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 without Replacemet oside populatio of elemets a a... a. y uodeed aagemet of elemets is called a uodeed sample of size. Two uodeed samples ae diffeet oly if oe cotais a elemet ot cotaied i the othe. oside uodeed sample of size. This sample ca be used to make Odeed samples ( pemutatios). ombiatios uodeed sample: (abc) emutatio odeed samples: (abc); (acb); (bac) (bca); (cab); (cba) The total umbe of odeed samples ( ) 6 Theefoe Numbe of odeed samples Numbe of uodeed samples Numbe of odeed samples pe uodeed sample () ( )...( ) * *... * ( ) ( ) Example box cotais 75 good I chips ad 5 defective chips ad chips ae selected at adom. Let: at least oe chip is defective Fid () S 00 o chip of sample is defective 75

2 Hece S idge diffeet bidge hads. 3 5 oke diffeet poke hads. 5 Fo example Let had of poke cotais five diffeet face values These face values ca be chose i to choose oe of fou suits. Thus o each cad: 3 ways ad coespodig to each cad we ae fee S oside distiguishable balls i cells. Defie specified cell cotais exactly K balls k a place balls i cells diffeet ways. S. K balls ca be chose i diffeet k ways. The emaiig (-k) balls ca be placed ito the emaiig - cells i k ways. k ( ) k ( k ) How may lies coect 5 poits with o 3 co-liea. To aswe this we must select poits (which defie a lie) fom the 5 give poits i.e.

3 3 5 Multiomial oefficiets Now coside the fpollowig situatio: set of distict items is to be divided ito distict goups of espective sizes whee i. How may diffeet divisios ae possible? To aswe thos we i ote that thee ae possible choices fo the fist goup; fo each choice of the fist goup thee ae possible choices fo the secod goup; fo each choice of the fist two goups thee ae possible choices fo the thid goup; ad so o. Hece it follows fom the geealized 3 vesio of the basic coutig piciple that thee ae So we ca state: If ( 3 ) ( ) 0 possible divisios. we defie by Thus epesets the umbe of possible divisios o distict objects ito distict goups of espective sizes police depatmet i a small city cosists of 0 offices. If the depatmet policy is to have 5 of the offices patollig the steets of the offices wokig full time at the statio ad 3 of the offices o eseve at the statio how may diffeet divisios of the 0b offices ito 3 goups ae possible?

4 4 Solutio: Thee ae divisios. Thee ae 0 boys who ae to be divided ito a team ad a team of 5 boys each. The team will play i oe league ad the team i aothe. How may diffeet divisios ae possible? Solutio: thee ae 0 53 possible divisios. I ode to play a game of basketball 0 boys at a playgoud divide themselves ito two teams of 5 each. How may diffeet divisiios ae possible? Solutio: Note that this example is diffeet fom the pevious oe because ow the opde of the two of the two teams is ievelat. That is thee is o ad V teams but j ust a divisio cosistig of goups of 5 boys each. That is thee ae twice () as may divisios i the fist case as i this case. If thee wee thee teams the thee would be 3 s may divisios tha if thee wee teams ad. Hece the disied aswe is 0/ 55 6 Summay: This is a good place to summaize. emutatio: ombiatio: Defiitio: The umbe of distict aagemets that ca be made fom the elemets of S usig of them at a time is deoted by ( ) ad called the umbe of pemutatios of thigs take at a time. Note that. Ode is impotat. Defiitio: The umbe of distict subsets of size that ca be fomed fom the elemets if S is deoted by ad is called umbe of combiatios of thigs take at a time. Note that. Ode is ot impotat (biomial ceeficiet) -Whe the sample cotais seveal sets of idetical elemets we have pemutatio with epetitio o udistiguishable sets. The umbe of pemutatios of objects of which ae alike ae alike... ae alike etc Hece ()... multi-omial... If we have objects - pemutatios exist. If ad ae alike the # of distiguishable pemutatio.

5 5 Example How may diffeet sigals each cosistig of 8 flags hug i a vetical lie ca be fomed fom a set of 4 idistiguishable ed flags thee idistiguishable white flags ad a blue flag? We seek the umbe of pemutatios of 8 objects of which 4 ae alike (the ed flags) ad 3 ae alike (the white flags). y the above theoem thee ae 8 8* 7 * 6 *5* 4 * 3* * 80 diffeet sigals * 3* ** 3* * How may ways ca people be divided ito 3 ows of 4 each. (Ode is ot impotat) ite-chagig ows oce selected Two sets of times ae icluded i a goup of eight people. How may ways ca six distiguishable people fom this goup be aaged i a ow? takig cae of pemutatios i the ow Example pai of six sided dice is thow util a 8 o appea. What is the pobability that a 8 appeas fist. 5/36 /36 5/36 5/36 /36 /36 othe 9/36 Othe 9/36 othe 9/36 8 appeas fist appeas fist

6 k 5 most pob k NowLet us look at distibutig balls ito sum whe all the balls ae udistiguishable. We kow distiguishable balls ca be distibuted to possible us i possible outcomes. Whe idistiguishable balls ae used the the balls put ito us is descibed by the outcome; (... th x x x ) whee x # balls i ι sum i So we have to fid the umbe of distict o-egative itege-valued vectos (x x x ) such that; X x...x I ode to fid this suppose we fist have idistiguishable objects lied-up ad we wat to divide them ito o-empty goups. 0x 0 x0 x... x 0x 0 object 0 with x deotig a space We eed to select - spaces of the - spaces x available betwee adjacet objects.

7 7 i.e. If 8 3 the the two divides ae 000/000/00 x x 3 x ad the vecto is Sice thee ae possible selectios we ca coclude. Theoem I Thee ae distict positive itege-valued vectos ( x x... x ) satisfyig x x... x x > 0 i To obtai the umbe of o-egative solutios (that is to allow empty cells) the umbe of oegative solutios of x x... x is idetical to the umbe of positive itege solutios of y y... y (Which we ca see by lettig yi xi i... ) Fom Theoem I above we get ( ) N with ( ) N d Theoem II is Thee ae (-) distict o-egative itege valued vectos x x... x satisfyig x x... x h How may ways ca we distibute 3 black idistiguishable balls ito two sum as alike the # of distict o-egative itege valued solutios of x x 3 ae possible ( 03)( 3.0) oside a ivesto has $0000 to ivest amog 4 possible ivestmets. How may diffeet ivestmet stategies ae possible? If: )all moey is to be ivested )ot all moey is to be ivested.

8 8 Solutio: Let x ι ; i 3 4 be the umbe of 000 s of dollas ivested i ivestmet i. a) The whe all moey is to be ivested x x x x 0 x ι Hece whe possible stategies 3* * 3* 3* *7 b) If ot all moey is to be ivested we let x 5 deote the amout kept i eseve. x x x3 x4 x5 0 5 I this case a stategy is a o-egative itege-valued vecto Example oside a set of ateas of which m ae defective ad -m ae fuctioal. ssume that the defective ad all the fuctioig ateas ae idistiguishable. How may liea odeig ae thee i which o two defective ateas ae cosecutive m< m. Solutio: Imagie -m fuctioal ateas lied up. 0-fuctioal ateas ;-defective ateas -m

9 9 Now if o two defectios ae to be cosecutive the the spaces betwee the fuctioal ateas must cotai at most oe defective atea That is i the -m possible positios betwee the -m fuctioal ateas we must select m of these to put the defective ateas. Hece: m m m < These ae the possible odeigs i which thee is at least oe fuctioal atea betwee two defective oes. Useful combiatio idetity is; i.e. oside object ad focus o object #. Now thee ae combiatio of size that cotais object. lso thee ae combiatio of size that do ot cotai object. Sice the ae combiatio of size. QED oof:

10 0 We ae ow eady to give the fomal axiomatic defiitio of pobability. Let S be the set of all outcomes i ε of a expeimet. ε is a set (o sub set of S of evet poits. Hece we have a pobability space. (SF) If we ca assig () obability Measue of evet i F such that the followig axioms will be satisfied: i) ) ( 0 ii) S - obability of cetai evet is iii) () obability of impossible evet is 0 iv)if 0 -o poits i commothe if 0 β We had two evets ad. ut QED a a a c a a

11 Geealizig: We ca exted to thee ad fou evets as follows ( ) ( a) ( c) ( D) ( D) ( D) ( D) ( D) ( D) ( D) ( D) ( D) * Εi ( Εi ) ( Εi Εi ) L L i< i ( Ε L Ε L... Ε L )... L < L < L ( Ε Ε... Ε )... The Summatio: ( Ε Ε Ε )... i { }. i i is take ove all possible subsets of size of the set Example oside tossig thee cois ad obsevig how they lad. The Space S ca be descibed 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 Fist coi is a head

12 { } { } { } ssumig equal pobability fo each eve

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