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UNCERTAINTY
Nature of uncertainty
Sources of uncertainty
Describing uncertainty
Concept of probability
History of probability
Describing probability
Typical probabilities
Risk and reliability
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Laplace’s Demon
A Philosophical Essay on Probabilities
Pierre-Simon Laplace (1814)
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
3
“
”
Solvay Conference, 1927
4
God does not play dice!
Einstein! Stop telling God what
to do.
2p
Describing Uncertainties
• Probability
• Most prevalent
• A number between 0 and 1
• Examples:
• Probability that an earthquake occurs with M≥8.0
• Probability that the drift of a structure exceeds 2%
• Fuzzy approaches
• Possibilistic methods
• Entropy
• A measure of the unpredictability of a phenomenon
• Very low entropy: Almost unpredictable
• Very high entropy: Very predictable
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Meaning of Probability
• A number between 0 and 1
• 0 No occurrence
• 1 Certain occurrence
Two schools of thought amongst Statisticians:
• Frequentist (Objectivist)
• Probability means the frequency of occurrence
• Bayesian (Subjectivist)
• Probability means the “degree of belief” in occurrence
• β-club in the structural reliability community
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( ) lim E
n
nP E
n
( ) Degree of belief in occurrence of“ ” P E E
History of Probability
1545: Gerolamo Cardano
1st paper on probabilities
1654: Blaise Pascal
First usage of probabilities to solve problems
in communication with Pierre de Fermat
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Pascal to Farmat, 1654 Monsieur,
Impatience has seized me as well as it has you, and although I am still abed, I cannot refrain from telling you that I received your letter in regard to the problem of the points yesterday evening from the hands of M. Carcavi, and that I admire it more than I can tell you. I do not have the leisure to write at length, but, in a word, you have found the two divisions of the points and of the dice with perfect justice. I am thoroughly satisfied as I can no longer doubt that I was wrong, seeing the admirable accord in which I find myself with you.
اظار از وی تان ستن، بستر در ز چراگ را، شوا ک وچاى ربد هرا صبری بی دستاى از دیرز اهتیازات، هسال هرد در را شوا اه ک کن خدداری هضع ایي
تحسیي را اه ایي کن، بیاى بتان آک از بیش ک ایي یس ودم، دریافت کارکای کاهل عدالت با را اهتیازات تقسین شی شوا کلو، یک در اها دارم، اطال هجال .هی کن اید را خد بد ام اشتبا بر ک دارم شکی دیگر ک طری ب خرسدم کاهال هي .یافت .هی بین شوا با ستدی هافقتی در اکى
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Pascal to Fermat, 1654 Let us suppose that the first of them has two (points) and the other one. They now play one throw of which the chances are such that if the first wins, he will win the entire wager that is at stake, that is to say 64 pistoles. If the other wins, they will be two to two and in consequence, if they wish to separate, it follows that each will take back his wager that is to say 32 pistoles. Consider then, Monsieur, that if the first wins, 64 will belong to him. If he loses, 32 will belong to him. Then if they do not wish to play this point, and separate without doing it, the first should say “I am sure of 32 pistoles, for even a loss gives them to me. As for the 32 others, perhaps I will have them and perhaps you will have them, the risk is equal. Therefore let us divide the 32 pistoles in half, and give me the 32 of which I am certain besides.” He will then have 48 pistoles and the other will have 16. Now let us suppose that the first has two points and the other none, and that they are beginning to play for a point. The chances are such that if the first wins, he will win all of the wager, 64 pistoles. If the other wins, behold they have come back to the preceding case in which the first has two points and the other one. But we have already shown that in this case 48 pistoles will belong to the one who has two points. Therefore if they do not wish to play this point, he should say, “If I win, I shall gain all, that is 64. If I lose, 48 will legitimately belong to me. Therefore give me the 48 that are certain to be mine, even if I lose, and let us divide the other 16 in half because there is as much chance that you will gain them as that I will.” Thus he will have 48 and 8, which is 56 pistoles.
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Pascal to Fermat, 1654
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2 1
64 0
32 32
Next game:
32+16=48 16
2 0
64 0
48 16
Next game:
48+8=56 8
48 16
History of Probability (Continued)
1657: Christiaan Huygens
1st book on probabilities
De ratiociniis in ludo aleae
(On Reasoning in Games of Chance)
1713: Jacob Bernoulli (one of six bothers!)
The Art of Guessing
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History of Probability (Continued)
1760: Thomas Bayes
Bayes’ Theorem
An Essay towards solving a Problem
in the Doctrine of Chances
1814: Pierre-Simon Laplace
Analytical Theories of Probability
1933: Andrey Kolmogorov
Formulated the axioms of the probability theory
Foundations of the Theory of Probability 12
Representation of Probabilities
• Probability itself!
• Chance, described by percentage
• Odds
• Outcomes “for” : Outcomes “against”
• Outcomes “for” in total outcomes
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10.333 or
3p p
33.3%
1: 2 (1 to 2)
:n m
in a
a b pb
1 in 3
n
pn m
Typical Probabilities
Death/h (10-9) Exposure h/y Prob./y (10-6)
• Alpine climbing 40,000 50 2,000
• Smoking 2,500 400 1,000
• Boating 1,500 80 120
• Air travel 1,200 20 24
• Car travel 700 300 210
• Train travel 80 200 16
• Coal mining (UK) 210 1500 315
• Construction work 70-200 2,200 154-440
• Building fires 1-3 8,000 8-24
• Structure failures 0.02 6,000 0.12
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SET THEORY
Definitions
Venn diagram
Operations
MECE
De Morgan rules
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Set Theory • Set Theory is the official language for defining events
• Kolmogorov employed this theory to formulate the axioms of the Probability Theory
• Set Theory Defines events
• Probability Theory Computes the probability of events
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Definitions
• Sample space: Set of all possible events S
• Discrete (finite or infinite)
• Continuous (always infinite)
• Sample point x
• Event: Collection of sample points E
• Complement of an event
• Certain event S
• Null event
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No damage, Light damage, Heavy damage, CollapseS
1,2,3,4,S
| 0S x x
E
1 2
3
1,2 | 4 6
Heavy damage, Collapse
E E x x
E
7 sets
Venn Diagram
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S
x E
Discrete
x .
E
S
Continuous
Operations
• Union
• Intersection
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“or” E1
E2
21 EE
“and” E1
E2
21 EE 1 2 1 2E E E E
1 2E E
Operation Rules
• Commutative rule
• Associative rule
• Distributive rule
• Intersection operations take precedence over union operations, unless specified otherwise by parenthesis 20
1221 EEEE 1221 EEEE
321321 EEEEEE 321321 EEEEEE
1 2 3 1 3 2 3E E E E E E E 3231321 EEEEEEE
Mutually Exclusive & Collectively Exhaustive (MECE) Events
• Mutually exclusive events
• Collectively exhaustive events
• For n sets:
21
E1 E2 E3 … En
1 2E E
1 2E E S
De Morgan’s Rules
• The complement of a union is equal to the intersection of the complements
• The complement of an intersection is equal to the union of the complements
• De Morgan’s rules convert “series system” problems to “parallel system” problems, and vice versa
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1 2 1 2n nE E E E E E1 2 1 2E E E E
nn EEEEEE 2121 1 2 1 2E E E E
De Morgan’s Rules
• Proof by Venn diagram
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E1 E2
21 EE 21EE
PROBABILITY THEORY
Axioms
Frequentist vs. Bayesian
Probability rules
Statistical dependence
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Axioms
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Formulated in 1933 by Andrey Nikolaevich Kolmogorov
1 2
1 2 1 2
For an event
0
For the certain event
1
For mutually exclusive events and
( ) ( )
( )
( )
E
P E
S
P S
E E
P E E P E P E
1
2
3
Probability Rules • Probability of the complement
• Union rule
• Inclusion-exclusion rule
• Conditional probability rule
• Multiplication rule
• Bayes’ rule
• Rule of total probability
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)(1)( EPEP
)()()()( 212121 EEPEPEPEEP
1 2 3 1 2 3 1 2
1 3 2 3 1 2 3
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
P E E E P E P E P E P E E
P E E P E E P E E E
)(
)()|(
2
2121
EP
EEPEEP
)()|()( 22121 EPEEPEEP
)()(
)|()|( AP
EP
AEPEAP
n
i
ii EPEAPAP
1
)()|()(
Complement Probability
• All rules are derived from the axioms
• Consider the complementary events E and E
• They are mutually exclusive and collectively exhaustive
• From second axiom:
• From third axiom:
• Hence
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1)()( EEPSP
)()()( EPEPEEP
)(1)( EPEP
Union Rule • Consider a Venn diagram with two events
• Subtract the area that we count twice when P(E1E2) is nonzero
• The inclusion-exclusion rule extends the Union Rule to more than two events
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E1 E2
)( 21EEP
)()()()( 212121 EEPEPEPEEP
1 2 3 1 2 3
1 2 1 3 2 3
1 2 3
( ) ( ) ( ) ( )
( ) ( ) ( )
( )
P E E E P E P E P E
P E E P E E P E E
P E E E
Conditional Probability Rule
• This rule may be justified by the frequency notion of probability
• E2 can be considered as the new outcome space
• The Multiplication rule is a direct consequence of the conditional probability rule:
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)|()(
)(21
2
12
2
12
2
21 EEPn
n
n
nn
n
EP
EEP
E1
E2
)(
)()|(
2
2121
EP
EEPEEP
1 21 2
2
( )( | )
( )
P E EP E E
P E
)()|()( 22121 EPEEPEEP
Bayes’ Rule
• Conditional probability rule
• Multiplication rule
• Combine to obtain Bayes’ rule
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1 21 2
2
( )( | )
( )
EP EP E E
P E
2 1 11 2( ) ( | ) ( )P E P E E P EE
2 11 2 1
2
( | )( | ) ( )
( )
P E EP E E P E
P E
“Posterior”
“Likelihood” “Normalization”
“Prior”
Bayes’ Theorem
• The Bayes’ theorem is describes the application of the Bayes’ rule
• The theorem expresses how a subjective degree of belief should rationally change to account for evidence
• This is Bayesian inference, which is fundamental to Bayesian statistics
• Suppose the probability distribution of the earthquake occurrence rate is described by f(λ)
• Upon observing another earthquake in the time interval x:
31 ( | )
( | ) ( )( )
xff f
f xx
2 11 2 1
2
( | )( | ) ( )
( )
P E EP E E P E
P E
Rule of Total Probability
• Extensively employed in the performance-based engineering framework put forward by PEER
• For a set of MECE events, Ei:
32
n
i
ii
n
i
i
n
n
EPEAP
AEP
AEAEAEP
EEEAP
ASPAP
1
1
21
21
)()|(
)(
)(
)( )(
E1 E2 E3
… En
A
The Monty Hall Problem
33
The Monty Hall Problem
34
The Monty Hall Problem
35
Should you switch?
The Monty Hall Problem
36
Each door has a 1 in 3 chance of hiding the grand prize. Suppose we begin by choosing door #1.
In this case Monty may open either door #2 or #3
In both of these cases, Monty is forced to reveal the only other zonk.
The Monty Hall Problem
37
So what happens when you switch?
In this case you were right the first time. You lose!
In both of these cases, you switch to the correct door. You win!
The Monty Hall Problem
38
Generalized Monty Hall Problem
39
