ELEMENTARY AND COMPOUND EVENTS PROBABILITY

Transcription

1 Euopea Joual of Basic ad Applied Scieces Vol. 5 No., 08 ELEMENTARY AND COMPOUND EVENTS PROBABILITY William W.S. Che Depatmet of Statistics The Geoge Washigto Uivesity Washigto D.C williamwsche@gmail.com ABSTRACT The objective of is pape is to eview some kow elemetay ad compoud evets. We will eview e samplig wi ad wiout eplacemet cases i ode to see e effects o e pobability. This will also chage e weightig pocess i suvey samplig. Fially, we will discuss two useful compoud evets, e Baach Matchbox Poblem ad e Poisso Negative Biomial Distibutio i compoud Evets. These two distibutios have bee foud useful i isk aalysis. All eoies ae accompaied wi examples to explai e meaig of ese eoies. Maematical Subject Classificatio: 6PXX Some Key Wods ad Phases: Baach Matchbox Poblem, Depedet ad Idepedet Evets, Elemetay ad Compoud Evets, Empty Box, Fid Empty Box, Poisso Negative Biomial Distibutio, Pobability, Wi o Wiout Replacemet..INTRODUCTION A evet A is said to be a compoud evet, if it ca be epeseted as e sum of two evets at ae bo diffeet fom A: A B C,B A,C A. Evets at do ot pemit ay such epesetatio ae said to be elemetay evets. Thee ae some iteestig elatioships betwee elemetay ad compoud evets. We povide a shot list, but o pove of ese facts: It ca be show at e poduct of two distict elemetay evets is 0; We kow at fo evey compoud evet B, ee exists a elemetay evet A such at A B holds; 3 I a algeba cosistig of a fiite umbe of evets,evey evet ca be epeseted as a sum of elemetay evets. This epesetatio is uique except fo e ode of e evets. 4 The umbe of evets of a fiite algeba of evets is ecessaily a powe of. Istead of povig ese facts we will discuss some of geeally useful evets at may elate to oe aeas study. The evets wi o wiout eplacemet could chage e pobability, ad us effect e weighig pocess i a sample suvey. I sectio 3, we give ee classical examples at have bee foud useful i isk aalysis. I geeal, it is ot easy to fid ei pobabilities. I example 3. ad 3., we study e Baach Matchbox Poblem ad moe complex situatio, fist empty but ot e fist oe foud empty. These situatios ca be exteded to eal life applicatios. I example 3.3, we had applied e Poisso distibutio compoud wi Negative Biomial Distibutio. The easo fo usig is distibutio is poved maematically, ad ot assumed. We give is pove fact i ou cocludig emaks.. Elemetay Evets Daw a odeed sample of size k, 0 k, wiout eplacemet fom a populatio of size. The total umbe Pogessive Academic Publishig, UK Page 9

2 Euopea Joual of Basic ad Applied Scieces Vol. 5 No., 08 of possible samples deoted by ( k, is ( k ( (...( k ( k is also kow as e pemutatio of objects take k at a time. By a pemutatio of objects we mea ay aagemet of ese objects. This implies at e total umbe of pemutatios of objects is (!. Howeve, if we coce e samplig wi eplacemet, e e total umbe of samples of size k wi k eplacemet fom a populatio of size is. We apply ese meods to e followig examples. Example. A elevato stats wi 0 passeges ad stops at 5 floos. Fid e pobability at o two passeges leave at e same floo. Let A: evet at o two passeges get off o e same floo. The e umbe of favoable evet A is 5 *4*3***0* 9* 8*7*6 * 0 ad sample space umbe 5. Theefoe e pobability of evet A is #( A 5* 4* 3* * * 0* 9* 8*7* 6 P( A #( Example. Suppose i a populatio of elemets, a adom sample of size is take. Fid e pobability at oe of k pescibed elemets is i e sample if e meod used is :(a samplig wiout eplacemet; (b samplig wi eplacemet. a. Give at e populatio size is, e sample size is, ad e samplig is doe wiout eplacemet. So e umbe of sample space is #( ( (...( Defie A: e evet at oe of k pescibed objects is i e sample. If e k pescibed objects ae excluded fom e sample, e e sample of objects must be selected fom e emaiig (-k objects. This ca be doe ( k ( ways. Thus, e umbe of favoed evets A ( k ( ad ( k ( P( A. ( b. I is case, e samplig is doe wi eplacemet ad e umbe of sample space #(. Defie A evet e same as pat (a. We have e umbe of favoed evet A: ( k k #( A ( k. So at P( A (. Example.3: Suppose balls ae distibuted ito boxes so at all of e possible aagemets ae equally likely. Compute e pobability at oly box is empty. The pobability space i is case cosists of equally likely poits. Let A be e evet at oly box is empty. This ca happe oly if e balls ae i e emaiig - boxes i such a mae at o box is empty. Thus, exactly oe of ese (- boxes must have two balls, ad e emaiig (- boxes must have exactly oe ball each. Let B j be e evet at box j, j=,3, has two balls, box has o balls, ad e emaiig (- boxes Pogessive Academic Publishig, UK Page 30

3 Euopea Joual of Basic ad Applied Scieces Vol. 5 No., 08 have exactly oe ball each. The e B j ae disjoit evets ad A U B j. To compute j P( B j obseve at e two balls put i box j ca be chose fom balls i ways. The (- balls i e emaiig (- boxes ca be eaaged i (- ways. Thus e umbe of distict ways at we ca put two balls ito box j, o ball i box, ad exactly oe ball i (! each of e emaiig boxes is (! so P( B j ad cosequetly ( (! (! P( A P( B j j P(A x0 4 0.x0 6 Based o above tabulatio, we ca see at as e sample size icease fom 5 to 0 ad e chace of evet A occu quickly dop to zeo. 3. Defie e Radom Vaiables Istead of defiig e evets, we sometime pefe to defie e adom vaiables i a sample space. I is sectio we itoduce e discete adom vaiable, amely e egative biomial distibutio. I a epeated idepedet Beoulli tial wi pobability p fo success util obtaiig e success, we defie e adom vaiable x : umbe of e tials util e success occus. Suppose e success occu at e (k tial, k=0,,,. This meas at ee ae k F s util e success. Thee ae exactly k F s i (k+- tials follows by a success at e (k tials, whee ese evets occu wi pobability k - k q p k - p eefoe p q k P( x k k k k - k P(,k, p p q k 0,,,3...,,,3,... k 0 oewise The above Negative Biomial Distibutio is also kow as e Pascal Distibutio. We foud it vey helpful i fidig some difficult pobabilities. The ext two examples will demostate is. Pogessive Academic Publishig, UK Page 3

4 Euopea Joual of Basic ad Applied Scieces Vol. 5 No., 08 Example 3. (Baach s Matchbox Poblem A maematicia caies two matchboxes, oe i each of e two pockets of his coat. Wheeve he wishes to light a cigaette, he chooses a pocket at adom ad uses a match fom e box i at pocket. Suppose each box cotais N matches to begi wi. Fid e pobability at ee ae k matches i e oe pocket whe he fids at e box i oe pocket is empty. (Remembe to diffeetiate betwee empty box is diffeet fom fids empty box Defie A : e evet at oe of e matchboxes is foud empty whe e oe box cotais exactly k matches. Let AR ( AL : e evet at e ight (left pocket is foud to cotai e empty match box while e left (ight pocket cotais a matchbox wi exactly k matches. The A A R A L ad sice P( AR P( AL, so P(A P(AR, so let us compute P( A R. We may also assume e chace to select e ight pocket o left pocket is e same. The p=0.5 fo success. Now, fidig, fo e fist time, e box i e ight pocket empty meas a (N+ chace fo success. Exactly k matches left i e left pocket meas at e left pocket has bee selected (N-ktimes. So we have had (N-k failues. Thus, if f(,k,p deotes e egative biomial desity, we have P( A R f ( N,N k, ad N - k N N k P( A f ( N,N k, ( ( N - k To show some sample example of computig P(A, we select some N, k ad calculate ei pobabilities follows: N k P(A Example 3. Fid e pobability at, at e momet whee e fist box is emptied ad is ot foud empty, e oe cotais exactly k matches whee k=,,.n. Usig is esult, fid e pobability x at e box fist emptied is ot e oe fist foud empty. Defie all evets e same as e example 3., except by chagig is foud empty to is empty. Similaly, dop e wod foud i defiig " AR ( AL ". Now, fidig e box i e ight pocket empty meas N success, ad exactly k matches left i e left pocket meas at e left pocket has bee selected (N-ktimes. So we have had (N-k failues. Thus, we have P( A R f ( N,N k, ad N - k - N N k N - k - N k P( A f ( N,N k, ( ( N - k N - To fid e pobability x at e box fist emptied is ot e oe fist foud empty: N N - k - N N N N N - k - x ( ( k N - k N - N 5 0 N N - k k N - X Pogessive Academic Publishig, UK Page 3

5 Euopea Joual of Basic ad Applied Scieces Vol. 5 No., 08 Fially, we would like to povide aoe useful example of compoud distibutio, i.e. Poisso distibutio compoud wi egative biomial distibutio. Example 3.3 Duig its flight peiod, e istumet compatmet of a spacecaft is eached e by elemetay paticles wi e pobability desity fuctio, f (,. The! coditioal pobability fo each paticle to hit a pe-assiged uit equals p. Fid e pobability at is uit will be hit by (a exactly k paticles; (b at least oe paticle. Let y adom vaiable e umbe to hit pe-assiged uit wi pobability p. P( y k P( N 0, y k P( N P( y k / N 0 e k k k k p ( p 0! k e (! p k ( ( p 0! ( k!( k! p e ( ( p p k ( ( k!( k! p e ( ( p ( p p P( k 0 e e e!(! p P( k P(0 e Thee is a tivial easo at we ae ot iteested i e case whe =0. We use e popety of gamma fuctio, (! ad (0, we ca calculate -!=. I is way we deive at e pobability o paticle hit e pe-assiged poit is e p ad at least oe paticle is p e. CONCLUDING REMARKS The expeimets esult i coutig e umbe of times paticula evets occu i give times o o give physical objects. Fo example, we could cout e umbe of phoe calls aivig at a switchboad betwee ad p.m., e umbe of customes at aive at a ticket widow betwee oo ad p.m., o e umbe of patiets aivig a hospital i a cetai day. Each cout ca be looked upo as a adom vaiable associated wi a appoximate Poisso Pocess wi paamete 0, povided e followig fou coditios ae satisfied.. Radom evets i o-ovelappig itevals ae idepedet.. I a ifiitesimal iteval of leg t, e pobability of occuece exactly evet is ( 3. I e iteval t, e pobability at o evet occus is (. 4. I e itevals t, e pobability at o moe evets occu is give by (. Usig e above fou assumptios, we ca show at of pobability P(t opeate duig time iteval t follow e Poisso Distibutio. Let P k ( be e pobability at k evets occu duig a time peiod of leg. The, Pogessive Academic Publishig, UK Page 33

6 Euopea Joual of Basic ad Applied Scieces Vol. 5 No., 08 P 0(t P 0( t P 0( P 0( t [ ( ] P 0( t P 0( t ( P 0( t dp 0 ( t P0 ( t dt l P 0( t P 0( t e sice P(0 as e iitialco ditio Pk ( t Pk ( t P 0( Pk ( t P ( Pk ( t [ ( ] Pk ( t [ ( ], Pk ( t Pk ( t ( Pk ( t Pk ( t [ Pk ( t Pk ( t ] d( P Thus, k ( t Pk ( t Pk ( t dt Fo k P ( t e ( ( Fo k P ( t e! ( Fo k P (t e, 0.! Thus we see at e assumptio (-(4 give above descibe a poisso law. Usually we use poisso distibutio to appoximatio of biomial distibutio whe p is vey small ad is compaatively lage. As see fom e above discussio it also aises whe we coside a sequece of adom evets occuig i time o space. REFERENCES [] Felle W. (957 A Itoductio to Pobability Theoy ad Its Applicatios. Volume I, secod editio, Joh Wiley & Sos, Ic. [] Felle W. (965 A Itoductio to Pobability Theoy ad Its Applicatios. Volume II, Joh Wiley & Sos, [3] Hoel P.G., Pot S.C. ad Stoe C.J.(97 Itoductio To Pobability Theoy. Houghto Miffli Compay. [4] Hoel P.G., Pot S.C. ad Stoe C.J.(97 Itoductio To Statistical Theoy. Houghto Miffli Compay. [5] Hoel P.G., Pot S.C. ad Stoe C.J.(97 Itoductio To Stochastic Pocesses. Houghto Miffli Compay. [6] Reyi A. (970 Pobability Theoy. No-Hollad, Publishig Compay-Amstedam. [7] Tucke A. (980 Applied Combiatoics. Joh Wiley & Sos. Pogessive Academic Publishig, UK Page 34