{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "Helvetica" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 24 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Helvetica " 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Helvetica" 1 14 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Helvetica" 1 14 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Tit le" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 18 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 40 "Stirling's Formula and \+ Some Coin Tossing" }}{PARA 0 "" 0 "" {TEXT 256 19 "Mth 232 Jan 15 2000 " }}{PARA 0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 32 "Filename: stirling_coin_toss.mws" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 18 "S tirling's Formula" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 3 "Let" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "S :=n->sqrt(2*Pi*n)*n^n*exp(-n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" SGR6#%\"nG6\"6$%)operatorG%&arrowGF(*(-%%sqrtG6#,$*&%#PiG\"\"\"9$F3\" \"#F3)F4F4F3-%$expG6#,$F4!\"\"F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "S(n);" "6#-%\"SG6#%\"nG" }{TEXT -1 35 " is a pretty good approximation to " }{XPPEDIT 18 0 "n!;" "6#-%*fa ctorialG6#%\"nG" }{TEXT -1 116 ". One can prove this and even estimate the error but we can get experimental evidence just by plotting the q uotient " }{XPPEDIT 18 0 "S(n)/n!;" "6#*&-%\"SG6#%\"nG\"\"\"-%*factori alG6#F'!\"\"" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(S(n)/n!,n=1..100,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7jn7$$\"\"\"\"\"!$\"1#*y&*)3q8A *!#;7$$\"1DJqX\\Vn5!#:$\"1%p:-GPmE*F-7$$\"1]iS\"*)p[8\"F1$\"1))f6Bm*pI *F-7$$\"1v$4r$[I-7F1$\"19nnL%)>V$*F-7$$\"1+D\"GyR(p7F1$\"1:*[=M[eP*F-7 $$\"1](=Un4YS\"F1$\"1m%)>q#4=F1$\"1$*G.&e(*Rb*F-7$$\"1++DJ\"f*y?F1$\"125&))G'))4'*F-7$$\" 1+D19*)p[BF1$\"1C[*)4$)Q`'*F-7$$\"1+](opQ%=EF1$\"1p+yRu>)o*F-7$$\"1+vo z%y\"))GF1$\"1%f:A=vmr*F-7$$\"1++]i#=z:$F1$\"1q=R01SS(*F-7$$\"1+vVL[r' 4%F1$\"1qE%fKH!*z*F-7$$\"1+]P/9^N]F1$\"1;()o'=#3O)*F-7$$\"1,D1MjG\"4'F 1$\"1y\"eVVWU')*F-7$$\"1++vj71ZrF1$\"1tIbsQ:%))*F-7$$\"1,+DYygs#*F1$\" 1vw!\\gm0\"**F-7$$\"1+vo2_!)Q6!#9$\"1)[Mrg4r#**F-7$$\"1+vVnR$\\L\"Faq$ \"1tE#['4yP**F-7$$\"1+v$z$R,Q:Faq$\"1Faq$\"1k*ox^?v&**F-7$$\"1++D#)ets@Faq$\"1R[B _?sh**F-7$$\"1+]7l7TiBFaq$\"1h5%pa*yk**F-7$$\"1++v@3%fd#Faq$\"1\"eL8D. x'**F-7$$\"1++D^rM!z#Faq$\"1t@;j4=q**F-7$$\"1++Dyt'p*HFaq$\"1llFlPBs** F-7$$\"1+v$*Q$)f%=$Faq$\"1.*\\IXnQ(**F-7$$\"1++D*p4xS$Faq$\"1SHBCidv** F-7$$\"1+++(G9nf$Faq$\"1D\"p37eo(**F-7$$\"1+vVa:d;QFaq$\"1]gP8(*=y**F- 7$$\"1++]*[#=6SFaq$\"1z'*[TnCz**F-7$$\"1+vVH:qCUFaq$\"1zpw\\XH!)**F-7$ $\"1+DJ[@-GWFaq$\"1q:Vy%)>\")**F-7$$\"1,]P%)e;SYFaq$\"17HIms0#)**F-7$$ \"1+v=e+)\\$[Faq$\"1U'pFezF)**F-7$$\"1+]7jM6X]Faq$\"18_*o@'\\$)**F-7$$ \"1+v$R,$Qj_Faq$\"1`:4h+=%)**F-7$$\"1+Dc*H(Q`aFaq$\"1=I;=3t%)**F-7$$\" 1+]PLrfecFaq$\"1Iz%y8%G&)**F-7$$\"1****\\r*)fqeFaq$\"1VqWw^\"e)**F-7$$ \"1**\\()49+ygFaq$\"1@&z5())H')**F-7$$\"1+vVLRnyiFaq$\"1SFUukt')**F-7$ $\"1+]7dl[,lFaq$\"16x_I2>()**F-7$$\"1****\\4Op,nFaq$\"1%>rA:tv)**F-7$$ \"1++DrtX:pFaq$\"1%3O=/dz)**F-7$$\"1+v$46f\"4rFaq$\"18&yq(\\G))**F-7$$ \"1++vvi#4K(Faq$\"1-(3?lB'))**F-7$$\"1+vVuE=?vFaq$\"1Rq!***F-7$$\"1+]7l*[%z\"*Faq$\"1J(z_!f#4***F-7$$\" 1+++*)[fv$*Faq$\"1K$Rxl:6***F-7$$\"1+vVats%e*Faq$\"1X-eA%48***F-7$$\"1 ,Dc3Q*[y*Faq$\"1S-,Er[\"***F-7$$\"$+\"F*$\"1v!yc;q;***F--%'COLOURG6&%$ RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$Q\"n6\"%!G-%*THICKNESSG6#\"\"#-%%VIE WG6$;F(Fg^l%(DEFAULTG" 1 2 0 1 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "As you can see the quotient is pretty close to 1. We can therefore use " }{XPPEDIT 18 0 "S(n);" "6#-%\"SG6#%\"nG" }{TEXT -1 56 " to obtain a rough estimate of the binomial coefficient " }{XPPEDIT 18 0 "C(n,k);" "6#-%\"CG6$%\"nG% \"kG" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "B:=( n,k)->S(n)/(S(k)*S(n-k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BGR6$ %\"nG%\"kG6\"6$%)operatorG%&arrowGF)*&-%\"SG6#9$\"\"\"*&-F/6#9%\"\"\"- F/6#,&F1\"\"\"F6!\"\"\"\"\"!\"\"F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "We expect " }{XPPEDIT 18 0 "B(n,k);" "6#-%\"BG6$%\"nG%\"k G" }{TEXT -1 38 " to be a pretty good approximation to " }{XPPEDIT 18 0 "C(n,k);" "6#-%\"CG6$%\"nG%\"kG" }{TEXT -1 12 " as long as " } {XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "k;" "6#% \"kG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "n-k;" "6#,&%\"nG\"\"\"%\"kG !\"\"" }{TEXT -1 64 " are all fairly large. We can get a feeling for i t by comparing " }{XPPEDIT 18 0 "B(2*m,m);" "6#-%\"BG6$*&\"\"#\"\"\"% \"mGF(F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C(2*m,m);" "6#-%\"CG6$*& \"\"#\"\"\"%\"mGF(F)" }{TEXT -1 11 " for large " }{XPPEDIT 18 0 "m;" " 6#%\"mG" }{TEXT -1 11 ". Note for " }{XPPEDIT 18 0 "m = 100;" "6#/%\"m G\"$+\"" }{TEXT -1 25 " these numbers are about " }{XPPEDIT 18 0 "10^5 9;" "6#*$\"#5\"#f" }{TEXT -1 130 ", so even if they are relatively clo se the difference may be a very large number. Thus we compare them by \+ looking at the quotient " }{XPPEDIT 18 0 "B(2*m,m)/C(2*m,m);" "6#*&-% \"BG6$*&\"\"#\"\"\"%\"mGF)F*F)-%\"CG6$*&\"\"#F)F*F)F*!\"\"" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(B(2*m,m)/binomi al(2*m,m),m=1..150,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7in7$$\"\"\"\"\"!$\"18b4n\"z$G6!#:7 $$\"1++++I\\,6F-$\"1'4ak\\Pl6\"F-7$$\"1+++**f)H?\"F-$\"1%*3Z!enm5\"F-7 $$\"1+++)**yWI\"F-$\"13Zyp(=$)4\"F-7$$\"1+++(*>(fS\"F-$\"1(3%GTt;\"4\" F-7$$\"1+++&*z&*3;F-$\"1f80B?cz5F-7$$\"1+++$*R%>\"=F-$\"1%3bc^b02\"F-7 $$\"1+++#**H\\,#F-$\"1#*HrgsOj5F-7$$\"1+++!*f\"z@#F-$\"1K]8G(*\\d5F-7$ $\"1+++))>!4U#F-$\"1:!*e<6i_5F-7$$\"1+++')z)Qi#F-$\"1VB[F=][5F-7$$\"1+ ++z>$eV$F-$\"1=2T;7#p.\"F-7$$\"1+++rfxZUF-$\"12:jebzH5F-7$$\"1+++DArgc F-$\"1@'**Qb)HA5F-7$$\"1+++y%[O2(F-$\"1]U(4R8y,\"F-7$$\"1+++mP;D5!#9$ \"1Uk1MHE75F-7$$\"1+++i#p]M\"Fap$\"15s%4OM$45F-7$$\"1+++.VXj;Fap$\"10e I\\;a25F-7$$\"1+++#*zje>Fap$\"1K;3(p,k+\"F-7$$\"1+++/OGkAFap$\"1*H-^KN b+\"F-7$$\"1+++LFQ!e#Fap$\"187X!pb[+\"F-7$$\"1+++H\"oa*GFap$\"1o:'y?EV +\"F-7$$\"1+++(=s&>KFap$\"1[^Y$*)*)Q+\"F-7$$\"1+++=K/0NFap$\"15qDNDd.5 F-7$$\"1+++&)fTEQFap$\"1s_'H,sK+\"F-7$$\"1+++R$3\"\\TFap$\"1\"Q(4lr,.5 F-7$$\"1+++;A3gWFap$\"1_E76l!G+\"F-7$$\"1+++)GwCu%Fap$\"1/E_\"=RE+\"F- 7$$\"1+++a,Fy]Fap$\"1Y#R,YkC+\"F-7$$\"1+++/;ti`Fap$\"1ZvZ'eLB+\"F-7$$ \"1+++'*yi$p&Fap$\"1Q:A?y>-5F-7$$\"1+++#=Fl)fFap$\"1/X.z,4-5F-7$$\"1++ +@T)yI'Fap$\"1Cb-\"f$)>+\"F-7$$\"1+++@>*Qh'Fap$\"1Q(4*Ffv $\"+2[P,5Fiv7$$\"+Dj@*R*Ffv$\"+y2L,5Fiv7$$\"+x0cM(*Ffv$\"+@ZG,5Fiv7$$ \"+I#)e.5!\"($\"+-jC,5Fiv7$$\"+c2wN5Fey$\"+nv?,5Fiv7$$\"+YQ\"\\1\"Fey$ \"+*[u6+\"Fiv7$$\"+uey'4\"Fey$\"+Q.9,5Fiv7$$\"+o\\xE6Fey$\"+s*46+\"Fiv 7$$\"+8A7e6Fey$\"+7*z5+\"Fiv7$$\"+u)p()=\"Fey$\"+d?0,5Fiv7$$\"+N^&3A\" Fey$\"+%RC5+\"Fiv7$$\"+bqv^7Fey$\"+$4**4+\"Fiv7$$\"+K(eLG\"Fey$\"+![u4 +\"Fiv7$$\"+I()p98Fey$\"+P7&4+\"Fiv7$$\"+Mp\\V8Fey$\"+Q3$4+\"Fiv7$$\"+ DH]w8Fey$\"+5&34+\"Fiv7$$\"+xQ-19Fey$\"+E%*)3+\"Fiv7$$\"+q$*\\P9Fey$\" +T*p3+\"Fiv7$$\"+Kain9Fey$\"+x?&3+\"Fiv7$$\"$]\"F*$\"+zO$3+\"Fiv-%'COL OURG6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$Q\"m6\"%!G-%*THICKNESSG6#\" \"#-%%VIEWG6$;F(Fd^l%(DEFAULTG" 1 2 0 1 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "We see f or " }{XPPEDIT 18 0 "100 <= m;" "6#1\"$+\"%\"mG" }{TEXT -1 37 " we hav e a fairly good approximation." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 260 19 "A Coin Toss Example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Suppose we tos s a fair coin " }{XPPEDIT 18 0 "2*m;" "6#*&\"\"#\"\"\"%\"mGF%" }{TEXT -1 82 " times. We may view a record of the outcomes as a sequence of H and T's of length " }{XPPEDIT 18 0 "2*m;" "6#*&\"\"#\"\"\"%\"mGF%" } {TEXT -1 84 ". Since the coin is fair each sequence has the same proba bility of occuring, namely " }{XPPEDIT 18 0 "2^(-2*m);" "6#)\"\"#,$*& \"\"#\"\"\"%\"mGF(!\"\"" }{TEXT -1 39 ". The number of sequences with \+ exactly " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 52 " heads is given \+ by the number of ways to select the " }{XPPEDIT 18 0 "k;" "6#%\"kG" } {TEXT -1 63 " positions in the the sequence that contain H's from the \+ total " }{XPPEDIT 18 0 "2*m;" "6#*&\"\"#\"\"\"%\"mGF%" }{TEXT -1 17 " \+ positions, thus " }{XPPEDIT 18 0 "C(m,k);" "6#-%\"CG6$%\"mG%\"kG" } {TEXT -1 11 " sequences." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 31 "Now let's find the probability " }{XPPEDIT 18 0 "p (m);" "6#-%\"pG6#%\"mG" }{TEXT -1 12 " of tossing " }{TEXT 262 7 "exac tly" }{TEXT -1 1 " " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 24 " head s in a sequence of " }{XPPEDIT 18 0 "2*m;" "6#*&\"\"#\"\"\"%\"mGF%" } {TEXT -1 42 " coin tosses. This probability is clearly " }{XPPEDIT 18 0 "C(2*m,m)*2^(-2*m);" "6#*&-%\"CG6$*&\"\"#\"\"\"%\"mGF)F*F))\"\"#,$*& \"\"#F)F*F)!\"\"F)" }{TEXT -1 19 ". Except for small " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 76 " this number is very time-consuming to co mpute, so we will approximate it by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p:=m->B(2*m,m)*2^(-2*m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGR6#%\"mG6\"6$%)operatorG%&arrowGF(*&-%\"BG6$,$9$ \"\"#F1\"\"\")F2,$F1!\"#F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Maple can simplify this expression a great deal" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p:=unapply(simplify(p(m)),m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGR6#%\"mG6\"6$%)operatorG%&arrowGF(*&\" \"\"F-*&-%%sqrtG6#%#PiGF--F06#9$F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Suppose now you toss a coin " }{XPPEDIT 18 0 "2*m;" "6 #*&\"\"#\"\"\"%\"mGF%" }{TEXT -1 74 " times and count the number of he ads. Suppose you perform this experiment " }{XPPEDIT 18 0 "N;" "6#%\"N G" }{TEXT -1 16 " times and that " }{XPPEDIT 18 0 "Q;" "6#%\"QG" } {TEXT -1 35 " of your sequences contain exactly " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 13 " heads. Then " }{XPPEDIT 18 0 "Q/N;" "6#*&%\"Q G\"\"\"%\"NG!\"\"" }{TEXT -1 51 " is a statistical approximation of th e probability " }{XPPEDIT 18 0 "p(m);" "6#-%\"pG6#%\"mG" }{TEXT -1 52 ". You could then use the formula above to estimate " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 0 "" }{TEXT -1 67 ". There's is something my sterious about tossing a coin to estimate " }{XPPEDIT 18 0 "pi;" "6#%# piG" }{TEXT -1 18 ", don't you think?" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "I hasten to add that this method of e stimating " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 25 " is not pract ical since " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 220 " both have to be inconveni ently large to obtain a good estimate. Of course, you can simulate the coin tossing with a computer, but then you have to worry about the qu ality of your random number generator. Why not try it?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "For example, suppose " }{XPPEDIT 18 0 "m = 100;" "6#/%\"mG\"$+\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "N = 10000;" "6#/%\"NG\"&++\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "Q = 563;" "6#/%\"QG\"$j&" }{TEXT -1 0 "" }{TEXT -1 6 ". Then" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "evalf(solve(563/10 000=1/(sqrt(x)*sqrt(100)),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ aE)[:$!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "This is a made-up e xample. You are unlikely to do that well with such small " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Finally, to wrap up, let's observe how close an appr oximation " }{XPPEDIT 18 0 "p(m);" "6#-%\"pG6#%\"mG" }{TEXT -1 42 " is to the actual theoretical probability " }{XPPEDIT 18 0 "q(m);" "6#-% \"qG6#%\"mG" }{TEXT -1 10 ", say for " }{XPPEDIT 18 0 "m = 1;" "6#/%\" mG\"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "m = 30;" "6#/%\"mG\"#I" } {TEXT -1 17 " (in this range " }{XPPEDIT 18 0 ".1 <= q(m);" "6#1$\"\" \"!\"\"-%\"qG6#%\"mG" }{TEXT -1 48 " so we are not dealing with very s mall numbers)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q:=m->bin omial(2*m,m)*2^(-2*m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qGR6#%\" mG6\"6$%)operatorG%&arrowGF(*&-%)binomialG6$,$9$\"\"#F1\"\"\")F2,$F1! \"#F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(q(m)-p( m),m=2..30,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7en7$$\"\"#\"\"!$!1aK9S!GUR#!#<7$$\"1LLeR+H w?!#:$!1+!>yF8vE#F-7$$\"1mm;z+e_@F1$!10')G6fb^@F-7$$\"1++v=,()GAF1$!1h [*\\4B^/#F-7$$\"1LLLe,;0BF1$!1U2%3Xdr%>F-7$$\"1++]P-udCF1$!14?V7hmF1$!1**z@sv;gSF [o7$$\"1++](pGKD(F1$!1Cmp$QIld$F[o7$$\"1MLL.LGiyF1$!1jx)>8w:<$F[o7$$\" 1nmmJ*Q()R)F1$!1*[#**)e-V(GF[o7$$\"1,++54m-!*F1$!1ZN4#=q9f#F[o7$$\"1++ +]E14'*F1$!1z]6ISC^BF[o7$$\"1+++4VM>5!#9$!1?J`td)G:#F[o7$$\"1nmT^;Ts5F fr$!1)eM,=fd*>F[o7$$\"1LLLaQ^N6Ffr$!1PvZ8VRK=F[o7$$\"1LLL(pp*)=\"Ffr$! 1$\\(*=F)o518Ffr$!1% \\fZWAk[\"F[o7$$\"1++v]Aem8Ffr$!1al=PRJ*Q\"F[o7$$\"1++Dsq3C9Ffr$!1KJ!y #>G18F[o7$$\"1LL$3K(3%[\"Ffr$!1dclWy5G7F[o7$$\"1LL3zi=R:Ffr$!1eEQls%H; \"F[o7$$\"1nm;\\zh)f\"Ffr$!1dhbq7)))4\"F[o7$$\"1nmT^2Ng;Ffr$!1?vVK>LQ5 F[o7$$\"1++DP$*397$$\"1nm;#[G@x\"Ffr$!16cwe_)*=%*F ev7$$\"1+++)f)3K=Ffr$!1Vc#>W0:'*)Fev7$$\"1++]tyu!*=Ffr$!1!=*>^to[&)Fev 7$$\"1++vQP]Z>Ffr$!1HO[KZcy\")Fev7$$\"1++]]9_5?Ffr$!1h'))3)o!zz(Fev7$$ \"1LLLnc9n?Ffr$!1E/@]IQ![(Fev7$$\"1+++0TgF@Ffr$!1t,DYmYkrFev7$$\"1nmTN %)Q#=#Ffr$!1Y%p7hKp*oFev7$$\"1+++z@GUAFfr$!13Ew8h)Hi'Fev7$$\"1LL3Cvj)H #Ffr$!1]\\T9'Fev7$$\"1LLL=!Q^T# Ffr$!1HqOhT3EfFev7$$\"1++voCVvCFfr$!1f8V;NF6dFev7$$\"1nmm!>.N`#Ffr$!1/ ln.eP;bFev7$$\"1nm;\"*)))Gf#Ffr$!1[bvAnEG`Fev7$$\"1ML3[Gy^EFfr$!1CV\"3 vS?:&Fev7$$\"1+++y-!fq#Ffr$!1P'zRm.&)*\\Fev7$$\"1ML$)f\\#zw#Ffr$!1]s-! )zpJ[Fev7$$\"1nmmu0SBGFfr$!1lL20T@!p%Fev7$$\"1++v]\"\\D)GFfr$!1Wm!em6o a%Fev7$$\"1++D&)=;RHFfr$!1P5CQ7F;WFev7$$\"#IF*$!1H)[Gk3GG%Fev-%'COLOUR G6&%$RGBG$\"#5!\"\"F*F*-%+AXESLABELSG6$Q\"m6\"%!G-%*THICKNESSG6#F)-%%V IEWG6$;F(F`]l%(DEFAULTG" 1 2 0 1 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Not too shabby!" } }}}{MARK "19 1 8" 2 }{VIEWOPTS 1 1 0 1 1 1803 }