Unordered Samples (Without Replacement)

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1 Uodeed Samples (Wthout Replacemet) Ay 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. Numbe of uodeed samples Numbe of odeed samples Numbe of uodeed samples pe uodeed sample ( ) ( )...( ) **...* ( ) ( )

2 Examples:. A box cotas 75 good IC chps ad 5 defectve chps, chps selected at adom. Let: A at least oe chp s defectve ( A) S A ( A) ( A) Bdge 635,03,559, 600 dffeet bdge hads. 5 oke,598, dffeet poke hads.

3 3 Let A had of poke cotas fve dffeet face values A 5 S 5 ( A) Let dstgushable balls be cells. Defe, A k specfed cell cotas exactlyk balls k ( ) k ( Ak )

4 Multomal Coeffcets Dstct tems ae dvded to dstct goups of espectve szes,,, whee. The umbe of dffeet dvsos ae? possble choces fo the fst goup; fo each thee ae possble choces fo the secod goup; fo each 3 possble choces fo the thd goup; ad so o. If, by wedefe,,,,, Thus,,, epesets the umbe of possble dvsos o dstct objects to dstct goups of espectve szes,

5 ,, Thus,,, epesets the umbe of possble dvsos o dstct objects to dstct goups of espectve szes,

6 Examples:. A polce depatmet polcy s to have 5 offces patollg the steets, offces wokg full tme at the stato, ad 3 o eseve at the stato, how may dffeet dvsos of 0 offces to 3 goups ae possble? - 0 Soluto: Thee ae 50 dvsos boys ae to be dvded to a A team ad a B team. A team wll play oe league ad B team aothe. How may dffeet dvsos ae possble? 0 Soluto: thee ae 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:. Thee s just a dvso cosstg of goups of 5 boys each If thee wee thee teams, the thee would be 3 As may dvsos tha f thee wee teams A, B ad C. Hece 0/ 55 6

7 Summay emutato: Combato: Defto: The umbe of dstct aagemets that ca be made fom the elemets of S, usg of them at a tme, s deoted by ( ) Ode s mpotat. Defto: The umbe of dstct subsets of sze that ca be fomed fom the elemets f S s deoted by Ode s ot mpotat. Whe the sample cotas seveal sets of detcal elemets, The umbe of pemutatos of objects of whch ae alke, ae alke, etc ( ),,......

8 Moe Examples. 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 tws 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

9 3. A 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 8 /36 /36 othe 9/36 Othe 9/36 othe 9/36 Let A 8 appeas fst ad B appeas fst. The ( B) ( A) k most pob k

10 Dstbutg Idstgushable Balls Numbe of dstct, o-egatve, tege-valued vectos, (x,x, x ), x x...x Dvde dstgushable objects to o-empty goups x x x x x Theoem I Thee ae dstct postve tege-valued vectos ( x, x,... x ) satsfyg x x..., x x > 0, Theoem II Thee ae dstct o-egatve tege valued vectos ( x x,... ), x satsfyg x x..., x x > 0, "

11 Examples. How may ways ca we dstbute 3 black, dstgushable balls to two us 3 4 ( 0,3)(,)(,)( 3.0) 3. Cosde a vesto has $0,000 to vest amog 4 possble vestmets. How may dffeet vestmet stateges ae possble If: (a) all moey s to be vested; (b) ot all moey s to be vested. a) The whe all moey s to be vested, possblestateges b) If ot all moey s to be vested, 3** 3* 3**

12 Examples A set of ateas, m of whch ae defectve ad -m ae fuctoal. Assume that the defectve ad all the fuctog ateas ae dstgushable. How may lea odegs ae thee whch o two defectve ateas ae cosecutve, <. Let 0-fuctoal ateas v-defectve ateas No two defectos ae to be cosecutve, the the spaces betwee the fuctoal ateas must cota at most oe defectve atea Hece: m m m <

13 A Useful combato detty s oof: ( ) ( ) ( ) ( ) ( ) ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

14 Axoms of obablty. Let S be the set of all outcomes ε of a expemet ε. A s a set (o sub set of S) of evet pots. Hece we have a pobablty space. (S,I,) ) 0 ( A) ) ( S) - obablty of ceta evet s ) ( 0 ) 0- obablty of mpossble evet s 0 v) If AB 0 the thee ae o pots commo, ad ( A B) ( A) ( B) f Aβ 0 ( A B) ( A) ( B) ( AB)

15 Geealzg: ( ) ( ) ( ) ( ) ( ) ( ) L L L L L L L L < < < Whee ( )... s take ove all possble subsets of sze of the set, {,,,}. Cosdeg thee evets ( ) ( ) ( ) ( ) ( ) (AC AB c B a C B A ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ABCD ACD ABC CD BD AD AC D C B A D C B A

16 Example Tossg thee cos. The Space, S, s descbed as s 3 t d d A H H H A H H T A 3 H T T A 4 T T T A 5 H T H A 6 T T H A 7 T H H A 8 T H T If we let A heads occu, ad B Fst co s a head the A B AB { A, A5, A7} { A, A, A3, A5} { A, A } Assumg equal pobablty fo each eve 5 ( A B) ( A) ( B) ( AB )

Randomized Load Balancing by Joining and Splitting Bins

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