1. Overview of basic probability

Transcription

1 13.42 Desg Prcples for Ocea Vehcles Prof. A.H. Techet Sprg Overvew of basc probablty Emprcally, probablty ca be defed as the umber of favorable outcomes dvded by the total umber of outcomes, other words, the chace that a evet wll occur. Formally, the probablty, p of a evet ca be descrbed as the ormalzed area of some evet wth a evet space, S, that cotas several outcomes (evets), A, whch ca clude the ull set,. The probablty of the evet space tself s equal to oe, hece ay other evet has a probablty ragg from zero (ull space) to oe (the whole space). Smple evets are those whch do ot share ay commo area wth a evet space,.e. they are o-overlappg, whereas composte evets overlap (see fgure 1). The probablty that a evet wll be the evet space s oe: ps ( ) = 1. A 1 A 2 A 1 A 2 A 3 A 4 S A 3 A 4 S Smple Evets A Composte Evets A 1. Smple ad composte evets wth evet space, S. 2004, 2005 A. H. Techet 1 Verso 3.1, updated 2/14/2005

2 DEFINE (see fgure 2 for graphcal represetato): UNION: The uo of two regos defes a evet that s ether A or B or both regos. INTERSECTION: The tersecto of two regos defes a evet must be both A ad B. COMPLEMENT: The complemet A s everythg the evet space that s ot A,.e. A. A B A B tersecto; A B uo; A B A A' ot A; A' 2. Uo, Itersecto, ad Complemet. 2004, 2005 A. H. Techet 2 Verso 3.1, updated 2/14/2005

3 1.1. Mutually Exclusve Evets are sad to be mutually exclusve f they have o outcomes commo. These are also called dsjot evets. EXAMPLE: Oe store carres sx kds of cookes. Three kds are made by Nabsco ad three by Keebler. The cookes made by Nabsco are ot made by Keebler. Observe the ext perso who comes to the store to buy cookes. They choose oe bag. It ca oly be made by ether Nabsco OR Keebler thus the probablty that they choose oe made by ether compay s zero. These evets are mutually exclusve. p( A B) = 0 Mutually Exclusve (1) AXIOMS: For ay evet A (1) pa ( ) 0 (2) ps ( ) =1 (all evets) (3) If A, A, A, L, A are a collecto of mutually exclusve evets the: pa ( 1 A 2 A 3 L A ) = =1 p ( A ) Probablty ca be see as the ormalzed Area of the evet, A. Sce pa ( ) = 1 pa ( ) 1 (2) the the probablty of the ull set s zero: p( ) = 1 p ( S ) = 0. (3) Ths holds sce the probablty of the evet space, S, s exactly oe. 2004, 2005 A. H. Techet 3 Verso 3.1, updated 2/14/2005

4 If A B 0 ad A B= A (B A ) where A ad (B A ) are mutually exclusve, the p( A B) = pa ( ) + p ( B ) pa ( B) (4) becomes p( A B) = p ( A ) + p ( B A ) (5) sce B s smply the uo of the part of B A wth the part of B ot A: B = (B A) (B A ). (6) These two parts are Mutually Exclusve thus we ca sum ther probabltes to get the probablty of B. So p( B ) = p ( B A) + p ( B A ) (7) Lookg back to equato 5 we ca substtute for p(b A ) wth p(b) p ( A B). Therefore, the probablty of the evet A B s equal the probablty of A plus the probablty of evet B mus the probablty that A B,.e. p( A B) = pa ( ) + p ( B ) pa ( B). (8) Example 1: Toss a far co. Evet A = heads ad evet B = tals. pa) ( = 05.;p( B ) = 05. Example 2: If A, B, ad C are the oly three evets S ad are mutually exclusve evets, where 2004, 2005 A. H. Techet 4 Verso 3.1, updated 2/14/2005

5 pa) ( = 49 / 100 ad pb ( ) = 48 / 100 the pc ( ) = 3 / 100. Example 3: Roll a sx-sded de. Sx possble outcomes, pa ( ) = 1 / 6. Probablty of rollg a eve umber: p(eve) = 1 / 2 = p(2) + p(4) + p(6). 2. Codtoal Probablty Codtoal probablty s defed as the probablty that a certa evet wll occur gve that a composte evet has also occurred. We wrte ths codtoal probablty as p( A B) ad say "probablty of A gve B". Gve that a composte evet, M (see fgure 3), has happeed what s the probablty that evet A also happeed? By statg that evet M has happeed we the have excluded all evets that do ot overlap wth M as possble outcomes. The mplcato s that ow the evet space has shruk from S to M. Therefore we must redefe the probabltes of the evets such that pm ( ) =1 ad all other evets have pa ( ) = 0 f M A = 0 but f M A 0 (.e. f A has some overlap wth M ) the 0 pa ( ) 1. The greater the overlap, the hgher the probablty of the evet. 2004, 2005 A. H. Techet 5 Verso 3.1, updated 2/14/2005

6 A 1 A 2 A 3 A 4 S M (composte evet) 3. Composte Evet M. Thus for ay two evets A ad B wth pb) ( > 0 the CONDITIONAL PROBABILITY of A gve B has occurred s defed as: p pa ( B ) = ( A B ) pb ( ) (9) whch s coveetly rewrtte as p ( A B) = p ( A B ) p ( ) B (10) ad s commoly referred to as the Multplcato Rule ad s ofte a easer form of equato 9. Example 1: A gas stato s tryg to determe what the average customer eeds from ther stato. The have determed the probablty that a customer wll check oly hs/her ol level or oly hs/her tre pressure ad also the probablty they wll check both. pcheck ( tres ) = p ( T ) = p ( check ol ) = p ( L ) = , 2005 A. H. Techet 6 Verso 3.1, updated 2/14/2005

7 p ( check both ) = p ( B ) = p ( T L) = Ther ext step s to determe the probablty that a perso checks ther ol gve they also checked ther tre pressure. (1) Choose a radom customer ad fd the probablty that a customer has checked hs tres gve he/she checked the ol: pt L) = pt ( L) = 0. ( 01 = 0. 1 (11) pl ( ) 0.1 (2) Choose a radom customer ad fd the probablty that a customer has checked hs ol gve he/she checked the tres: pl ( T ) = pl ( T ) 0. = 01 = (12) pt ( ) 0.2 Example 2: What s the probablty that the outcome of a roll of a dce s 2 ( A ) gve that 2 the outcome s eve? Let the complex evet M be the occurrece of all possble eve umbers. Sce a de s sx sded wth three possble eve umbers, 2,4, ad 6, the probablty that a eve umber wll be rolled s 0.5: M = A 2 A A 4 6 pm ( ) =1 / 2 The probablty that M tersects wth evet A, rollg a two, s 2 pm ( A ) = p ( A ) =1 / 6, (13) , 2005 A. H. Techet 7 Verso 3.1, updated 2/14/2005

8 sce there are three possble eve umbers ad a 50% chace of rollg a eve umber. Thus the probablty that a perso wll roll a two gve that a eve umber s rolled s ( 2 / p( A M ) = pm A = 16 2 = 1 / 3 pm ( ) 1 /2 = 33 % (14) 3. Law of Total Probablty & Bayes Theorem If all evets, A ( = 1: ), are mutually exclusve ad exhaustve, the for ay other evet B, p( B ) = p ( B A ) p ( A ) + L + p ( B A ) p ( A ) (15) 1 1 or p( B ) = p ( B A ) p ( A =1 ). (16) Ths s the Law of Total Probablty. We ca prove ths smply by lookg at the codtoal probablty of each of the evets. Proof: Sce the evets, A, are mutually exclusve ad exhaustve, order for evet B to occur t must exst cojucto wth exactly oe of the evets A. B = ( A 1 ad B ) or ( A 2 ad B ) or Lor ( A ad B ) = ( A 1 B) L ( A B) =1 =1 p ( B A ) p ( A pb ( ) = p ( A B) = ) Ths formulato leads us to Bayes Theorem. Let A, A 2, A 3, L, A k be a collecto of mutually exclusve evets wth pa ( k ) > 0 for all , 2005 A. H. Techet 8 Verso 3.1, updated 2/14/2005

9 k = 1 :. The for ay evet B wth pb ( ) > 0 we get pa ( B) = k p( A k B) pb ( ) (17) usg the multplcato rule ad the Law to total Probablty we get BAYES THEOREM p( A B) = k p( B A k ) p ( A k ) = 1 p( B A ) p ( A ) ; for k = 1, 2, 3,..., 4. Examples Let s look at a few complex examples. Example 1: Suke Treasure! You ve bee hred by a salvage compay to determe whch of two regos the compay should look to fd a shp that sak a hurrcae wth treasure worth bllos. Ths s very exctg, so you pull out your probablty book ad get to work. You kow that group X s usg a sde sca soar that has a success rate of fdg objects equal to 70% (probablty that t fds a object s 0.7) ad that group Y has equpmet that has a success rate of 60%. You also kow that meteorologsts predct that there s a 80% probablty that suke shp ad her treasure le Rego I, at the edge of the cotetal shelf, ad 20% probablty that t s Rego II, beyod the shelf. 2004, 2005 A. H. Techet 9 Verso 3.1, updated 2/14/2005

10 px ( ) = P = 0. 7 ; py ( ) = P = 0. 6 ; pi ( ) = P = 0. 8 ; pii ( ) = P I = X Y I I 0. 2 The two groups have bee searchg the regos for some tme ow. What s the probablty that the treasure s rego I, call ths evet A, gve that perso Y dd ot I fd t rego I, call ths evet B? p( A B) = p(treasure s I Perso I Y dd ot fd t I) = p ( Y dd t fd t I t s I ) p ( t s I ) p( B A I ) p ( A I )+ p ( B A I ) p ( A I ) = (1 P Y ) P I (1 P Y ) P I + P II = = 062. = 62% (0.8) s I P I (0.6) PY 1-PY (0.4) Y fds I =P Y P I =0.48 Y does ot fd =(1-P Y )P I I gve t s I =0.32 P II s II Y dd ot fd t I =P II = 0.2 (0.2) Probablty that Y fds t I f t s I: P Y P I Probablty that Y does t fd t I evethough t s I: (1-P Y )P I Probablty that Y does ot fd t I gve t s II: P II 2004, 2005 A. H. Techet 10 Verso 3.1, updated 2/14/2005

11 4. Probablty tree for the suke treasure problem. The ext day you dscover that addto to Y, X dd ot fd the treasure area I. So what s the probablty that X dd t fd t after Y dd t fd t ad the probablty that t s area I eve though both partes faled to locate t? See the probablty tree fgure 5 (0.8) PI P II (0.2) (0.6) PY 1-PY (0.4) (0.7) PX 1-P X (0.3) Total probablty that X does ot fd t Area I gve that Y dd ot fd t there ether: P II + (1-P X )(1-P Y )P I = Probablty that t s I evethough both X ad Y dd t fd t: P(I X,Y do t fd t) = (1-P X )(1-P Y )P I = P II+ (1-P X )(1-P Y )P I 5. Probablty tree for the suke treasure problem gve that Y dd ot fd t area I. 2004, 2005 A. H. Techet 11 Verso 3.1, updated 2/14/2005

12 5. Recap of Probablty For evets A & B the p( A B) = pa ( ) + p ( B ) pa ( B) space S : pa ( B ) = p( B ) p( A B ) If A s wholly cotaed B the p( A B) = p ( A ) Mutually exclusve evets p( A B) = p ( A ) + p ( B ) A,B: pa ( B) = 0 pa ( B ) = 0 6. Probablty gve multple trals Cosder a group of objects. We ca determe the umber of possble ways to pck k objects from ths group at radom, wthout keepg track of the order whch we pck the objects, as = k k!!( k )! Smply stated: choose k. 2004, 2005 A. H. Techet 12 Verso 3.1, updated 2/14/2005

13 If we perform a expermet wth a success probablty equal to p, the q = 1 p s the probablty of falure. If we repeat the expermet tmes the the probablty of k successes depedet trals, aga payg o atteto to the order whch the successes are obtaed, s k k p( k ) = p k q (18) Thus the sgle evet cotag ( k) falures ad k successes has the probablty k k p q ad there are possble combatos. k 7. Useful Refereces There are may probablty text books each slghtly dfferet. Addtoal textbooks are lsted below. Bertsaks ad Tskls: Text book used for Devore, J "Probablty ad Statstcs for Egeers ad Scetsts" Tratafyllou ad Chryssostomds, (1980) "Evromet Descrpto, Force Predcto ad Statstcs for Desg Applcatos Ocea Egeerg" Course Supplemet. 2004, 2005 A. H. Techet 13 Verso 3.1, updated 2/14/2005