Why need probabilistic approach? Rain probability
How does that affect our behaviour?? ?
Why need probabilistic approach? Rain probability
How does that affect our behaviour?
? ?
Uncertainties in Engineering
Natural Hazards
Material Properties
Design Models
Construction Errors
Absolute Safety Not Guaranteed
Engineers need to:
model, analyze, update uncertainties
evaluate probability of failure
Questions
What is acceptable failure probability?
- stadium vs shed
Questions
Should one want to be conservative if a perfectly safe system is possible?
- overbooking in airlines
- parking permits
Questions
Should one minimize risk if money is not a problem?
- system consideration –
e.g.dam
Trade-off Decision Analysis
Risk vs. consequence
System risk
Formal analysis of uncertainties and probability
Not all problems can be solved by analysis of data
Set Theory
Sample space: collection of all possibilities
Sample point: each possibility
Event: subset of sample space
Probability Theory
Union: either E1 or E2 occur E1∪E2
Intersection:
both E1 and E2 occur E1∩ E2 or E1 E2
Examples
A B No communication between A and B = E1E2 C B A No communication between A and B = E3∪E1E2 1 2 E1 = road 1 closed
E2 = road 2 closed E3 = road 3 closed 1 2 3
Example - pair of footings
1 2 E1 = 1 settles
Ē1 = 1 does not settle
E2 = 2 settles
Ē2 = 2 does not settle Settlement occurs = E1∪E2 Tilting occurs = E1Ē2∪Ē1E2
de Morgan’s rule
1 2 Water Supply E = failure in water supply = E1∪E2
no failure in water supply = Ē = E1∪E2
E1 = pipe 1 breaks
E2 = pipe 2 breaks
Event of “no failure” Extension to n events
de Morgan #2
Basis of Probability Estimation
judgment + limited observation Subjective assumption e.g. P(Q) = 1/2
Relative frequency e.g. P(Q)=502/1000
Bayesian (a)+(b)
Probability of Union in general E1 E2
Using de Morgan’s rule
P (intersection) conditional
probability
or
Statistical independence if E1 and E2 are s.i. or s.i.
Example: E2 = flood in 廣西 on June E3 = flood in 哈爾濱 on June E1 = flood in 廣東 on June P(E1) = 0.1; P(E2)=0.1; P(E3) = 0.1 E1 and E2 are not s.i. E1 and E3 are s.i.
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