Consider unordered sample of size r. This sample can be used to make r! Ordered samples (r! permutations). unordered sample

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1 Uodeed Samples wthout Replacemet osde populato of elemets a,,..., a a. y uodeed aagemet of elemets s called a uodeed sample of sze. Two uodeed samples ae dffeet oly f oe cotas a elemet ot cotaed the othe. osde uodeed sample of sze. Ths sample ca be used to make Odeed samples ( pemutatos). ombatos uodeed sample: (a,b,c) emutato odeed samples: (a,b,c); (a,c,b); (b,a,c) (b,c,a); (c,a,b); (c,b,a) The total umbe of odeed samples ( ) 6 Theefoe, Numbe of odeed samples Numbe of uodeed samples Numbe of odeed samples pe uodeed sample ( ) ( )...( ) * *... * ( ) ( ) box cotas 75 good I chps ad 5 defectve chps, ad chps ae selected at adom. Let: at least oe chp s defectve Fd () S 00 o chp of sample s defectve 75

2 ( ) Hece, S ( ) ( ) dge 635,03,559, 600 dffeet bdge hads. 3 5 oke,598, 960 dffeet poke hads. 5 Fo example, Let had of poke cotas fve dffeet face values These face values ca be chose to choose oe of fou suts. Thus, o each cad: 3 ways, ad coespodg to each cad we ae fee S 5 ( ) osde dstgushable balls cells. Defe, specfed cell cotas exactly K balls k a place balls cells dffeet ways. S. K balls ca be chose dffeet k ways. The emag (-k) balls ca be placed to the emag - cells ( ) k ways. k ( ) k ( k ) How may les coect 5 pots wth o 3 co-lea. To aswe ths we must select pots (whch defe a le) fom the 5 gve pots,.e.

3 3 5 Multomal oeffcets Now, cosde the followg stuato: set of dstct tems s to be dvded to dstct goups of espectve szes,,, whee. How may dffeet dvsos ae possble? To aswe ths, we ote that thee ae possble choces fo the fst goup; fo each choce of the fst goup thee ae possble choces fo the secod goup; fo each choce of the fst two goups thee ae possble choces fo the thd goup; ad so o. Hece t follows fom the geealzed 3 veso of the basc coutg pcple that thee ae So we ca state: If F HG IF KJ HG ( 3 ) ( ) ( ) 0 possble dvsos. I KJ F HG,,, we defe by,, I KJ Thus,,, epesets the umbe of possble dvsos o dstct objects to dstct goups of espectve szes, polce depatmet a small cty cossts of 0 offces. If the depatmet polcy s to have 5 of the offces patollg the steets, of the offces wokg full tme at the stato, ad 3 of the offces o eseve at the stato, how may dffeet dvsos of the 0b offces to 3 goups ae possble?

4 4 Soluto: Thee ae dvsos. Thee ae 0 boys who ae to be dvded to a team ad a team of 5 boys each. The team wll play oe league ad the team aothe. How may dffeet dvsos ae possble? Soluto: thee ae 0 55 possble dvsos. I ode to play a game of basketball, 0 boys at a playgoud dvde themselves to two teams of 5 each. How may dffeet dvsos ae possble? Soluto: Note that ths example s dffeet fom the pevous oe because ow the ode of the two of the two teams s elevat. That s, thee s o ad teams but just a dvso cosstg of goups of 5 boys each. That s, thee ae twce () as may dvsos the fst case as ths case. If thee wee thee teams, the thee would be 3 s may dvsos tha f thee wee teams, ad. Hece the desed aswe s 0/ 55 6 Summay: Ths s a good place to summaze. emutato: ombato: Defto: The umbe of dstct aagemets that ca be made fom the elemets of S, usg of them at a tme, s deoted by ( ) ad called the umbe of pemutatos of thgs take at a tme. Note that. Ode s mpotat. Defto: The umbe of dstct subsets of sze that ca be fomed fom the elemets f S s deoted by, ad s called umbe of combatos of thgs take at a tme. Note that. Ode s ot mpotat, (bomal coeffcet) -Whe the sample cotas seveal sets of detcal elemets, we have pemutato wth epetto, o udstgushable sets. The umbe of pemutatos of objects of whch ae alke, ae alke,... ae alke, etc Hece ( ),..., mult-omal... If we have objects, - pemutatos exst. If ad ae alke the # of dstgushable pemutatos.

5 5 How may dffeet sgals, each cosstg of 8 flags hug a vetcal le, ca be fomed fom a set of 4 dstgushable ed flags, thee dstgushable whte flags, ad a blue flag? We seek the umbe of pemutatos of 8 objects of whch 4 ae alke (the ed flags) ad 3 ae alke (the whte flags). y the above theoem, thee ae, 8 8* 7 * 6 *5* 4 * 3* * 80 dffeet sgals * 3* ** 3* * How may ways ca people be dvded to 3 ows of 4 each? (Ode s ot mpotat) ,4, te-chagg ows oce selected Two sets of tmes ae cluded a goup of eght people. How may ways ca sx dstgushable people fom ths goup be aaged a ow? takg cae of pemutatos the ow dace class cossts of studets, 0 wome qd me. f 5 me ad 5 wome ae to be chose ad the paed off, how may esults ae possble? 0 Soluto: Thee ae 5 5 possble choces of the 5 me ad 5 wome. They ca the be paed up 5 Ways, suce f we abtaly ode the e the the fst ma ca be paed wth ay of the 5 wome. Thewxt wth ay of the emag 4, ad so o. Hece 0 thee ae possble esults. I howmay ways ca detcal balls be dstbuted to us so that the th u cotas at least m balls, fo each,,? ssume that Soluto: The umbe of tege solutos of x x, x m Is the same as the umbe of oegatve solutos of m.

6 6 y y m, y 0. oposto 6. ou text gves the esult m. pa of sx sded dce s thow utl a 8 o appea. What s the pobablty that a 8 appeas fst? 5/36 / /36 5/36 /36 /36 othe 9/36 Othe 9/36 othe 9/36 8 appeas fst appeas fst ( ) 5 36 ( ) k 5 most pob k

7 7 Now, Let us look at dstbutg balls to sum whe all the balls ae udstgushable. We kow dstgushable balls ca be dstbuted to possble us possble outcomes. Whe dstgushable balls ae used, the the balls put to us s descbed by the outcome; (,,... th x x x ), Whee x # balls ι sum So we have to fd the umbe of dstct, o-egatve, tege-valued vectos, (x,x, x ), such that; X x...x I ode to fd ths, suppose we fst have dstgushable objects led-up, ad we wat to dvde them to o-empty goups. 0x 0 x0 x... x 0x 0 object, 0, wth x deotg a space We eed to select - spaces of the - spaces, x, avalable betwee adjacet objects..e. If 8, 3, the the two dvdes ae 000/000/00 x, x 3, x 3,3, 3 3 ad the vecto s ( ) ( ) Sce thee ae possble selectos, we ca coclude. Theoem I Thee ae dstct postve tege-valued vectos ( x, x,... x ) satsfyg x x..., x x > 0,

8 8 To obta the umbe of o-egatve solutos (that s, to allow empty cells), the umbe of oegatve solutos of x x... x s detcal to the umbe of postve tege solutos of y y... y (Whch we ca see by lettg y x,,... ) Fom Theoem I, above, we get ( ) N, wth ( )( ) N d, Theoem II s Thee ae (-) dstct o-egatve tege valued vectos x x... x satsfyg x x... x ( ) h, How may ways ca we dstbute 3 black, dstgushable balls to two sum as alke the # of dstct o-egatve tege valued solutos of x x 3 ae possble. 3, ( 0,3)(, )(,)( 3.0) osde a vesto has $0,000 to vest amog 4 possble vestmets. How may dffeet vestmet stateges ae possble? If: ) all moey s to be vested ) ot all moey s to be vested. Soluto: Let x ι ;,, 3, 4, be the umbe of 000 s of dollas vested vestmet. a) The whe all moey s to be vested, x x x x 0 x ι Hece, whe

9 9 0, 4, possble stateges 3* * 3* 3* * 7 b) If ot all moey s to be vested, we let x 5 deote the amout kept eseve. x, x, x3, x4, x5 0, 5 I ths case, a stategy s a o-egatve tege-valued vecto ( ) osde a set of ateas, of whch m ae defectve ad -m ae fuctoal. ssume that the defectve ad all the fuctog ateas ae dstgushable. How may lea odeg ae thee whch o two defectve ateas ae cosecutve, m< m. Soluto: Image -m fuctoal ateas led up. 0-fuctoal ateas ϖ-defectve ateas -m Now, f o two defectos ae to be cosecutve, the the spaces betwee the fuctoal ateas must cota at most oe defectve atea That s the -m possble postos betwee the -m fuctoal ateas, we must select m of these to put the defectve ateas. Hece: m m < m These ae the possble odegs whch thee s at least oe fuctoal atea betwee two defectve oes.

10 0 Useful combato detty s;.e. osde object ad focus o object #. Now, thee ae combato of sze that cotas object. lso, thee ae combatos of sze that do ot cota object. Sce the ae combato of sze. QED oof: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) We ae ow eady to gve the fomal axomatc defto of pobablty. Let S be the set of all outcomes ε of a expemet. ε s a set (o sub set of S of evet pots. Hece we have a pobablty space. (S,F,) If we ca assg () obablty Measue of evet F such that the followg axoms wll be satsfed: ) ) ( 0

11 ) ( ) S - obablty of ceta evet s ) ( ) obablty of mpossble evet s 0 v) If 0 -o pots commo, the ( ) ( ) ( ) f 0 β ( ) ( ) ( ) ( ) We had two evets ad. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ut, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) QED a a a c a a We ca exted to thee ad fou evets as follows, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) c a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D D D D D D D D D Geealzg: ( ) ( ) ( ) ( ) ( ) ( ) L L L L L L L L < < <

12 The Summato: ( )... {,,,}. s take ove all possble subsets of sze of the set, osde tossg thee cos ad obsevg how they lad. The Space, S, ca be descbed as follows; st d 3d H H H H H T 3 H T T 4 T T T 5 H T H 6 T T H 7 T H H 8 T H T Let heads occu Fst co s a head {, 5, 7} {,, 3, 5} {, } 5 ssumg equal pobablty fo each eve ( ) ( )... 8 ( ) 3, ( ) 4, ( ) ( ) ( ) ( ) ( ) ( ) 5 8

13 3 u cotas balls, of whch oe s specal. If k of these balls ae wthdaw oe at a tme, wth each selecto beg equally lkely to be ay of the balls that em,a at the tme, what s the pobablty that the specal ball s chose? Soluto: Sce all the balls ae detcal, the the set of k balls ca be oe of the sets of k balls. So k k k {specal ball s selected} k Ths could also have bee obtaed by lettg deote the evet that the specal ball s the th ball to be chose,, k. The, sce each oe of the balls s equally lkely to be the th ball chose, t follows that ( )/. Hece, sce these evets ae clealy mutually exlusve, the k k k {specal ball s selected} ( ) It could have bee agued that ()/, by otg that thee ae ( ) ( k ) /( k) equally lkely outcomes of the expemet, of whch ( )( ) ( )()( ) ( k ) ( )/( k) specal ball beg the th oe chose. Fom ths t follows that ( ) ( ) esult the football team cossts of 0 offesve ad 0 defesve playes. The playes ae to be paed goups of fo the pupose of detemg oommates. If the pag s doe at adom, what s the pobablty that thee ae o offesve-defesve oommate pas? What d s the pobablty that thee ae offesve-defesve pas, I,, 0? Soluto We have 40 playe that must be paed oto goups of, Hece thee ae 40 (40) 0,,..., () ways od sotg the playes to odeed pas.(.e. thee s a fst pa, secod pa, etc) To get uodeed pas we smply dvde by 0. Now, a dvso wll esult o offesve-defesve pas f the offesve (ad cosequetly, the defesve) playes ae paed amog themselves. It should be (0) obvous that thee ae such dvso fo both offesve ad defesve 0 (0) playes. Hece, the pobabolty of o offesve-defesve pag, O, s

14 4 O (0) (0) [(0)] 0 3 (40) [(0)] (40) 0 (0) To deteme, the pobablty that thee ae offesve-defesve pas, we 0 ote thee ae ways of selectg the 4 playes volved ( offesve ad defesve). These 4 playes ca the be paed up () as. The emag 0- (offesve ad defesve) playes ae the paed amog themselves, t follows that thee ae 0 (0 ) ( ) 0 (0 ) dvsos whch lead to offesve-defesve pas. Hece 0 (0 ) ( ) 0 (0 ), 0,,...,0 (40) 0 (0) aothe example whch llustates the xoms of pobablty ad the mathematcal dextety eeded fo solvg poblems s as follows. Suppose that each of N me at a paty thows hs hat to the cete of th e oom, The hats ae fst mxed up ad the each ma adomly selects a hat. What s the pobablty that a) oe of the me selects hs ow hat; (b) exactly k of the me select the ow hat? Soluto: (a) We fst go the back doo ad calculate the pobablty of the complemetay evet of at least oe ma s selectg hs ow hat. Let E,,, N, be the evet that the th ma selects hs ow hat. Now by the geeal expesso fo the uo of N o-mutually exclusve evets, the pobablty that at least oe of the me selects hs owhat s ( ) ( EE ) E < N N E E ( ) EE ( ) ( ) ( ) EE E < < N ( ) EE ( EN )

15 5 osde ths expemet esultg a vecto whose elemets epeset the umbe of the hat take by the th ma. Fo example, the vecto (E E E ) s the vecto fo the evet that each ma selects hs ow hat.. The thee ae N possble such vectos Now, the evet that each of me,,,,, selects hs ow hat ca occu ay of (N-)x(N--)x x3xx(n-) possble ways, sce thee s oly oe way fo the me to select the ow hat, the, of the N- me emag, the fst ca select ay of (N-) hats, the secod ca select (N--), ad so o. So, assumg all N possbltes equally lkely, N ( N ) otug, thee ae tems EE { ) E, so N N( N ) EE ( ) E < < ( N ) N ad thus, N N E ( ) 3 N So, the pobablty that oe of the me selects hs ow hat s N ( ) 3 N d fo lage values of N, ths ca be appoxmated by e I othe wods, fo lage N, the pobablty that oe of the me selects hs ow hat s appoxmately.37. Note, t does ot go to fty, as mght be expected. (b) osde the evet that exactly k of the me select the ow hats, The umbe of ways that oly these k me ca select the ow hats s the sam as the umbe of ways whch the othe N- me ca select amog the emag hats such that oe of them select the ow hats. ut Nk ( ) s the pobablty we just foud fo ot 3 ( N k) oe of the N-k me selectg the ow hat.. The the umbe of ways whch the set of me selectg the ow hats s Nk ( N k) ( ) 3 ( N k)

16 6 Sce thee ae N ways to select k me fom N, Nk ( N k) ( ) 3 ( N k) {Exactly k me select the ow hats} N Nk ( ) 3 ( N k) k Fo lage N, ths ca be appoxmated by e. k