2. Sample Space: The set of all possible outcomes of a random experiment is called the sample space. It is usually denoted by S or Ω.

Transcription

1 Ut: Rado expeet saple space evets classcal defto of pobablty ad the theoes of total ad copoud pobablty based o ths defto axoatc appoach to the oto of pobablty potat theoes based o ths appoach codtoal pobablty ad depedet evets ay s theoe. 5 as DFINITION OF VRIOU TRM:. Rado xpeet: expeet whch ca be epeated ude essetally detcal codtos ad whose all possble esults ae ow advace but whose esult ay patcula stace caot be deftely pedcted s called a ado expeet. g. If we toss a co we ay ethe get a head o a tal but we caot pedct ou esult a patcula stace advace. Thus tossg a co s a ado expeet.. aple pace: The set of all possble outcoes of a ado expeet s called the saple space. It s usually deoted by o Ω. g. The saple space fo the expeet 'tossg two cos sultaeously' s = { HH HT TH TT }. vet: evet s the collecto of all possble outcoes of a ado expeet whch ae favoable to a pheoeo o happeg elated to the expeet. vey subset of the saple space defes a evet. g. I the ' tossg two cos sultaeously' expeet let be the evet of gettg at least oe head the = { HH HT TH}. The evet happes f the outcoes of the udelyg ado expeet belogs to the subset. 4. xhaustve vets: set of evets --- s sad to be exhaustve f evey possble outcoe of the udelyg expeet belogs to oe of the 's.e.. g. I the ' tossg two cos sultaeously' expeet the set of evets { } whee s the evet ' gettg at least oe head' ad s the evet ' gettg at least oe tal' s exhaustve. 5.Mutually xclusve vets: Two evets ae sad to be utually exclusve f the happeg of ay oe of the pevets the happeg of the othe. If ad ae two utually exclusve evets the. u Mahata Kalabo College

2 g. I the above utually exclusve evets ad we have = { HH HT TH } ad = {HT TH TT }. s ad ae ot utually exclusve. Now f C s the evet 'gettg exactly two tals' the C = { TT }. Now as C ad C ae utually exclusve evets. 6. Idepedet vets: Two o oe tha two evets ae sad to be depedet f the happeg o o happeg of a evet s ot affected by the occuece of ay oe of the eag evets. g. If we daw a cad fo a pac of well-shuffled cads ad eplace t befoe dawg the d cad the esult of d daw s depedet of the fst daw. ut howeve f the st cad daw s ot eplaced the the d daw s ot depedet of the st daw. 7.Tals: tals s a sgle pefoace of a ado expeet. If we toss a co oce we get oe tal of the co tossg expeet f we toss twce we get two tals ad so o. CLICL OR RIORI DFINITION OF ROILITY: If a tal esult exhaustve utually exclusve ad equally lely cases ad f of the ae favoable to the happeg of a evet the the pobablty p of the happeg of the evet s gve by p = = Clealy ube of cases favoable to s - so = = - = lso sce 0 ad ad ad0 Deets: Classcal defto fals f. xhaustve ube of cases s fte coutable o ucoutable.. The outcoes ae ot equally lely. u Mahata Kalabo College

3 VON MII' TTITICL MRICL DFINITION OF ROILITY: If a tal s epeated a ube of tes ude essetally hoogeeous ad detcal codtos the the ltg value of the ato of the ube of tes the evets happes to the ube of tal as the ube of tals becoes deftely lage s called the pobablty of happeg of ths evet e. f tals a evet occus tes the p = = l Hee t s assue that the lt exsts ad uque. Clealy 0 p Deets: tatstcal defto eove ost of the defcecy of classcal defto but ths defto s ot sutable fo atheatcal vew pot as calculato of pobablty usg ths ethod eeds actual coputato ad uecal data. Ths defto wll poduce esult oly whe the pcple of statstcal egulaty s satsfed. The assupto that the codto of the expeet s epoducble ay ot be tue f codtos chages ove te. XIOMTRIC DFINITION OF ROILITY: Let be a saple space ad Ω be the class of all evets σ feld. Let be a eal valued fucto defe o Ω the s called a pobablty fucto ad s called the pobablty of the evet f the followg axos hold:. No Negetvty: 0. Cetaty : Fo the ceta evet =. Fo ay two dsjot evets ad = + a. Fo ay fte sequece of utually dsjots evets = If satsfes the above axos the saple space s called a pobablty space. RUL OF ROILITY:. Rule of ddto: Theoe of total pobablty: If ---- ae -utually exclusve evets the the pobablty of o o o s the su of the pobabltes of ---- e. u Mahata Kalabo College

4 4 u Mahata Kalabo College = oof: Let a fte saple space of equally lely outcoes of a ado expeet cossts of saple pots of whch ae favoable fo the evet ad ae favoable fo the evet. Now ce ad ae utually exclusve Let ths be tue fo utually exclusve evets Theefoe.... uppose... + ae + utually exclusve evets of. Now }... {... Theefoe by atheatcal ducto fo ay tege Note : Whe... ae utually exclusve ad exhaustve evets the 'Note : ce ad ae utually exclusve ad exhaustve evets + = Note : Fo ay two evets ad ad ae utually exclusve ad exhaustve evets. lso = theefoe

5 5 u Mahata Kalabo College 5 lso Note 4: y De Moga's Law c c = c c = - Note 5: If be possble evet ad be ay evet the = = + = 0 'Geealsed Theoe o Total obablty: If ad ae ay two evets o a saple space the oof: y defto of pobablty = Theoe '06'08: If ae ay thee evets the show that oof: We have Now Copoud obablty: The pobablty of occuece of two o oe evets sultaeously s teed as copoud pobablty. If... ae evets a sae saple space the the copoud pobablty s deoted by... o.... It s also ow as jot pobablty.

6 6 Codtoal obablty: If ad ae two evets the sapl space of a ado expeet such that > 0. Now the pobablty of occuece of evet subject to the codto that evet has aleady occued s called the codtoal pobablty of gve. It s deoted by /. Theoe o copoud obablty: The pobablty of sultaeous occuece of two evets of a saple space s equal to the poduct of the pobablty of oe of the evets by the codtoal pobablty of the othe; gve that the fst oe has aleady occued e. f ad ae two evets the saple space of a ado expeet the = / = / oof: Let be a fte saple space of equally lely outcoes of ado expeet ad ad ae ay two evets. If has aleady occued the the ube of saple pots favoable fo the evet =. Theefoe / / laly f s aleady occued the / Fo ad / = / f >0 = / f >0 Note: Fo thee evets C C = C = C/ = /C/ Theoes ased o xoatc ppoach: 6 u Mahata Kalabo College

7 7 Theoe : The possble evet has pobablty zeo e. = 0. oof: Let be ay evet the saple space of a ado expeet ad be the possble evet. Now we have = = = + [ sce ad ae utually exclusve] = 0. 'Theoe : Fo ay evet c = -. oof: Let be ay evet the saple space. Fo the ceta evet we ow fo axo of pobablty that =. Now we have c = c = + c = [ s ad c ae utually dsjot ] c = -. '09Theoe : If ad ae evets a pobablty space such that the ad hece - = -. oof: We have [ce ad - ae dsjot evets] ga by axo of pobablty we ow that 0 0. Theoe 4: Fo ay evet 0 oof: Let be ay evet the pobablty space. the theoe we eed to pove that. We have = 0 Thus we have 0. It s clea that 0. Thus to pove [ce = ad ad - ae utually dsjot evets] 7 Theoe 5: If ad ae ay two evets the = +. u Mahata Kalabo College

8 8 u Mahata Kalabo College 8 oof: Let ad be ay two evets the pobablty space. Now we have = [ce ad - ae dsjot evets] = + - [ ce - = - ] Theoe 6 : Fo ay two evets ad - = -. oof: We have - = - { - = -} Codtoal obablty: uppose s a evet a saple space wth > 0. Now the pobablty that a evet occus oce has aleady occued s called codtoal pobablty of gve. It s deoted by /. Theoe: If ad ae two evets a equpobable space the /. oof: uppose s s a equpobable space ad ad be the ubes of eleets the evet ad espectvely. Now / = The elatve pobablty of w.. t. the educed space. =

9 9 u Mahata Kalabo College 9 Idepedet vet: Two evets ad a pobablty space ae sad to be depedet f the occuece of oe of the does ot fluece the occuece of the othe. Moe specfcally ad ae depedet f / = o / =. Theefoe = Note: Thee evets ad C ae depedet f the followg two codtos ae hold: = C =C C =C C =C. Theoe of Total obablty: Let be a evet a saple space ad let --- be utually dsjot evets whose uo s e. exhaustve the / oof: Hee we have } {... j ad j fo ad j Now fo ay evet of =... We cla that fo j ad j ae dsjot fo f j the thee exst j j j x x ad x x x Whch s a cotadcto as ad j ae dsjot. / [by ultplcato theoe.]

10 0 Thus fo we have / aye's Theoe: Let be ay evet a saple space ad... be dsjot evets whose uo s the fo ay =... / / / oof: y the theoe of total pobablty we have / / / / '09 x. : cad s daw at ado fo a oday dec of plyg cads. Fd the pobablty that a a ace b a jac of heath c a thee ob clubs o a sx of daods d a heat e ay sut except heats f a te o a spade g ethe a fou o a club. ' x.: fa de s tossed twce. Fd the pobablty of gettg 4 5 o 6 o the fst toss ad o 4 o the d toss. '07 x. : Fd the pobablty of ot gettg a 7 o total o ethe of two tosses of a pa of fa dce. ' x. 4: Two cads ae daw fo a well-shuffled oday dec of 5 cads. Fd the pobablty that they ae both aces f the fst cad s a eplaced b ot eplaced. '09 x. 5: Thee balls ae daw at ado fo a box cotag 6 ed balls 4 whte balls 5 blue balls. Fd the pobablty that they ae daw the ode ed whte ad blue f each ball s a eplaced b ot eplaced. 0 u Mahata Kalabo College

11 '06 x. 6: Fo each + evets 0... such that 0... > 0. ove that 0... = 0 / 0 / 0... / x. 7: Thee ae thee bags fst cotag whte ed gee balls secod cotas whte ed gee ball ad the thd cotas whte ed gee balls. Two balls ae. daw fo a beg chose at ado thee ae foud to be oe whte ad oe ed. how that the pobablty that the ball so daw coe fo the secod beg s 6/. '0 x 8: Fo the two utually exclusve evets ad = ad = fd. '07 x. 9: Thee aches M M M poduce detcal tes. Of the espectve output 5% 4% ad % of tes ae faulty. O a ceta day M has poduced 0% ad M the eade. te selected at ado s foud to be faulty. pplyg aye's theoe show that the chace that t was poduced by the ache wth the lagest output s '08 x. 0: If a evet ust esult oe of utually exclusve evets the pove that = / + / + /. x. : Ida plays ad 5 atches agast asta Laa ad West Ides espectvely. obablty that Ida ws a atch agast asta Laa ad West Ides ae espectvely ad 0.4. If Ida ws a atch show that the pobablty that t was agast asta s /47. '08 x. : bag cotas balls oe of whch s whte. The pobablty that ad soea tuth s p ad p espectvely. Oe ball s daw fo the bag ad ad both asset that t s whte. how that the pobablty that the daw ball s actually whte s p p p p p p x. : I a ceta college thee wee thee caddates fo the posto of pcpal D. Chalha D. Das ad D. Hazaa whose chaces of gettg the appotet ae the popooto 4:: espectvely. The pobablty that D. Chalha f selected would toduce post gaduate classes the college s 0. ad that of D. Das ad D. Hazaa ae 0.5 ad 0.8 espectvely. Fd the pobablty of toducg post gaduate classes the college. Fd the pobablty that D. Chalha wll be selected as pcpal gve that.g. classes wee toduce the college. u Mahata Kalabo College

12 '0 x. 4: ove the followg laws assug that each case the codtoal pobablty ae defed: = the /= If ad ae dsjot ad > 0 the / = 0. ' x. 5: Gve = 0.0 = 0.78 = 0.6. Fd / /. '06 x. 6: bag cotas 40 tcets ubeed of whch 4 ae daw at ado ad aaged ascedg ode t < t < t <t 4. how that the pobablty of t beg 5 s 44/99. ' x7: bag cotas 6 whte ad 9 blac balls. fou balls ae daw at a te. Fd the pobablty fo the fst daw to gve 4 whte ad the secod daw to gve 4 blac balls f the balls ae ot eplaced befoe the d daw. 5 ' x 8: If = φ the show that. ' x. 9: The cotets of us I II ad III ae as follows: whte blac ad ed balls ; whte blac ad ed balls ; 4 whte 5 blac ad ed balls. Oe u s chose at ado ad two balls ae daw. They happe to be whte ad ed. What s the pobablty that they coe fo u I II o III. 7 u Mahata Kalabo College