The Dice Game of Craps
 Define the following probabilities







Let
Let
Let
Let
Let
Let
Let
1
2
3
4
5
6
1
1,1
1,2
1,3
1,4
1,5
1,6
2
2,1
2,2
2,3
2,4
2,5
2,6
Dice 1
3
3,1
3,2
3,3
3,4
3,5
3,6
4
4,1
4,2
4,3
4,4
4,5
4,6
5
5,1
5,2
5,3
5,4
5,5
5,6
A= probability of 7 or 11 on 1st roll
B= probability of 4 on 1st roll and then another 4 before a 7
C= probability of 5 on 1st roll and then another 5 before a 7
D= probability of 6 on 1st roll and then another 6 before a 7
E= probability of 8 on 1st roll and then another 8 before a 7
F= probability of 9 on 1st roll and then another 9 before a 7
G= probability of 10 on 1st roll and then another 10 before a 7
Dice 2
 These are mutually exclusive events so the
probability of winning is
P( A)  P( B)  P(C )  P( D)  P( E )  P( F )  P(G )
or P( win )  P( A)  P( B)  P(C )  P( D)  P( E )  P( F )  P(G )  0.4929
 The House always wins…
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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6
6,1
6,2
6,3
6,4
6,5
6,6
Probability Rules- Conditional Probability
 The probability of A given that B has already
occurred is called a conditional probability
P AB
 Conditional Probability is calculated from
P( A  and  B) P B A  P( A)
P AB 

P( B)
P( B)
 This can be written as
P( A  and  B)  P A B  P( B)  P B A  P( A)
 If A and B are Independent thenP
A B  P(A) so that
P( A  and  B)  P( A)  P( B)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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Dice Example 1.6
Dice 1
3
3,1
3,2
3,3
3,4
A B  1 / 33,5
3,6
1
2
4
5
6
1
1,1
2,1
4,1
5,1
6,1
2
1,2
2,2
4,2
5,2
6,2
Dice 2
3
1,3
2,3
4,3
5,3
6,3
4
1,4
2,4
4,4
5,4
6,4
5
1,5
2,5P
4,5
5,5
6,5
6
1,6
2,6
4,6
5,6
6,6
 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 P A B  1 / 3
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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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?
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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Medical testing and False positives….What do
we know? What do we want to know?
 Define the events as


A:
B:
person has the disease
person tests positive
 The known information can be written as

P( A)  0.001

P B A  0.99

Probability of a positive result for a
person with the disease
P B not  A  0.02
 We want to know
L Berkley Davis
Copyright 2009
1/1000 has the disease
Probability of a false positive for a
person without the disease
P A B  what ?
MER301: Engineering Reliability
Lecture 1
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Medical testing and False positives….
B
not B

A
not A
A and B
not A and B
A and not B
not A and not B
Sample space is four mutually exclusive events…
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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Medical testing and False positives….
A
B
not B
A and B
A and not B
P(A)
not A
Sum
not A and B
P(B)
not A and not B P(not B)
P(not A)
1
 Sample space is four mutually exclusive events..the
known quantities are
P( A)  0.001
P(notA)  0.999
P( A  and  B)  P B A  P( A)  0.99  0.001  0.00099
P(notA  and  B)  P B notA  P(notA)  0.02  0.999  0.01998
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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Medical testing and False positives….
B
not B
A
not A
Sum
0.00099
0.01998
0.02097
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
P( A  and  notB)  0.001  0.00099  0.00001
P(notA  and  notB)  0.999  0.01998  0.97902
P(notB)  0.00001  0.97902  0.979034
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
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Medical testing and False positives….
B
not B
A
not A
Sum
0.00099
0.01998
0.02097
0.00001
0.97902
0.97903
0.001
0.999
1
 The probability of actually having the disease given
a positive test is then
P AB 
L Berkley Davis
Copyright 2009
P( A  and  B) 0.00099

 0.0472
P( B)
0.02097
MER301: Engineering Reliability
Lecture 1
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Medical testing and False positives….
B
not B
A
not A
Sum
1
20
21
0
979
979
1
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.
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 1
34
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


L Berkley Davis
Copyright 2009
Addition/Multiplication
Conditional Probabiity
MER301: Engineering Reliability
Lecture 1
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