Chapter 2: Probability

Transcription

1 hapter : roaility A {FF}, B {MM}, {MF, FM, MM} The, A B 0/, B {MM}, B {MF, FM}, A B {FF,MM}, A, B a A B A B c A B d A B A B 4 a 8

2 hapter : roaility 9 5 a A B A B A B B A A B A B B A B B B A A c A B A B A B B 0/ The result follos from part a d B A B A B B 0/ The result follos from part 6 A {,,,,,, 4,, 5,, 6,,,4,,4,,4, 4,4, 5,4, 6,4,,6,,6,,6, 4,6, 5,6, 6,6} {,,,4,,6, 4,, 4,4, 4,6, 6,, 6,4, 6,6} A B {,, 4,, 6,,,4, 4,4, 6,4,,6, 4,6, 6,6} A B {,,,, 5,,,4,,4, 5,4,,6,,6, 5,6} A B everythig ut {,,,4,,6,,,,4,,6, 5,, 5,4, 5,6} A A 7 A {to males} {M, M, M,M, M,M } B {at least oe female} {M,W, M,W, M,W, M,W, M,W, M,W, {W,W } B {o females} A 8 a c 8 9 {A, B, AB, O, A-, B-, AB-, O-} A B A B 0/ A B A 0 a {A, B, AB, O} {A} 04, {B} 00, {AB} 004, {O} 045 c {A} or {B} {A} {B} 05, sice the evets are mutually exclusive a Let L {left ter}, {right tur}, {cotiues straight} vehicle turs L / / / 4 a eeds glasses eeds glasses ut does t use them 4 c uses glasses Let B ushig defect, shaft defect a B B or c exactly oe defect d either defect B or a {, T, T, TT} if the coi is fair, all evets have proaility 5 c A {T, T}, B {T, T, } d A 5, B 75, A B A 5, A B B 75, A B

3 0 hapter : roaility 6 a Let, deote the empty cas ad W, W deote the cas filled ith ater Thus, {, W, W, W, W, W W } f this a merely a guess, the evets are equally likely o, W W /6 7 a {,, L,,, L, L, L, LL} 5/9 c 5/9 8a eote the four cadidates as A, A, A, ad M ice order is ot importat, the outcomes are {A A, A A, A M, A A, A M, A M} assumig equally likely outcomes, all have proaility /6 c miority hired A M A M A M 5 a Let represet a customer seekig style, ad represet a customer seekig style The sample space cosists of the folloig 6 four-tuples:,,,,,,,,,,,,,,, f the styles are equally i demad, the orderig should e equally likely o, the proaility is /6 c A /6 4 a Three patiets eter the hospital ad radomly choose statios,, or for service The, the sample space cotais the folloig 7 three-tuples:,,,,,,,,,,,,,,,,,,,,,,,,,, A {,,,,, } c f the statios are selected at radom, each sample poit is equally likely A 6/7 5 The total umer of flights is There are! 6 orderigs 7 a There are 6! 70 possile itieraries the 70 orderigs, exactly 60 have ever efore a Fracisco ad 60 have a Fracisco efore ever o, the proaility is 5 8 By the multiplicatio rule, a By the multiplicatio rule, there are 66 6 possile roles efie the evet A {,6,,5,,4, 4,, 5,, 6,} The, A 6/6 40 a By the multiplicatio rule, the dealer must stock autos To have each of these i every oe of the eight colors, he must stock 8*40 0 autos

4 hapter : roaility 4 f the first digit caot e zero, there are 9 possile values For the remaiig six, there are 0 possile values Thus, the total umer is * There are three differet positios to fill usig te egieers The, there are 0!/! 0*9*8 70 differet ays to fill the positios 50 5 There are 9,600 ays to choose the iers Each of these is equally likely 4 a There are 4 ays for the orgaizers to i all of the prizes The proaility is 4/ There are 76 ays the orgaizers ca i to prizes ad oe of the other 46 people to i the third prize o, the proaility is 76/ c 440 The proaility is 440/ d 5,80 The proaility is 580/ a choosig three of the five firms, order is importat o 60 sample poits 4 f F is aarded a cotract, there are ays the other cotracts ca e assiged ice there are possile cotracts, there are 6 total umer of ays to aard F a cotract o, the proaility is 6/ The studet ca solve all of the prolems if the teacher selects 5 of the 6 prolems that the 6 0 studet ca do The proaility is There are 6 ays to dra to cards from the deck The proaility is 4*/ There are,598,960 ays to dra five cards from the deck a There are 4 ays to dra three Aces ad to Kigs o, the proaility is 4/598960

5 hapter : roaility There are 56 types of full house hads From part a aove there are 4 differet ays each type of full house had ca e made o, the proaility is 56*4/ There are,598,960 ays to dra five cards from the deck a o, the proaility is 04/ ! 6 6 5/ ! 5/ a After assigig a ethic group memer to each type of jo, there are 6 laorers remaiig for the other jos Let a e the umer of ays that oe ethic group ca e assiged to each type of jo The: 4 6 a The proaility is a / t does t matter ho the ethic group memers are assiged to jos type,, ad Let a e the umer of ays that o ethic memer gets assiged to a type 4 jo The: a The proaility is a AB / / BA /5 /5 c A A B 5/5-5/7 d AA B, sice A has occurred e A B A B 5- /7 7 ote that A 06 ad AM 4/4 06 o, A ad M are idepedet imilarly, A F 4/6 04 A, so A ad F are idepedet

6 hapter : roaility 75 Give the first to cards dra are spades, there are spades left i the deck Thus, the proaility is ote: this is also equal to Give the first three cards dra are spades, there are 0 spades left i the deck Thus, 0 the proaility is 008 ote: this is also equal to c Give the first four cards dra are spades, there are 9 spades left i the deck Thus, 9 the proaility is 0875 ote: this is also equal to efie the evets: U: jo is usatisfactory A: plumer A does the jo a UA A U/A AUU/A 5*/4 05 From part a aove, UA Assume AB A The: A B ABB AB BA B A/A AB/A B Assume BA B The: A B BAA BA AB A B/B AB/B A Assume A B BA The results follo from aove 80 f A B, A B A AB, uless B i hich case B 8 A A B A/ A B A, sice A ad B are ME evets A B [ B A ] A [ B ] B A A A 85 A B A B / B B B B B A A o, A ad B are idepedet B

7 4 hapter : roaility [ A B ] A B B B B A B A / A From the aove, A A A ad B are idepedet o B A [ A ] B A B B o, A A A ad B are idepedet 90 a /50/ at least oe ijury o ijuries i 50 jumps 49/ Your fried is ot correct 9 Let deote a hit ad let M deote a miss The, she is the game i three trials ith the evets, M, ad M f she egis ith her right had, the proaility she is the game, assumig idepedece, is efie the evets A: device A detects smoke B: device B detects smoke a A B smoke is udetected - A B art a is foud usig the Additio ule arts ad c use emorga s Las a c A B B A B d A B / B B 97 a curret flos all three relays are ope 0999 Let A e the evet that curret flos ad B e the evet that relay closed properly The, BA B A/A B/A 9/ ote that B A 0 Let A e the evet the item gets past the first ispector ad B the evet it gets past the secod ispector The, A 0 ad BA 05 The A B efie the evets: : disease us cotracted : disease is cotracted The, 0, 05, ad 00 a / / 09 This is similar to Ex 06 Apply Ex 08: 095 A B A B ice the evets have the same proaility, 095 A Thus, A 098

8 hapter : roaility 5 0 efie the evets: : item is from lie : item is from lie : item is ot defective The, a aircraft udetected all three fail to detect all three detect aircraft 98 4 Let T {detects truth} ad L {detects lie} The sample space is TT, TL, LT, LL ice oe suspect is guilty, assume the guilty suspect is questioed first: a LL LT TL d By idepedece, By the complemet rule, system orks system fails 0 9 a efie the evets: A: otai a sum of B: do ot otai a sum of or 7 ice there are 6 possile rolls, A /6 ad B 8/6 Otaiig a sum of efore a sum of 7 ca happe o the st roll, the d roll, the rd roll, etc Usig the evets aove, e ca rite these as A, BA, BBA, BBBA, etc The proaility of otaiig a sum of efore a sum of 7 is give y A BA [B] A [B] A ere, e are usig the fact that the rolls are idepedet This is a ifiite sum, ad it follos as a geometric series Thus, /6 8/6/6 8/6 /6 /4 imilar to part a efie : otai a sum of 4 : do ot otai a sum of 4 or 7 The, /6 ad 7/6 The proaility of otaiig a 4 efore a 7 is / 0 eote the evets G: good refrigerator : defective refrigerator a f the last defective refrigerator is foud o the 4 th test, this meas the first defective refrigerator as foud o the st, d, or rd test o, the possiilities are GG, GG, 4 ad GG o, the proaility is The proailities associated ith the other 6 5 to evets are idetical to the first o, the desired proaility is ere, the secod defective refrigerator must e foud o the d, rd, or 4 th test efie: A : secod defective foud o d test A : secod defective foud o rd test A : secod defective foud o 4 th test Also, A 5 from part a ote that A {G, G} learly, A Thus, A o, A A A /5 c efie: B : secod defective foud o rd test B : secod defective foud o 4 th test learly, B /4 ad B /4/ /4 o, B B /

9 6 hapter : roaility 4 efie the evets for the voter: : democrat : repulica F: favors issue 7/ F F F F 5 efie the evets for the perso: : has the disease : test idicates the disease Thus, 9, 9, 0, ad 99 Thus, / 9 efie the evets: : positive respose M: male respodet F: female respodet F 7, M 4, M 5 Usig Bayes rule, F F M M M M M 04 0 efie the evets: : cotract lug cacer : orked i a shipyard Thus,, 4, ad 0004 Usig Bayes rule, For i,,, let F i represet the evet that the plae is foud i regio i ad i e the complemet Also i is the evet the plae is i regio i The F i i α i ad i / for all i The, a α α α α imilarly, α ad c α efie the evets: G: studet guesses : studet is correct G G G G G G G efie F as failure to lear The, FA, FB, A 7, B By Bayes rule, AF 4/7 5 Let M major airlie, private airlie, commercial airlie, B travel for usiess a B BMM B B B B 6 08 c B B /B 8/57 058

10 hapter : roaility 7 d B efie the evets: A: the player is B i : a sum of i o first toss k : otai a sum of k efore otaiig a 7 o, A A B i A B A B A B i 6 A B7 B7 6 A B B 6 A B 4 4 B7 4 B7 6 We have that 0 Also,, o, 6 usig idepedece Ex 9 imilarly, , 6 8 5, ad 0 9 Thus, A B 5 5 A B9 45, A B6 A B8 96, A B0 6 uttig all of this together, A 049 6From Ex, Y 0 0 ad Y 98 The evet Y are the evets FF, FF, ad FF, each havig proaility 0 98 o, Y 0 98 imilarly, Y The total umer of ays to select from 6 refrigerators is 0 The total umer of 4 ays to select y defectives ad y odefectives is, y 0,, o, y y 4 0 Y 0 4/0, Y 4/0, ad Y /0 0 8The evets Y, Y, ad Y 4 ere foud i Ex 0 to have proailities /5, /5, ad /5 respectively The evet Y 5 ca occur i four ays: GGG GGG GGG GGG Each of these possiilities has proaility /5, so that Y 5 4/5 By the complemet rule, Y 6 5/5 9Each positio has proaility /4, so every orderig of to positios from to spis has 4 proaility /6 The values for Y are, Y /4 o, Y /4 6 B A B A 40ice B B A B A, A B A B B B

11 8 hapter : roaility 4a {6 possiilities of draig 0 to 4 of the sample poits} c A B {E, E, E, E 4 }, A B {E }, A B 0/, A B {E, E 4 } 4All 8 orderigs are possile, so the total umer of orderigs is 8! 5 4There are ays to dra 5 cards from the deck For each suit, there are ays to select 5 cards ice there are 4 suits, the proaility is The gamler ill have a full house if he is dealt {to kigs} or {a ace ad a kig} there are 47 cards remaiig i the deck, to of hich are aces ad three are kigs The proailities of these to evets are ad, respectively o, the proaility of a full house is ote that 495 each supplier has at least oe compoet tested is give y / Let A e the evet that the perso has symptom A ad defie B similarly The a A B A B 04 A B A B 06 c A B B A B/ B / Y 0 04, Y , Y 0 48The proaility that team A is i 5 games is p 4 p ad the proaility that team B is i 5 games is p p 4 ice there are 4 ays that each team ca i i 5 games, the proaility is 4[p 4 p p p 4 ] 49Let deote the evet that the specime turs red ad deote the evet that the specime cotais itrates

12 hapter : roaility 9 a Usig Bayes rule, 95/ Usig Bayes rule, 0 5Let Y the umer of pairs chose The, the possile values are 0,, ad a There are ays to choose 4 socks from 0 ad there are ays to pick 4 o-matchig socks o, Y 0 80/0 Geeralizig the aove, the proaility is r r r 5a A A B c 00 d Usig the result from Ex 80, a i 5686/ ii / ii 0560/ iv /5 08 f the U populatio i 00 as ko, this could e used to divide ito the total umer of deaths i 00 to give a proaility 54Let deote death due to lug cacer ad deote eig a smoker Thus: Thus, 00 55Let W deote the eve that the first all is hite ad B deote the evet that the secod all is lack The: W W B W W B W W B B W 56ote that 0/, ad ad 0/ are disjoit o, 0/ ad therefore 0/ 0

13 0 hapter : roaility 57There are 0 odefective ad defective tues that have ee dra from the machie, ad umer of distict arragemets is 66 a The proaility of oservig the specific arragemet is /66 There are to such arragemets that cosist of rus additio to hat as give i part a, the other is Thus, the proaility of to rus is /66 / 58We must fid, sice the miimum value for is d the to s occurs o cosecutive trials ut ot i positios ad or ad, there are 9 such arragemets The oly other case is a defective i positio ad, so that comiig ith Ex 60 ith, there are possiilities o, /66 59There are 9! ays for the attedat to park the cars There are! ays to park the expesive cars together ad there are 7 ays the expesive cars ca e ext to each other i the 9 spaces o, the proaility is 7!/9! / 60Let A e the evet that curret flos i desig A ad let B e defied similarly esig A ill fuctio if or & or 4 operate esig B ill fuctio if & or & 4 operate eote the evet i {relay i operates properly}, i,,, 4 o, usig idepedece ad the additio rule, A B o, desig A has the higher proaility 6Usig the otatio from Ex 6, 4 A 4 A/ A ote that 4 A 4, sice the evet 4 represets a path for the curret to flo The proaility of this aove evet is 9 8, ad the coditioal proaility is i questio is 8/ Usig the otatio from Ex 6, 4 B 4 B/ B 4 B The proaility of the aove evet is o, the coditioal proaility i questio is 889/ There are 70 ays to choose the tires f the est tire the customer has is raked #, 4 5 the other three tires are from raks 4, 5, 6, 7, 8 There are 0 ays to select three tires from these five, so that the proaility is 0/70 /7

14 hapter : roaility 7 64f Y, the customer chose the est tire There are 5 ays to choose the remaiig tires, so Y 5/ f Y, the customer chose the secod est tire There are 0 ays to choose the remaiig tires, so Y 0/70 /7 Usig the same logic, Y 4 4/70 ad so Y 5 /70 65 a The to other tires picked y the customer must have raks 4, 5, or 6 o, there are ays to do this o, the proaility is /70 There are four ays the rage ca e 4: # to #5, # to #6, # to #7, ad #4 to #8 Each has proaility /70 as foud i part a o, 4 /70 c imilar to parts a ad, 5/70, 5 8/70, 6 0/70, ad 7 5/70 66a For each eer driker, there are 4! 4 ays to rak the eers o there are 4,84 total sample poits order to achieve a comied score of 4 our less, the give eer may receive at most oe score of to ad the rest eig oe osider rad A f a eer driker assigs a oe to A there are still! 6 ays to rak the other rads o, there are 6 ays for rad A to e assiged all oes imilarly, rad A ca e assiged to oes ad oe to i! ays Thus, some eer may ear a total rak less tha or equal to four i 4[6! ] 456 ays o, the proaility is 456/ There are 5 ays to select three ames from seve f the first ame o the list is 6 icluded, the other to ames ca e picked 5 ays o, the proaility is 5/5 /7 68t is stated that the proaility that kyla ill hit someoe is ucoditioally /50, ithout regard to here that perso lives f oe ats to ko the proaility coditio o livig i a certai area, it is ot possile to determie 69Oly A B A B is true for ay evets A ad B

15 hapter : roaility 70efie the evets: : item is defective : item goes through ispectio Thus, 6, ad Thus, 5 7Let A athlete disqualified previously B athlete disqualified ext term The, e ko B A 5, B A 5, A To fid B, use the la of total proaility: B ote that A B 5 But, A B 5 5 o, they are ot mutually idepedet 7a [ A B ] [ A B ] A B A B A B A B [ A B A B] A B imilar to part a aove 74a o ijury i 50 jumps 49/ at least oe ijury i 50 jumps o ijury i 50 jumps 066 c o ijury i jumps 49/50 060, so is at most 5 75efie the evets: E: perso is exposed to the flu F: perso gets the flu osider to employees, oe of ho is ioculated ad oe ot The proaility of iterest is the proaility that at least oe cotracts the flu osider the complemet: at least oe gets the flu either employee gets the flu For the ioculated employee: F F E F E For the o-ioculated employee: F F E F E o, at least oe gets the flu a The gamlers reak eve if each i three times ad lose three times osiderig the 6 possile sequeces of is ad losses, there are 0 possile orderigs ice each has proaility 6, the proaility of reakig eve is order for this evet to occur, the gamler Joes must have $ at trial 9 ad must i o trial 0 o, i the ie remaiig trials, seve is ad to losses must e 9 placed o, there are 6 ays to do this oever, this icludes cases here Joes ould i efore the 0 th trial o, Joes ca oly i the game o a eve trial sice he must gai $6 cluded i the 6 possiilities, there are three ays Joes could

16 hapter : roaility i o trial 6: WWWWWWWLL, WWWWWWLLW, WWWWWWLWL, ad there are six ays Joes could i o trial 8: LWWWWWWWL, WLWWWWWWL, WWLWWWWWL, WWWLWWWWL, WWWWLWWWL, WWWWWLWWL o, these ie cases must e removed from the 6 o, the proaility is a f the patrolma starts i the ceter of the 6x6 square grid, there are 4 8 possile paths to take Oly four of these ill result i reachig the oudary ice all possile paths are equally likely, the proaility is 4/4 8 /4 7 Assume the patrolma egis y alkig orth There are ie possile paths that ill rig him ack to the startig poit:,,, EW, WE, WE, EW, EW, WE By symmetry, there are ie possile paths for each of orth, south, east, ad est as the startig directio Thus, there are 6 paths i total that result i returig to the startig poit o, the proaility is 6/4 8 9/4 7 78We ill represet the alls as 0 s ad create the oxes y placig ars etee the 0 s For example if there are 6 alls ad 4 oxes, the arragemet represets oe all i ox, to alls i ox, o alls i ox, ad three alls i ox 4 ote that six 0 s ere eed ut oly ars geeral, 0 s ad ars are eeded to represet each possile placemet of alls i oxes Thus, there are ays to arrage the 0 s ad ars o, if o to ars are placed ext to each other, o ox ill e empty o, the ars must e placed i the spaces etee the 0 s The total umer of ays to do this is, so that the proaility is as give i the prolem