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1 SCIENCE WORLD JOURNAL VOL (NO4) ISSN SHORT COMMUNICATION ANALYSING THE APPROXIMATION MODEL TO BIRTHDAY PROBLEM *CHOJI, D.N. & DEME, A.C. Depatment of Mathematcs Unvesty of Jos, Ngea *(Coespondng autho) Felle (968) stated that the mathematcal theoy of pobablty gans pactcal value and an ntutve meanng n connecton wth eal o conceptual expements such as tossng a con 00 tmes, thowng thee dce, fequency of accdents, o detemnng fom a goup of people those wth the same bthdays. All these descptons ae athe vague, and, n ode to ende the theoy meanngful, we have to agee on what we mean by possble esults of expement o obsevaton n queston. Consde an expement of detemnng fom a goup of people, the pobablty of two people n the goup havng the same bthday. We wll stat by fst assumng that thee ae days n a yea f leap yeas ae neglected. Ths assumpton s good enough as t taes fou yeas befoe a leap yea s ealzed and on condton that such a yea s also not dvsble by 00. Secondly, that each day has an equal chance of beng a bthday. If we have fo nstance a goup of say 70 people, t wll be cetan (a pobablty of one) that at least two people wll have the same bthday snce thee ae only possble bthdays to go aound. Howeve, f on the othe hand, thee wee only people n the goup, the chances that those people shae the same bthday to be qute small (a pobablty close to 0). But when the numbe n a goup nceases upwad fom the pobablty nceases that at least people shae a common bthday. The assumptons smplfy the theoy wthout affectng ts applcablty. The hstoy of pobablty (and of mathematcs n geneal) showng a stmulatng nteplay of theoy and applcatons; theoetcal pogess opens new felds of applcatons, and n tun applcatons lead to new poblems and futful eseach. The theoy of pobablty s now appled n many dvese felds and flexblty of a geneal theoy s equed to povde appopate tools fo so geat a vaety of needs (Felle, 968; Levne & Bue, 97). One of the aeas pobablty can be found useful s that of bthday poblems. We shall be concened wth how combnatons and pemutatons ae put to use, and the eo analyss nvolved. A smple pogam n the appendx has been developed to pedct many stuatons wth egads to bthdays. MATHEMATICAL DERIVATION Tang a andom sample of sze say wth eplacement fom a populaton of sze n wll esult n n possble ways. In ode to obtan the pobablty of the event that no element appeas twce (equvalent to samplng wthout eplacement we obtan n pemutaton, that s n p. If we assume that all aangements have equal pobablty, we obtan the pobablty of no epetton n ou sample s n n( n )...( n ) () X ) p n Such that fo ou bthday poblem we assume that the yeas ae of equal length that s n= days neglectng the Febuay 9 of leap yeas. Secondly we assume that the bth ates ae constant thoughout the yea. We can then obtan the pobablty that all bthdays ae dffeent equals x ) P 64 6 ( )..... () whch can be expessed as p x ) ( p p ( )( )( )( )( ( ) )...( ) ( ) ) )...( () (4) Equaton 4 depcts the pobablty that at least two people shae a bthday and so we use small p fo the case of shang the same bthday. We can deve a good numecal appoxmaton to p when the s small. Ths can be done by neglectng all coss poducts (what ths means s that though ( ) ae small such that the poduct of any two of them s vey small, fo example Cho & Deme (SWJ):-7 Analysng The Appoxmaton Model

2 SCIENCE WORLD JOURNAL VOL (NO4) ISSN ( ) (( ) ), whch s a small value. In essence n ( a )... a (5) That s neglectng tems shown n Equatons 6 to 8... a a (6) whch s postve n value negatve n value 4 l4... a a a... a a a a l (7) (8) postve n value and the le. By ths we wll have that P ( x ) [... ( )] ( ) (()) (9) Snce the sum of the fst - tems n a sees n Athmetc Pogesson s easly devable to be (-)/. Ths mples that the pobablty that at least two people n a goup of people wll have the same bthday wll be p ( )/(()) Suppose we would le to now the maxmum such that p</ (case of medan). Snce thee ae possble bthdays, t wll be temptng to suggest that we would need ust about half ths numbe, whch s 8. Howeve, we eque = fo the above to happen (Felle, 968; Ross, 976; Snell, 987). To show ths, we obseved the pobablty p usng Equaton that n a goup of people = people, thee s no duplcaton of bthdays, wll be less than one half, that s P ( ).( ).( ( ) )...( ) and tang logs of both sdes and usng the fact that log(-x) -x fo small x, we have log() / +/ (-)/ = (-)/70 whch has a soluton =. Equatons and 9 gve close esults that fo a gven numbe people havng dffeent bthdays. Fo example when =0 Equaton gves p=0.88 but Equaton 9 gves p=0.877, a dffeence n value of Howeve when s becomng lage an appoxmaton can be obtaned by Equaton. log(p) [ ( (-) ]/ = (-)/70 () Such that fo =0 the Equaton gves the pobablty of 0.07 whee as by Equaton p=0.94. A bette appoxmaton when s becomng bgge s we ncease moe tems of the expanson, e.g., pa coss poducts, tple coss poducts e.t.c. ERROR ANALYSIS Ou concen wll be to see how the appoxmaton model wll tend to dffe fom the theoetcal pobablty model. We wll study the stuaton that at least two people wll shae a gven bthday, that s to say a gven numbe people wll not all have dffeent bthdays. The dagam n Fg. shows how Equatons 4 (the bthday model) p and Equaton 0 (the appoxmaton model) p dsplay the pobabltes that fo a gven Cho & Deme (SWJ):-7 Analysng The Appoxmaton Model 4

3 SCIENCE WORLD JOURNAL VOL (NO4) ISSN numbe of people wth at least two of them shang a bthday and the eo ncued d as a esult of the appoxmaton. The appoxmaton n Equaton 0 that s p s good enough when dealng wth goups of people below 6. Fg. shows that as the numbe of people n a goup becomes bgge, t wll eque that we ncease moe tems of the expanson n Equaton 4 fo a bette appoxmaton, that s t wll eque us to consde Equatons 5, 6, 7, e.t.c. Table dsplays the vaous esults unde dffeent numbe of people n a goup. The pobablty p ndcates that obtaned due to Equaton 4, p that due to Equaton 0 when tang nto consdeaton Equaton 5, we obtan pobablty p as p and p4 as p p...( ).( ) 4 p...( ).( ).( ) whle the dffeences (eos) of these p values fom p values ae d= p- p, d= p- p, d= p4- p. The models wee coded n Pascal pogammng language n ode to geneate Table. The data n Table wee used to obtan Fgs. and usng Mcosoft Excel. The pogam s pesented as Begn clsc; wteln; :=; wteln( p p p p4 d d d ); wteln( ); Repeat c:=; w:=0; s:=0; t:=0; fo :=-+ to do c:=c*/; p:=-c; p:=*(-)/70; fo := to - do fo := to - do s:=s+(/*/); fo := to - do fo := to - do fo := to - do t:=t+(/*/*/); fo v:=4 to - do fo := to v- do fo := to - do fo := to - do b:=b+(/*/*/*v/); p:=p-s; p4:=p+t; d:=p-p; d:=p-p; d:=p4-p; wteln(:6,' ',p:6:6,' ',p:6:6,' ',p:6:6,' ',p4:6:6,' ',d:6:6,' :=+; untl =m; end. ',d:6:6,' ',d:6:6); Fom Table and Fg., f we do not want to ncu an eo (dffeence between p wth othes) of not moe than.05, we wll not wsh to use p as explaned above when the numbe n a goup does exceed 6. Whle fo p the numbe should not exceed. Fo p4 the numbe should not exceed 9. The p estmate fals to estmate well when the numbe n a goup becomes lage as fo example 8 we see that the pobablty exceeds, whch s outageous. Ths s so because the poduct (-) = (8)(7) exceeds 70. p neve exceeds, but that t stats to declne when the numbe n a goup s moe than 8 whch fals to appoxmate p. p4 s bette of the estmates, though t also exceeds pobablty of when the numbe n a goup exceeds 5. The ds ae qute smalle n value depctng how good the estmate p4 of p. The d successvely nceases to moe than when the numbe n a goup goes to beyond 40 due to the eason gven of p above. Cho & Deme (SWJ):-7 Analysng The Appoxmaton Model 5

4 SCIENCE WORLD JOURNAL VOL (NO4) ISSN Fg. PROBABILITY OF AT LEAST TWO PEOPLE SHARING SAME BIRTHDAY PROBABILITY NUMBER OF PEOPLE IN A GROUP p p p PROBABILITY Fg. PROBABILITY OF AT LEAST TWO PEOPLE SHARING A BIRTHDAY NUMBER OF PEOPLE IN A GROUP p p p p d d d Cho & Deme (SWJ):-7 Analysng The Appoxmaton Model 6

5 SCIENCE WORLD JOURNAL VOL (NO4) ISSN TABLE. PROBABILITIES OF AT LEAST TWO PEOPLE IN A GROUP OF SIZE N SHARING SAME BIRTHDAY UNDER VARIOUS SITUATIONS ALONG WITH THEIR RESPECTIVE DEVIATIONS. p p p p4 d d d Fom the above wo, t s ealzed that the fewe the people n a goup the geate the chance that they ae bon on dffeent days of the yea. When numbe of people n a goup s as small as, the pobablty that at least two of them shae a bthday s geate than ½. Ths shows that we do not need have half the days of the yea to attend the pobablty ½. The appoxmaton model gves good estmates when the numbe n a goup s small, but eques moe tems of the expanson when the numbe n a goup becomes lage. Levne, G. & Bue, C. J. (97). Mathematcal Model Technques Fo Leanng Theoes. Academc Pess, New Yo. Ross, S. (976). A Fst Couse n Pobablty. Macmllan Publshng Co. New Yo. Snell, J. L. (987). Intoducton to Pobablty. Random House/ Bhause. Mathematcs Sees New Yo. REFERENCES Felle, W. (968). An Intoducton to Pobablty Theoy and Its Applcatons. John Wley and Sons, New Yo. Cho & Deme (SWJ):-7 Analysng The Appoxmaton Model 7