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September 27
th
Lecture
CEE 3000
Incorporating Uncertainty in System Analysis
Axioms (fundamentals of probability theory)
Simple calculations to characterize the uncertainty in systems
Probability – likelihood of an event accruing relative to the set of all possible
events.
1.
Sample Space set of all possible events
{all possible events}
0 =< P(E1) =< 1
P(s) = 1
s= sample space
assumptions
1. discrete random events
2. equally likely outcomes
Sdie = {1 2 3 4 5 6}
^
^
Unbiased
discrete random events
P(1) = P(2) = …P(6) = 1/6
P(odd) = P(1) + P(3) + P(5)= 3/6
½
P(even) = P(2) + P(4) + P(6) = 3/6
½
*we can add bc they are mutually exclusive
2.
Mutually Exclusive Events
Events that cannot happen at the same time
E1∩E2 = Ø
P(E1∩E2) = Ø even
P(E1 or E2)
P(E1UE2) = P(E1) + P(E2)
P(Ø) = 0
3. Overlapping Events
Events that can occur simultaneously
E.g. Flip a coin and a die at the same time
Die
Coin
H
T
1
H
T
2
2H
2T
3
3H
3T
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4H
4T
5
5H
5T
6
6H
6T
P(E1UE2) = P(E1) + P(E2) – P(E1∩E2)
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 Fall '07
 Meyer

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