Quantitative Techniques

FPST 4333

Review 2

Dr. Q. Wang, PhD, PE, CSP Dale F. Janes Endowed Professor

Associate Professor and Program Director

1. Probability

3

Probability Axioms

1.

2. Total probability:

3. For mutually exclusive A, B:

0 Pr A 1

Pr A OR B Pr A Pr B

Pr S 1

These axioms are the basis for all probability calculations.

4

Intersection of A and B

If A and B are (statistically) independent, the

probability that A and B occur simultaneously

is the product of probabilities that A and B

occur individually.

Pr(A ∩ B) = Pr(A AND B) = Pr(A) Pr(B)

In this case, the Pr(A) is not affected by the value

of Pr(B) and Pr(B) is not affected by Pr(A).

5

If A and B are statistically dependent, the probability

that A and B occur simultaneously is

Pr(A ∩ B) = Pr(A|B) Pr(B)

Pr(A ∩ B) = Pr(B|A) Pr(A)

which is the general expression. Then if A and B are

independent

Pr(A ∩ B) = Pr(A) Pr(B)

Intersection of Dependent A and B

Pr(A ∩ B) = Pr(B) Pr(A)

6

Union of A and B

Expression for calculating the probability of A∪B:

Pr(A∪B) = Pr(A) + Pr(B) – Pr(A∩B)

can be rewritten, based on earlier, as:

1. Pr(A∪B) = Pr(A) + Pr(B) – Pr(A) Pr(B|A),

= Pr(A) + Pr(B) – Pr(B) Pr(A|B),

or if A and B are statistically independent, as:

2. Pr(A∪B) = Pr(A) + Pr(B) – Pr(A) Pr(B)

Exposure Time

Length of time a component is

effectively exposed to failure during

system operation.

λ = 1/mean time to failure (MTTF) or the

failure rate.

R(t) = e-λt

P(t) = 1- e-λt = λt μ = MTTF = 1/λ

Exponential Reliability

8

R(t) et

F(t) 1et

t, 0.1 year

Pro

babili

ty

λ = 1/yr

Slope : f (t) et

9

Example

For a component with a failure rate of

0.05 per year, find the reliability over a

10 year period and the probability of

failure in 10 years.

2. FTA

Logical Connection Between Events

AND: The resulting output event requires the simultaneous occurrence of all input event.

e.g. Event C will occur only if both events A and B occur simultaneously, which is represented (for independent events, A, B) by

A • B = C

OR: The resulting output event requires the occurrence of any individual input event

e.g. C will occur if either A or B occurs, which is represented (when A, B each has low probability of occurring) by

A + B = C

A B

C

A B

C

AND-gate

OR-gate

Cut Sets

Identify Cut Sets

13

Data Requirement

FT Reduction with Boolean Algebra, 2

Boolean Identities

A • A = A A AND A equals A

A + A = A A OR A equals A

A + A • B = A A OR (A AND B) equals A

B

Fault Tree Reduction

T = A • (B+C+D+E) • (B+C+F+G+H)

T = A•(B•B + B•C + B•F + B•G + B•H

+ C•B + C•C + C•F + C•G + C•H

+ D•B + D•C + D•F+D•G + D•H

+ E•B + E•C + E•F + E•G + E•H)

T = A•(B+C+D•F+D•G+D•H+E•F+E•G+E•H)

T = A•{B + C + (D + E)•(F + G + H)}

Pitfall

Plan ahead

AND gate overconfidence

The effect of a repeated event

Gate calculation

MOE Error

FTA Reduction

3. ETA

Event Tree Structure

Initiating

Event

Safety Functions Outcomes

Event 1

(P1)

Event 2

(P2)

Event 3

(P3)

Initiating

Failure Rates

Loss of cooling: 1 event/year frequency

Hardware safety functions: Failure probability on

demand = 0.01 failure/demand

Operator notices high Temp 3/4 times; Operator

adjusts coolant flow 3/4 times. Failure probability

(for each)= 0.25 failure/demand

Operator shuts down system 9/10 times. Failure

probability = 0.10 failure/demand

Add the occurrence probabilities

ET Frequency Computations

Obtain frequency of a downstream event by multiplying the probability of the event times the frequency of the initiating event.

Freq of an event =

Probability of the event x Frequency of initial event

Hot Oil Heating System

Initiating event

Exam II

50 minutes

Cover Statistics, FTA, ETA

Multiple choices

Analysis

Calculations