Engineering Risk Benefit Analysis

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1 Engneerng Rsk Beneft Analyss.55, 2.943, 3.577, 6.938, 0.86, 3.62, 6.862, 22.82, ESD.72, ESD.72 RPRA 2. Elements of Probablty Theory George E. Apostolaks Massachusetts Insttute of Technology Sprng 2007 RPRA 2. Elements of Probablty Theory

2 Probablty: Axomatc Formulaton The probablty of an event A s a number that satsfes the followng axoms (Kolmogorov): 0 P(A) P(certan event) For two mutually exclusve events A and B: P(A or B) P(A) + P(B) RPRA 2. Elements of Probablty Theory 2

3 Relatve-frequency nterpretaton Imagne a large number n of repettons of the experment of whch A s a possble outcome. If A occurs k tmes, then ts relatve frequency s: It s postulated that: lm n k n P(A) k n RPRA 2. Elements of Probablty Theory 3

4 Degree-of-belef (Bayesan) nterpretaton o need for dentcal trals. The concept of lkelhood s prmtve,.e., t s meanngful to compare the lkelhood of two events. P(A) < P(B) smply means that the assessor judges B to be more lkely than A. Subjectve probabltes must be coherent,.e., must satsfy the mathematcal theory of probablty and must be consstent wth the assessor s knowledge and belefs. RPRA 2. Elements of Probablty Theory 4

5 Basc rules of probablty: egaton S Venn Dagram E E E E S P(E) + P(E) P(S) P(E) P(E) RPRA 2. Elements of Probablty Theory 5

6 RPRA 2. Elements of Probablty Theory 6 Basc rules of probablty: Unon ( ) ( ) ( ) I U K j j A P A A P A P A P ( ) A P P UA Rare-Event Approxmaton:

7 Unon (cont d) For two events: P(A B) P(A) + P(B) P(AB) For mutually exclusve events: P(A B) P(A) + P(B) RPRA 2. Elements of Probablty Theory 7

8 Example: Far De Sample Space: {, 2, 3, 4, 5, 6} (dscrete) Far : The outcomes are equally lkely (/6). P(even) P(2 4 6) ½ (mutually exclusve) RPRA 2. Elements of Probablty Theory 8

9 RPRA 2. Elements of Probablty Theory 9 Unon of mnmal cut sets j j T M M M M X ) (... Rare-event approxmaton: ( ) ( ) T M P X P From RPRA, slde j j T M P... P(M M ) P(M ) ) (X P ) (

10 Upper and lower bounds + T ) P(M ) P(MM j) ( P(M ) j + P (X ) P ( X ) T P ( M ) The frst term,.e., sum, gves an upper bound. P(X T ) P(M ) P(MM j) j + The frst two terms, gve a lower bound. RPRA 2. Elements of Probablty Theory 0

11 Condtonal probablty P ( A B) P P ( AB) ( B) ( AB) P( A B) P( B) P( B / A) P( A) P For ndependent events: P ( AB) P( A) P( B) ( A) P( B) P B/ Learnng that A s true has no mpact on our probablty of B. RPRA 2. Elements of Probablty Theory

12 Example: 2-out-of-4 System 2 M X X 2 X 3 M 2 X 2 X 3 X 4 M 3 X 3 X 4 X M 4 X X 2 X X T ( M ) ( M 2 ) ( M 3 ) ( M 4 ) X T (X X 2 X 3 + X 2 X 3 X 4 + X 3 X 4 X + X X 2 X 4 ) - 3X X 2 X 3 X 4 RPRA 2. Elements of Probablty Theory 2

13 2-out-of-4 System (cont d) P(X T ) P(X X 2 X 3 + X 2 X 3 X 4 + X 3 X 4 X + X X 2 X 4 ) 3P(X X 2 X 3 X 4 ) Assume that the components are ndependent and nomnally dentcal wth falure probablty q. Then, P(X T ) 4q 3 3q 4 Rare-event approxmaton: P(X T ) 4q 3 RPRA 2. Elements of Probablty Theory 3

14 Updatng probabltes () The events, H,..., are mutually exclusve and exhaustve,.e., H H j Ø, for j, H S, the sample space. Ther probabltes are P(H ). Gven an event E, we can always wrte H H2 E H 3 P(E) P(E/ H )P(H ) RPRA 2. Elements of Probablty Theory 4

15 Updatng probabltes (2) Evdence E becomes avalable. What are the new (updated) probabltes P(H /E)? Start wth the defnton of condtonal probabltes, slde. P(EH ) P(E / H )P(H ) P(H / E)P(E) P(H / E) P(E / H )P(H P(E) ) Usng the expresson on slde 4 for P(E), we get RPRA 2. Elements of Probablty Theory 5

16 Bayes Theorem Lkelhood of the Evdence P Posteror Probablty ( H E) P ( E H ) P( H ) P ( E H ) P( H ) Pror Probablty RPRA 2. Elements of Probablty Theory 6

17 Example: Let s Make A Deal Suppose that you are on a TV game show and the host has offered you what's behnd any one of three doors. You are told that behnd one of the doors s a Ferrar, but behnd each of the other two doors s a Yugo. You select door A. At ths tme, the host opens up door B and reveals a Yugo. He offers you a deal. You can keep door A or you can trade t for door C. What do you do? RPRA 2. Elements of Probablty Theory 7

18 Let s Make A Deal: Soluton () Settng up the problem n mathematcal terms: A {The Ferrar s behnd Door A} B {The Ferrar s behnd Door B} C {The Ferrar s behnd Door C} The events A, B, C are mutually exclusve and exhaustve. P(A) P(B) P(C) /3 E {The host opens door B and a Yugo s behnd t} What s P(A/E)? Bayes Theorem RPRA 2. Elements of Probablty Theory 8

19 P Let s Make A Deal: Soluton (2) ( A E ) P( E A ) P( A ) P( E A ) P( A ) + P( E B ) P( B ) + P( E C ) P( C ) But P(E/B) 0 (A Yugo s behnd door B). P(E/C) (The host must open door B, f the Ferrar s behnd door C; he cannot open door A under any crcumstances). RPRA 2. Elements of Probablty Theory 9

20 Let s Make A Deal: Soluton (3) Let P(A/E) x and P(E/A) p Bayes' theorem gves: x p + p Therefore For P(E/A) p /2 (the host opens door B randomly, f the Ferrar s behnd door A) P(A/E) x /3 P(A) (the evdence has had no mpact) RPRA 2. Elements of Probablty Theory 20

21 Let s Make A Deal: Soluton (4) Snce P(A/E) + P(C/E) P(C/E) - P(A/E) 2/3 The player should swtch to door C For P(E/A) p (the host always opens door B, f the Ferrar s behnd door A) P(A/E) /2 P(C/E) /2, swtchng to door C does not offer any advantage. RPRA 2. Elements of Probablty Theory 2

22 Random Varables Sample Space: The set of all possble outcomes of an experment. Random Varable: A functon that maps sample ponts onto the real lne. Example: For a de S {,2,3,4,5,6} For the con: S {H, T} {0, } RPRA 2. Elements of Probablty Theory 22

23 Events We say that {X x} s an event, where x s any number on the real lne. For example (de experment): {X 3.6} {, 2, 3} { or 2 or 3} {X 96} S (the certan event) {X -62} (the mpossble event) RPRA 2. Elements of Probablty Theory 23

24 Sample Spaces The SS for the de s an example of a dscrete sample space and X s a dscrete random varable (DRV). A SS s dscrete f t has a fnte or countably nfnte number of sample ponts. A SS s contnuous f t has an nfnte (and uncountable) number of sample ponts. The correspondng RV s a contnuous random varable (CRV). Example: {T t} {falure occurs before t} RPRA 2. Elements of Probablty Theory 24

25 Cumulatve Dstrbuton Functon (CDF) The cumulatve dstrbuton functon (CDF) s F(x) Pr[X x] Ths s true for both DRV and CRV. Propertes:. F(x) s a non-decreasng functon of x. 2. F(- ) 0 3. F( ) RPRA 2. Elements of Probablty Theory 25

26 CDF for the De Experment F(x) / x RPRA 2. Elements of Probablty Theory 26

27 Probablty Mass Functon (pmf) For DRV: probablty mass functon P( X x ) p F P(S) ( x) p for all x x, p normalzaton Example: For the de, p /6 and F(2.3) P( 2) p RPRA 2. Elements of Probablty Theory 27 6 p

28 Probablty Densty Functon (pdf) f(x)dx P{x < X < x+dx} ( x) f P(S) df ( x) dx F( ) f(s)ds F(x) x RPRA 2. Elements of Probablty Theory 28 f(s)ds normalzaton

29 f Example of a pdf () Determne k so that 2 ( x) kx, for 0 x ( x) 0, f otherwse s a pdf. Answer: The normalzaton condton gves: kx dx k RPRA 2. Elements of Probablty Theory 29

30 Example of a pdf (2) F(x) F ( x) 3 x F(0.875) F(0.75) x 2 dx f(x) 3 P{0.75 < X < 0.875} RPRA 2. Elements of Probablty Theory 30

31 Moments Expected (or mean, or average) value [ ] E X m xf j x ( ) x j p dx Varance (standard devaton σ ) E [ ] ( ) 2 X m σ 2 j CRV DRV ( ) ( ) j x ( x m) j m 2 f 2 x p dx RPRA 2. Elements of Probablty Theory 3 j CRV DRV

32 Percentles Medan: The value x m for whch F(x m ) 0.50 For CRV we defne the 00γ percentle as that value of x for whch x γ f x ( ) dx γ RPRA 2. Elements of Probablty Theory 32

33 Example m 3x dx σ F ( x 0 75) 3. x dx σ ( x ) x 0. 5 x 0 79 m m m. 3 x x x x RPRA 2. Elements of Probablty Theory 33