RECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S

Transcription

1 Fomulae Fo u Pobablty By OP Gupta [Ida Awad We, ] Impotat Tems, Deftos & Fomulae 01 Bascs Of Pobablty: Let S ad E be the sample space ad a evet a expemet espectvely Numbe of favouable evets E The, Pobablty = Total umbe of elemetay evets S Also as E S E S o 0 E S 0 E S S S S 0 PE 1 Hece, f PE deotes the pobablty of occuece of a evet E the, PE = 1 P E such that Note that P E deotes the pobablty of o-occuece of the evet E P E P E ca also be epeseted as 0 P E 1 ad 02 Mutually Exclusve o Dsjot Evets: Two evets A ad B ae sad to be mutually exclusve f occuece of oe pevets the occuece of the othe S 1,2,3,4,5,6 Cosde a example of thowg a de We have the sample spaces as, Suppose A the evet of occuece of a umbe geate tha 4 5,6, B the evet of occuece of a odd umbe 1,3,5 ad C the evet of occuece of a eve umbe 2,4,6 I these evets, the evets B ad C ae mutually exclusve evets but A ad B ae ot mutually exclusve evets because they ca occu togethe (whe the umbe 5 comes up) Smlaly A ad C ae ot mutually exclusve evets as they ca also occu togethe (whe the umbe 6 comes up) If A ad B ae mutually exhaustve evets the we always have, PA B 0 As A B P A B P A PB If A, B ad C ae mutually exhaustve evets the we always have, P A B C P A P C 03 Idepedet Evets: Two evets ae depedet f the occuece of oe does ot affect the occuece of the othe Cosde a example of dawg two balls oe by oe wth eplacemet fom a bag cotag 3 ed ad 2 black balls Suppose A the evet of occuece of a ed ball fst daw, B the evet of occuece of a black ball the secod daw 3 2 P A The, RECAPITULATION & CONDITIONAL PROBABILITY 5 5 Hee pobablty of occuece of evet B s ot affected by occuece o o-occuece of the evet A Hece evet A ad B ae depedet evets Page - [1] wwwtheopguptawodpesscom

2 Lst Of Fomulae fo Class XII By OP Gupta (Electocs & Commucatos Egeeg) But f the two balls would have bee daw oe by oe wthout eplacemet, the the pobablty of 2 occuece of a black ball secod daw whe a ed ball has bee daw fst daw 4 Also f a ed ball s ot daw the fst daw, the the pobablty of occuece of a black ball the 1 secod daw (Afte a black ball s daw thee ae oly 4 balls left the bag) I ths case 4 the evet of dawg a ed ball the fst daw ad the evet of dawg a black ball the secod daw ae ot depedet Fo depedet evets A ad B, we always have P A B PA PB Fo depedet evets A, B ad C, we always have P A B C PA PB PC Also we have fo depedet evets A ad B, P A B 1 PA Sce fo depedet evets A ad B we have P A B PA PB, so the codtoal pobablty (dscussed late ths chapte) of evet A whe B has aleady occued s gve as, P A B P A B e, P A B PA PA B P A 04 Exhaustve Evets: Two o moe evets say A, B ad C of a expemet ae sad to be exhaustve evets f, a) the uo s the total sample space e A B C S b) the evets A, B ad C ae dsjot pas e A B, B C ad C A c) PA PB P C 1 Cosde a example of thowg a de We have S 1,2,3,4,5,6 Suppose A the evet of occuece of a eve umbe 2,4,6 the evet of occuece of a odd umbe the evet of gettg a umbe multple of 3, B 1,3,5 ad C 3,6 I these evets, the evets A ad B ae exhaustve evets as A B S but the evets A ad C o the evets B ad C ae ot exhaustve evets as A C S ad smlaly B C S 05 Codtoal Pobablty: By the codtoal pobablty we mea the pobablty of occuece of evet A whe B has aleady occued You ca ote that case of occuece of evet A whe B has aleady occued, the evet B acts as the sample space ad A B acts as the favouable evet The codtoal pobablty of occuece of evet A whe B has aleady occued s sometmes also called as pobablty of occuece of evet A wt B PA B PA B PA B, B e 0 PB A, A e P A 0 P A PA B PA B, PB 0 PB PA B PA B, PB 0 PB 06 Useful fomulae: P A B P A B, PB 0 P A B P A B 1, B a) PA B PA P A B e P A o B PA P A ad B b) P(A B C) P(A) P(B) P(C) P(A B) P(B C) P(C A) P(A B C) c) P A B Poly B P B A P B but ot A P A B Page - [2] wwwtheopguptawodpesscom

3 MATHEMATICS Lst Of Fomulae fo Class XII By OP Gupta ( ) d) P A B Poly A P A B P A but ot B P A P A B e) PA B Pethe A o B 1 PA B Sut Clubs (Black cad) Damods (Red cad) Heats (Red cad) Spades (Black cad) Pctoal Descpto Of The Playg Cads: SET OF 52 PLAYING CARDS! C, C C such that!!! ( 1)( 2)54321 Also, 0! 1 Face Cads 07 Evets ad Symbolc epesetatos: Vebal descpto of the evet Equvalet set otato Evet A A Not A A o A A o B (occuece of atleast oe of A ad B ) A B o A B A ad B (smultaeous occuece of both A ad B ) A B o A B A but ot B ( A occus but B does ot) A B o A B Nethe A o B A B At least oe of A, B o C A B C All the thee of A, B ad C A B C Impotat Tems, Deftos & Fomulae 01 Bayes Theoem: If E 3,, E ae o- empty evets costtutg a patto of sample spaces S e, E 3,, E ae pa wse dsjot ad E1 E2 E 3 E S ad A s ay evet of o-zeo pobablty the, P E PA E PE A, 1,2,3,, P E P A E Fo example, P E A 1 TOTAL PROBABILITY & BAYES' THEOREM j1 j j P E 1PA E1 P E P A E P E P A E P E P A E Bayes Theoem s also kow as the fomula fo the pobablty of causes If E 3,, E fom a patto of S ad A be ay evet the, PA P E 1PA E1 PE 2 PA E 2 P E PA E P(E A) P(E )P(A E ) Page - [3] wwwtheopguptawodpesscom

4 Lst Of Fomulae fo Class XII By OP Gupta (Electocs & Commucatos Egeeg) The pobabltes PE 1, PE 2,, PE whch ae kow befoe the expemet takes place ae called po pobabltes ad PA E ae called posteo pobabltes Impotat Tems, Deftos & Fomulae 01 Beoull Tals: Tals of a adom expemet ae called Beoull tals, f they satsfy the followg fou codtos: a) The tals should be fte umbes b) The tals should be depedet of each othe c) Each of the tal yelds exactly two outcomes e success o falue d) The pobablty of success o the falue emas the same each of the tal 02 Bomal Dstbuto: Let E be a evet Let p pobablty of success oe tal (e, occuece of evet E oe tal) ad, q 1 p pobablty of falue oe tal (e,ooccuece of evet E oe tal) Let X = umbe of successes (e, umbe of tmes evet E occus tals) The, Pobablty of X successes tals s gve by the elato, P(X ) P( ) C p q whee 0,1,2,3,, ; p pobablty of success oe tal ad q 1 p pobablty of falue oe tal P(X ) o P( ) If a expemet s epeated tmes ude the smla codtos, we say that tals of the expemet have bee made tals Hee The esult P(X ) P( ) C p q ca be used oly whe: () the pobablty of success each tal s the same () each tal must suely esult ethe a success o a falue C s also called pobablty of occuece of evet E exactly tmes!! ( )! Note that C p q s the ( 1) th Mea P( ) p 0 Vaace P( ) Mea pq q p tem the bomal expaso of Stadad Devato pq p Recuece fomula, P( x + 1) P( ) 1 q A Bomal Dstbuto wth Beoull tals ad pobablty of success each tal as p s deoted by B, p Hee ad p ae kow as the paametes of bomal dstbuto The expesso P x dstbuto BERNOULLI TRIALS & BINOMIAL DISTRIBUTION o P s called the pobablty fucto of the bomal Page - [4] wwwtheopguptawodpesscom

5 MATHEMATICS Lst Of Fomulae fo Class XII By OP Gupta ( ) Impotat Tems, Deftos & Fomulae 01 Radom Vaable: A adom vaable s a eal valued fucto defed ove the sample space of a expemet I othe wods, a adom vaable s a eal-valued fucto whose doma s the sample space of a adom expemet A adom vaable s usually deoted by uppe case lettes X, Y, Z etc Dscete adom vaable: It s a adom vaable whch ca take oly fte o coutably fte umbe of values Cotuous adom vaable: It s a adom vaable whch ca take ay value betwee two gve lmts s called a cotuous adom vaable 02 Pobablty Dstbuto Of A Radom Vaable: If the values of a adom vaable togethe wth the coespodg pobabltes ae gve, the ths descpto s called a pobablty dstbuto of the adom vaable Mea o Expectato of a adom vaable X x P Vaace ( ) P x 1 Stadad Devato Vaace PROBABILITY DISTRIBUTION 1 Clck o the followg lk to go fo a pleasat supse: H, All! I hope ths textue may have poved beefcal fo you Whle gog though ths mateal, f you otced ay eo(s) o, somethg whch does t make sese to you, please bg t my otce though SMS o Call at o Emal at theopgupta@gmalcom Wth lots of Love & Blessgs! - OP Gupta Electocs & Commucatos Egeeg, Ida Awad We wwwtheopguptawodpesscom Page - [5] wwwtheopguptawodpesscom