{VERSION 2 3 "APPLE_PPC_MAC" "2.3" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 0 1 0 
0 1 0 0 0 0 }{CSTYLE "" -1 256 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 
0 }{CSTYLE "" -1 257 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE 
"" -1 258 "Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "
Geneva" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Geneva" 1 
12 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Geneva" 1 12 0 0 0 1 
0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Geneva" 
1 10 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 
1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 
1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 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 "R3 \+
Font 0" -1 256 1 {CSTYLE "" -1 -1 "Monaco" 1 9 0 0 255 1 2 2 2 0 0 0 
0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 
{CSTYLE "" -1 -1 "Monaco" 1 9 255 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 
-1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 598 "PROGRAM:  BertrandsPar
adox\nCALLING SEQUENCE:  BertrandsParadox(n, show)\nPARAMETERS:\n     \+
n - an integer\n     show - an integer from 1 to 4 inclusive\nSYNOPSIS
:\n     -  This program illustrates Bertrand's Paradox by choosing n r
andom chords of a\n        circle three times, each time using a diffe
rent one of the three coordinatizations\n        described in the book
.  For each coordinatization, the program prints the\n        proporti
on of chosen chords with length greater than sqrt(3).  The variable \"
show\"\n\011       allows the user to choose which method is illustrat
ed.  If show=1, then a picture " }}{PARA 256 "" 0 "" {TEXT 257 86 "   \+
     is shown\011in which chords are chosen by picking the rectangular
 coordinates of " }}{PARA 256 "" 0 "" {TEXT 258 86 "        their midp
oints at\011random.  If show=2, the picture represents choosing chords
 " }}{PARA 256 "" 0 "" {TEXT 259 85 "        by choosing the polar coo
rdinates of the midpoints at random.  If show=3, the" }}{PARA 256 "" 
0 "" {TEXT 261 86 "        picture describes holding one endpoint of t
he chord constant and choosing the " }}{PARA 256 "" 0 "" {TEXT 260 
205 "        other endpoint at random on the circle.  If show=4, only \+
the proportions are given.\nRETURNED VALUES:\n     -  none\nLOCATION:
\n     Folder:  Chapter 2\n     File:      \"BertrandsParadox.Chpt2.ma
p.r4\"\n\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1519 "BertrandsParadox:=p
roc(n,show)\n\011local circ,i,test,t1,t2,U,q,V,theta,\n\011\011\011x0,
x1,y0,y1,plotlist,l1,l2,l3,total1,total2,total3,j,r,k,W,alpha:\n\011wi
th(stats):\n\011with(plots):\n with(plottools,line):\n\011circ:=[plots
[polarplot](1,scaling=CONSTRAINED)]:\n\011U:=random[uniform[-1,1]]('ge
nerator'):\n\011V:=random[discreteuniform[0,1]]('generator'):\n\011W:=
random[uniform[0,evalf(2*Pi)]]('generator'):\n\011plotlist:=[]:\n\011t
otal1:=0:  total2:=0:  total3:=0:\n\011for i from 1 to n do\n\011\011t
est:=true:\n\011\011while test do\n\011\011\011t1:=U():  \n   t2:=U():
\n\011\011\011if (t1^2 + t2^2) <= 1 then test:=false fi:\n\011\011od:
\n\011\011if show=1 then\n\011\011\011q:=sqrt(t1^2 + t2^2):\n\011\011
\011if V()=0\n\011\011\011\011then theta:=arctan(t2/t1):\n\011\011\011
\011else theta:=arctan(t2/t1) + evalf(Pi): \n\011\011\011fi:\n\011\011
\011x0:=cos(theta-arccos(q)):  x1:=cos(theta+arccos(q)):\n\011\011\011
y0:=sin(theta-arccos(q)):  y1:=sin(theta+arccos(q)):\n\011\011\011plot
list:=[op(plotlist),line([x0,y0],[x1,y1])]:\n\011\011fi:\n\011\011l1:=
2*sqrt(1-(t1^2 + t2^2)):\n\011\011if l1>evalf(sqrt(3)) then total1:=to
tal1 + 1 fi:\n\011od:\n\011for j from 1 to n do\n\011\011r:=U():\n\011
\011if show=2 then\n\011\011\011plotlist:=[op(plotlist),line([sqrt(1-r
^2),r],\n\011\011\011\011\011\011[-sqrt(1-r^2),r])]:\n\011\011fi:\n
\011\011l2:=2*sqrt(1-r^2):\n\011\011if l2>evalf(sqrt(3)) then total2:=
total2 + 1 fi:\011\011\n\011od:\n\011for k from 1 to n do\n\011\011alp
ha:=W():\n\011\011if show=3 then plotlist:=[op(plotlist),\n\011\011
\011\011\011\011line([1,0],[cos(alpha),sin(alpha)])]:\n\011\011fi:\n
\011\011l3:=sqrt(2-(2*cos(alpha))):\n\011\011if l3>evalf(sqrt(3)) then
 total3:=total3 + 1 fi:\011\n\011od:\n\011lprint(evalf(total1/n),evalf
(total2/n),evalf(total3/n)): \n\011if (show=1) or (show=2) or (show=3)
\n\011 \011then display([op(circ),op(plotlist)])\n\011fi: \nend:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "BertrandsParadox(50,1);" }}
{PARA 6 "" 1 "" {TEXT -1 39 ".1600000000   .4000000000   .3800000000" 
}}{PARA 13 "" 1 "" {INLPLOT "6V-%'CURVESG6$7S7$$!\"\"\"\"!$\"1BmIq&o?5
%!#D7$$!1R6_uBO1**!#;$!12S\\y\"y_O\"F17$$!1'Gi)*4-Qn*F1$!1jmib*)GLDF17
$$!1'4@/['e[#*F1$!1,D1@.6.QF17$$!1!*H&3#42`')F1$!1lR!f$4U7]F17$$!19yQN
?D/zF1$!1t0ymceDhF17$$!1A_`rTd#3(F1$!1Tq)zpU&fqF17$$!10Yc\\dX;hF1$!1.A
,Z!>8\"zF17$$!1pt&)f:x5]F1$!1TPrOh-a')F17$$!1y<Y['R,#QF1$!1%)H\\4bcT#*
F17$$!1u0>!G'QDDF1$!1aaWL\"oen*F17$$!17*zm#o8X8F1$!19=>jt64**F17$$\"1J
X(>nl5(f!#>$!0e=t@)******!#:7$$\"1#ox^N#[i8F1$!1p$y\")GZn!**F17$$\"178
SY@=YEF1$!1/uE]D`V'*F17$$\"13bEF*3Jx$F1$!1N0PPl'3E*F17$$\"1\"3!3q_JU]F
1$!1Nq9v]oN')F17$$\"1?m3t%*[RgF1$!1-c(3/I-(zF17$$\"1\\6(QM;$*3(F1$!1Xc
Fj?x_qF17$$\"1+)RI?/U!zF1$!1nSb(QZc7'F17$$\"1u&ext1$f')F1$!1,a/8/k,]F1
7$$\"1)*)e:FO4B*F1$!1h_tBHvXQF17$$\"1>*[$3Nij'*F1$!1y(y0$Q%=d#F17$$\"1
@*e)=+.2**F1$!1vf64]Ug8F17$$\"1Q0F4>&*****F1$!14rpY)485$!#=7$$\"1&)oF>
!Q%3**F1$\"1$eM@&=8]8F17$$\"17BHoa1u'*F1$\"1orICIGKDF17$$\"1%\\yX@iKE*
F1$\"1c'=D`Bsw$F17$$\"1z$RT8'=u')F1$\"1;`*G)3zv\\F17$$\"1!RqPxng%zF1$
\"18-xlUCrgF17$$\"1Lc(*Rfc5rF1$\"1=9:4yMJqF17$$\"1bWQLke[gF1$\"1fkK&eG
L'zF17$$\"1?*3=U)o!*\\F1$\"1Qic4Mil')F17$$\"1E&>b[\"ysPF1$\"1$o(eQ***4
E*F17$$\"17DD&el'3EF1$\"1]=!\\p[Pl*F17$$\"1A*3$znc\"H\"F1$\"1%eQD*>C;*
*F17$$\"1>GdRQQdIFcs$\"1l5\">E`*****F17$$!1:Vc\\Ks(G\"F1$\"1zHhX=u;**F
17$$!1>!Q<MT]b#F1$\"1yl^I'z!o'*F17$$!1#fp<Q)zNQF1$\"1uTu$>x]B*F17$$!1)
*eas0I.]F1$\"1=%3EhZ$e')F17$$!1?2C'R]$4hF1$\"1_]N%H2o\"zF17$$!1K'yNQ+$
*4(F1$\"1`2]j?sUqF17$$!1Zp?\"Ry,!zF1$\"19+(\\bQ38'F17$$!1X\"pT^PVn)F1$
\"1#4Qh(p_v\\F17$$!1ZCj^4+D#*F1$\"1a8\\!eo*fQF17$$!1-U:o))oa'*F1$\"1![
NO%Q=0EF17$$!1-tSNb&p!**F1$\"1v3@5t'4O\"F17$F($!1BmIq&o?5%F--%'COLOURG
6&%$RGBG$\"#5F)F*F*-F$6#7$7$$\"+2ta%e#!#5$\"+/PBg'*Ff[l7$$!+:yI<**Ff[l
$\"++sN$G\"Ff[l-F$6#7$7$$\"+:&3!o%)Ff[l$\"+c15>`Ff[l7$$!+G(y?c'Ff[l$\"
+l;!ea(Ff[l-F$6#7$7$$\"+aLvS*)Ff[l$\"+ch;zWFf[l7$$!+SR>H)*Ff[l$!+eeOS=
Ff[l-F$6#7$7$$\"+&\\?1***Ff[l$!+H::IV!#67$$\"+&)>,QRFf[l$\"+`j&>>*Ff[l
-F$6#7$7$$\"+%oJ'H)*Ff[l$!+xj-Q=Ff[l7$$\"+ehV!>$Ff[l$\"+O.Sx%*Ff[l-F$6
#7$7$$!+%fgFh&Ff[l$!+<lGw#)Ff[l7$$!+qORGDF`^l$!+-J!o***Ff[l-F$6#7$7$$
\"+%f)fH**Ff[l$\"+-L^%=\"Ff[l7$$!+Fqe;MFf[l$\"+d4C)R*Ff[l-F$6#7$7$$!++
$>Lg*Ff[l$\"+WVf)y#Ff[l7$$!+;wQ-dFf[l$!+W+z9#)Ff[l-F$6#7$7$$!+PRnU#*Ff
[l$!+sqX<QFf[l7$$!+'*=8B9F`^l$!+(H()*)***Ff[l-F$6#7$7$$!+#fqf-'Ff[l$\"
+GgX!)zFf[l7$$!+Q\"Q\"*)**Ff[l$!+TdgfYF`^l-F$6#7$7$$\"+(=T(****Ff[l$\"
+J5d%>(!#77$$\"+`y#eI&Ff[l$\"+67Lw%)Ff[l-F$6#7$7$$\"+q4Z6&*F`^l$!+*=jY
&**Ff[l7$$\"+aT$zN%Ff[l$!+0sY+!*Ff[l-F$6#7$7$$!+[?lF_Ff[l$!+)4nZ_)Ff[l
7$$\"+Q#\\Z(oFf[l$!+q63isFf[l-F$6#7$7$$!+TJ]s'*Ff[l$\"+uSCQDFf[l7$$!+V
B$\\j#F`^l$!+jz_'***Ff[l-F$6#7$7$$!+yi=*G\"Ff[l$!+d<b;**Ff[l7$$\"+$oIW
))*Ff[l$!+%[Df^\"Ff[l-F$6#7$7$$!+8()GQdF\\dl$!+f`$)****Ff[l7$$\"+AFcE*
*Ff[l$\"+(3\"p47Ff[l-F$6#7$7$$!+?F\"*)G(Ff[l$!+=ZHYoFf[l7$$!+B'>YO#F`^
l$!+'*Q?(***Ff[l-F$6#7$7$$\"+M8:icF`^l$\"+_r&R)**Ff[l7$$!+gu$[3)Ff[l$!
+k]=&)eFf[l-F$6#7$7$$!+1D5>NFf[l$\"+quLg$*Ff[l7$$!+>F<4zFf[l$!+24B>hFf
[l-F$6#7$7$$!+pa<&>*Ff[l$\"+]&*[IRFf[l7$$!+L#>*=OFf[l$!+w0?A$*Ff[l-F$6
#7$7$$\"+$Q*z#y*Ff[l$!+W6)G2#Ff[l7$$\"+$H9:^%Ff[l$\"+YTZC*)Ff[l-F$6#7$
7$$!+%[r5;*Ff[l$\"+k_L4SFf[l7$$!+***zyK)Ff[l$!+0/#f`&Ff[l-F$6#7$7$$!+&
>1J)))Ff[l$!++7V#f%Ff[l7$$\"+\"y81.\"Ff[l$!+%)*\\n%**Ff[l-F$6#7$7$$!+H
c/rqFf[l$\"+&***3rqFf[l7$$\"+_L.zjF\\dl$!+Qlz****Ff[l-F$6#7$7$$\"+eEG%
)**Ff[l$\"+A2Y/cF`^l7$$!+x\\HhEF`^l$\"+F\"ek***Ff[l-F$6#7$7$$!+&z#GnvF
f[l$!+'R&HPlFf[l7$$\"+7_2\\#*Ff[l$!+#e@>!QFf[l-F$6#7$7$$\"+*>;6S*Ff[l$
!+8vm3MFf[l7$$!+%Q/q/(Ff[l$\"+F'\\]4(Ff[l-F$6#7$7$$\"+Cj@'Q\"Ff[l$\"+g
TX.**Ff[l7$$!+Ndy\")**Ff[l$!+pi&G.'F`^l-F$6#7$7$$\"+Ffc_UFf[l$!+:$G20*
Ff[l7$$\"+HQ1n(*Ff[l$\"+=:!e9#Ff[l-F$6#7$7$$!+o\"[pN\"Ff[l$!+Mo]2**Ff[
l7$$\"+g?%G/*Ff[l$\"+a*4$pUFf[l-F$6#7$7$$!+\"*\\w(G'Ff[l$!+;3'ex(Ff[l7
$$\"+-*G()p(Ff[l$!+BD(>Q'Ff[l-F$6#7$7$$!+?)3R9\"Ff[l$!+)=eV$**Ff[l7$$
\"+vxn@qFf[l$\"+`_6?rFf[l-F$6#7$7$$!+YRqE]Ff[l$!+::yW')Ff[l7$$\"+GjP$p
)Ff[l$!+jl=U\\Ff[l-F$6#7$7$$\"+i*pPx)Ff[l$\"+Cj,)z%Ff[l7$$!+.SK'o&Ff[l
$\"+Qv\"fA)Ff[l-F$6#7$7$$!+-PW%***Ff[l$\"++I4LLF`^l7$$!++i%\\&fF`^l$!+
hND#)**Ff[l-F$6#7$7$$!+8K7z**Ff[l$\"+)>F$ekF`^l7$$\"+b$>^%HFf[l$!+6!yk
b*Ff[l-F$6#7$7$$\"+j5(3A)Ff[l$!+*Q<Op&Ff[l7$$\"++\"4]C&Ff[l$\"+o))49&)
Ff[l-F$6#7$7$$!+nO\\$3(Ff[l$!+#3?'eqFf[l7$$!+DQ!eq\"Ff[l$!+RwV`)*Ff[l-
F$6#7$7$$\"+RpN7OF\\dl$\"+aZ$*****Ff[l7$$!+X<SRJFf[l$\"+lwU%\\*Ff[l-F$
6#7$7$$\"+z9^ZdFf[l$!+fCG$=)Ff[l7$$!+7#4z7&F`^l$\"+>O%o)**Ff[l-F$6#7$7
$$!+\"[H7>%Ff[l$!+w[Hz!*Ff[l7$$\"+W#QJr*Ff[l$!+BH,yBFf[l-F$6#7$7$$!+2s
#f4#Ff[l$!+lx)yx*Ff[l7$$\"+3&Qv***Ff[l$!+jJk=AF`^l-F$6#7$7$$!+g&\\****
*Ff[l$\"+]U9wJF\\dl7$$!+<w`2QFf[l$!+%pknC*Ff[l-F$6#7$7$$\"+N`w;!*F`^l$
\"+4gEf**Ff[l7$$!+(G*pxhFf[l$\"+s*)ejyFf[l-F$6#7$7$$\"+8\\uT**Ff[l$!+D
c#y2\"Ff[l7$$\"+vS?NBFf[l$\"+4!>Ns*Ff[l-F$6#7$7$$!+:8!*pWFf[l$\"+\")))
QX*)Ff[l7$$!+RcG)z*Ff[l$!+$**)R)*>Ff[l-F$6#7$7$$\"+nj(p6%Ff[l$!+7]?8\"
*Ff[l7$$\"+hTa\"3)Ff[l$!+6mq*)eFf[l-F$6#7$7$$!+<I+7'*Ff[l$!+KP^eFFf[l7
$$\"+`YjD**Ff[l$!+@DG<7Ff[l-F$6#7$7$$\"+hk.W\"*Ff[l$\"+:i.[SFf[l7$$!+n
m_xaFf[l$\"+YJSm$)Ff[l-F$6#7$7$$\"+%y1&zOFf[l$\"+)3`%)H*Ff[l7$$!+/')f)
\\(Ff[l$\"+tp'fh'Ff[l-%(SCALINGG6#%,CONSTRAINEDG" 2 254 254 254 2 0 1 
0 2 9 0 4 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "BertrandsParadox(50,2);" }}{PARA 6 "" 1 "" {TEXT -1 39 ".24000
00000   .5200000000   .4200000000" }}{PARA 13 "" 1 "" {INLPLOT "6V-%'C
URVESG6$7S7$$!\"\"\"\"!$\"1BmIq&o?5%!#D7$$!1R6_uBO1**!#;$!12S\\y\"y_O
\"F17$$!1'Gi)*4-Qn*F1$!1jmib*)GLDF17$$!1'4@/['e[#*F1$!1,D1@.6.QF17$$!1
!*H&3#42`')F1$!1lR!f$4U7]F17$$!19yQN?D/zF1$!1t0ymceDhF17$$!1A_`rTd#3(F
1$!1Tq)zpU&fqF17$$!10Yc\\dX;hF1$!1.A,Z!>8\"zF17$$!1pt&)f:x5]F1$!1TPrOh
-a')F17$$!1y<Y['R,#QF1$!1%)H\\4bcT#*F17$$!1u0>!G'QDDF1$!1aaWL\"oen*F17
$$!17*zm#o8X8F1$!19=>jt64**F17$$\"1JX(>nl5(f!#>$!0e=t@)******!#:7$$\"1
#ox^N#[i8F1$!1p$y\")GZn!**F17$$\"178SY@=YEF1$!1/uE]D`V'*F17$$\"13bEF*3
Jx$F1$!1N0PPl'3E*F17$$\"1\"3!3q_JU]F1$!1Nq9v]oN')F17$$\"1?m3t%*[RgF1$!
1-c(3/I-(zF17$$\"1\\6(QM;$*3(F1$!1XcFj?x_qF17$$\"1+)RI?/U!zF1$!1nSb(QZ
c7'F17$$\"1u&ext1$f')F1$!1,a/8/k,]F17$$\"1)*)e:FO4B*F1$!1h_tBHvXQF17$$
\"1>*[$3Nij'*F1$!1y(y0$Q%=d#F17$$\"1@*e)=+.2**F1$!1vf64]Ug8F17$$\"1Q0F
4>&*****F1$!14rpY)485$!#=7$$\"1&)oF>!Q%3**F1$\"1$eM@&=8]8F17$$\"17BHoa
1u'*F1$\"1orICIGKDF17$$\"1%\\yX@iKE*F1$\"1c'=D`Bsw$F17$$\"1z$RT8'=u')F
1$\"1;`*G)3zv\\F17$$\"1!RqPxng%zF1$\"18-xlUCrgF17$$\"1Lc(*Rfc5rF1$\"1=
9:4yMJqF17$$\"1bWQLke[gF1$\"1fkK&eGL'zF17$$\"1?*3=U)o!*\\F1$\"1Qic4Mil
')F17$$\"1E&>b[\"ysPF1$\"1$o(eQ***4E*F17$$\"17DD&el'3EF1$\"1]=!\\p[Pl*
F17$$\"1A*3$znc\"H\"F1$\"1%eQD*>C;**F17$$\"1>GdRQQdIFcs$\"1l5\">E`****
*F17$$!1:Vc\\Ks(G\"F1$\"1zHhX=u;**F17$$!1>!Q<MT]b#F1$\"1yl^I'z!o'*F17$
$!1#fp<Q)zNQF1$\"1uTu$>x]B*F17$$!1)*eas0I.]F1$\"1=%3EhZ$e')F17$$!1?2C'
R]$4hF1$\"1_]N%H2o\"zF17$$!1K'yNQ+$*4(F1$\"1`2]j?sUqF17$$!1Zp?\"Ry,!zF
1$\"19+(\\bQ38'F17$$!1X\"pT^PVn)F1$\"1#4Qh(p_v\\F17$$!1ZCj^4+D#*F1$\"1
a8\\!eo*fQF17$$!1-U:o))oa'*F1$\"1![NO%Q=0EF17$$!1-tSNb&p!**F1$\"1v3@5t
'4O\"F17$F($!1BmIq&o?5%F--%'COLOURG6&%$RGBG$\"#5F)F*F*-F$6#7$7$$\"+Drb
w**!#5$\")GIVo!\"*7$$!+Drbw**Ff[lFg[l-F$6#7$7$$\"+Z'Q=&)*Ff[l$\"*m9]r
\"Fi[l7$$!+Z'Q=&)*Ff[lFc\\l-F$6#7$7$$\"+%ptcF&Ff[l$!+j>8&\\)Ff[l7$$!+%
ptcF&Ff[lF^]l-F$6#7$7$$\"+KY&pS)Ff[l$\"*2#3:aFi[l7$$!+KY&pS)Ff[lFi]l-F
$6#7$7$$\"+b1I%f(Ff[l$!+_%*)e]'Ff[l7$$!+b1I%f(Ff[lFd^l-F$6#7$7$$\"+z'4
eT&Ff[l$\"*#f[1%)Fi[l7$$!+z'4eT&Ff[lF__l-F$6#7$7$$\"+dO#**p)Ff[l$\"*#>
lI\\Fi[l7$$!+dO#**p)Ff[lFj_l-F$6#7$7$$\"+iBGW_Ff[l$!+Cla9&)Ff[l7$$!+iB
GW_Ff[lFe`l-F$6#7$7$$\"+)Q!oznFf[l$\"*\"Q\"4N(Fi[l7$$!+)Q!oznFf[lF`al-
F$6#7$7$$\"+\">'y<ZFf[l$!+x.<<))Ff[l7$$!+\">'y<ZFf[lF[bl-F$6#7$7$$\"+K
c$Gm*Ff[l$!+IC![d#Ff[l7$$!+Kc$Gm*Ff[lFfbl-F$6#7$7$$\"+!=ya;*Ff[l$!+[6D
**RFf[l7$$!+!=ya;*Ff[lFacl-F$6#7$7$$\"+/^YG\")Ff[l$\"*SzZ#eFi[l7$$!+/^
YG\")Ff[lF\\dl-F$6#7$7$$\"+*HcAH'Ff[l$!+!)oAsxFf[l7$$!+*HcAH'Ff[lFgdl-
F$6#7$7$$\"+A)*o?))Ff[l$\"*s-7r%Fi[l7$$!+A)*o?))Ff[lFbel-F$6#7$7$$\"+(
*zFyZFf[l$\"*CNXy)Fi[l7$$!+(*zFyZFf[lF]fl-F$6#7$7$$\"+2G)em&Ff[l$!+Q/,
S#)Ff[l7$$!+2G)em&Ff[lFhfl-F$6#7$7$$\"+C)*)fM%Ff[l$\"*vSi+*Fi[l7$$!+C)
*)fM%Ff[lFcgl-F$6#7$7$$\"+l!yuq%Ff[l$\"*=yE#))Fi[l7$$!+l!yuq%Ff[lF^hl-
F$6#7$7$$\"+'o9J#*)Ff[l$\"*^-U^%Fi[l7$$!+'o9J#*)Ff[lFihl-F$6#7$7$$\"+e
&Q#y%*Ff[l$\"*3Xz=$Fi[l7$$!+e&Q#y%*Ff[lFdil-F$6#7$7$$\"+Wic*R*Ff[l$!+k
#RHT$Ff[l7$$!+Wic*R*Ff[lF_jl-F$6#7$7$$\"+<N/a**Ff[l$\")R5w&*Fi[l7$$!+<
N/a**Ff[lFjjl-F$6#7$7$$\"+=d1J#*Ff[l$\"*?Ua%QFi[l7$$!+=d1J#*Ff[lFe[m-F
$6#7$7$$\"+C*4J,)Ff[l$\"*?![#)fFi[l7$$!+C*4J,)Ff[lF`\\m-F$6#7$7$$\"+S7
>r()Ff[l$!+K)GF![Ff[l7$$!+S7>r()Ff[lF[]m-F$6#7$7$$\"+^)oys(Ff[l$\"*0cm
M'Fi[l7$$!+^)oys(Ff[lFf]m-F$6#7$7$$\"+'GvUs*Ff[l$\"*y_?L#Fi[l7$$!+'GvU
s*Ff[lFa^m-F$6#7$7$$\"+$3`O0$Ff[l$\"*!GNA&*Fi[l7$$!+$3`O0$Ff[lF\\_m-F$
6#7$7$$\"+;I@n**Ff[l$!*?N64)Ff[l7$$!+;I@n**Ff[lFg_m-F$6#7$7$$\"+M2\\;t
Ff[l$\"*v9o\"oFi[l7$$!+M2\\;tFf[lFb`m-F$6#7$7$$\"+l6wsGFf[l$\"*8y%y&*F
i[l7$$!+l6wsGFf[lF]am-F$6#7$7$$\"+:(>_x)Ff[l$\"*Kk`z%Fi[l7$$!+:(>_x)Ff
[lFham-F$6#7$7$$\"+7F7Z(*Ff[l$!+whjMAFf[l7$$!+7F7Z(*Ff[lFcbm-F$6#7$7$$
\"+(4\\'R*)Ff[l$\"*mp8[%Fi[l7$$!+(4\\'R*)Ff[lF^cm-F$6#7$7$$\"+q9y9\"*F
f[l$\"**\\[8TFi[l7$$!+q9y9\"*Ff[lFicm-F$6#7$7$$\"+x#H,6)Ff[l$\"*J#G]eF
i[l7$$!+x#H,6)Ff[lFddm-F$6#7$7$$\"+`cX%***Ff[l$!*c2&HLFf[l7$$!+`cX%***
Ff[lF_em-F$6#7$7$$\"+$Qdho'Ff[l$\"*#43OuFi[l7$$!+$Qdho'Ff[lFjem-F$6#7$
7$$\"+`Wi:#*Ff[l$\"*j,B)QFi[l7$$!+`Wi:#*Ff[lFefm-F$6#7$7$$\"+Zk\"Q)**F
f[l$!*W2po&Ff[l7$$!+Zk\"Q)**Ff[lF`gm-F$6#7$7$$\"+gVQmZFf[l$!+jU*4z)Ff[
l7$$!+gVQmZFf[lF[hm-F$6#7$7$$\"+\\Acm\"*Ff[l$\"*wln*RFi[l7$$!+\\Acm\"*
Ff[lFfhm-F$6#7$7$$\"+D(=d#)*Ff[l$\"*pI)e=Fi[l7$$!+D(=d#)*Ff[lFaim-F$6#
7$7$$\"+Q*GyM&Ff[l$!+3]*)\\%)Ff[l7$$!+Q*GyM&Ff[lF\\jm-F$6#7$7$$\"+Ya!)
Q**Ff[l$!+IBg/6Ff[l7$$!+Ya!)Q**Ff[lFgjm-F$6#7$7$$\"+M.vpqFf[l$!+W]QsqF
f[l7$$!+M.vpqFf[lFb[n-F$6#7$7$$\"+uJLT)*Ff[l$\"*)eIu<Fi[l7$$!+uJLT)*Ff
[lF]\\n-F$6#7$7$$\"+([Y8'))Ff[l$!+K'yUj%Ff[l7$$!+([Y8'))Ff[lFh\\n-F$6#
7$7$$\"+lSZZfFf[l$!+Oj7R!)Ff[l7$$!+lSZZfFf[lFc]n-%(SCALINGG6#%,CONSTRA
INEDG" 2 254 254 254 2 0 1 0 2 9 0 4 1 1.000000 45.000000 45.000000 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "BertrandsParadox(50,3);" }}{PARA 6 
"" 1 "" {TEXT -1 39 ".2000000000   .5200000000   .4600000000" }}{PARA 
13 "" 1 "" {INLPLOT "6V-%'CURVESG6$7S7$$!\"\"\"\"!$\"1BmIq&o?5%!#D7$$!
1R6_uBO1**!#;$!12S\\y\"y_O\"F17$$!1'Gi)*4-Qn*F1$!1jmib*)GLDF17$$!1'4@/
['e[#*F1$!1,D1@.6.QF17$$!1!*H&3#42`')F1$!1lR!f$4U7]F17$$!19yQN?D/zF1$!
1t0ymceDhF17$$!1A_`rTd#3(F1$!1Tq)zpU&fqF17$$!10Yc\\dX;hF1$!1.A,Z!>8\"z
F17$$!1pt&)f:x5]F1$!1TPrOh-a')F17$$!1y<Y['R,#QF1$!1%)H\\4bcT#*F17$$!1u
0>!G'QDDF1$!1aaWL\"oen*F17$$!17*zm#o8X8F1$!19=>jt64**F17$$\"1JX(>nl5(f
!#>$!0e=t@)******!#:7$$\"1#ox^N#[i8F1$!1p$y\")GZn!**F17$$\"178SY@=YEF1
$!1/uE]D`V'*F17$$\"13bEF*3Jx$F1$!1N0PPl'3E*F17$$\"1\"3!3q_JU]F1$!1Nq9v
]oN')F17$$\"1?m3t%*[RgF1$!1-c(3/I-(zF17$$\"1\\6(QM;$*3(F1$!1XcFj?x_qF1
7$$\"1+)RI?/U!zF1$!1nSb(QZc7'F17$$\"1u&ext1$f')F1$!1,a/8/k,]F17$$\"1)*
)e:FO4B*F1$!1h_tBHvXQF17$$\"1>*[$3Nij'*F1$!1y(y0$Q%=d#F17$$\"1@*e)=+.2
**F1$!1vf64]Ug8F17$$\"1Q0F4>&*****F1$!14rpY)485$!#=7$$\"1&)oF>!Q%3**F1
$\"1$eM@&=8]8F17$$\"17BHoa1u'*F1$\"1orICIGKDF17$$\"1%\\yX@iKE*F1$\"1c'
=D`Bsw$F17$$\"1z$RT8'=u')F1$\"1;`*G)3zv\\F17$$\"1!RqPxng%zF1$\"18-xlUC
rgF17$$\"1Lc(*Rfc5rF1$\"1=9:4yMJqF17$$\"1bWQLke[gF1$\"1fkK&eGL'zF17$$
\"1?*3=U)o!*\\F1$\"1Qic4Mil')F17$$\"1E&>b[\"ysPF1$\"1$o(eQ***4E*F17$$
\"17DD&el'3EF1$\"1]=!\\p[Pl*F17$$\"1A*3$znc\"H\"F1$\"1%eQD*>C;**F17$$
\"1>GdRQQdIFcs$\"1l5\">E`*****F17$$!1:Vc\\Ks(G\"F1$\"1zHhX=u;**F17$$!1
>!Q<MT]b#F1$\"1yl^I'z!o'*F17$$!1#fp<Q)zNQF1$\"1uTu$>x]B*F17$$!1)*eas0I
.]F1$\"1=%3EhZ$e')F17$$!1?2C'R]$4hF1$\"1_]N%H2o\"zF17$$!1K'yNQ+$*4(F1$
\"1`2]j?sUqF17$$!1Zp?\"Ry,!zF1$\"19+(\\bQ38'F17$$!1X\"pT^PVn)F1$\"1#4Q
h(p_v\\F17$$!1ZCj^4+D#*F1$\"1a8\\!eo*fQF17$$!1-U:o))oa'*F1$\"1![NO%Q=0
EF17$$!1-tSNb&p!**F1$\"1v3@5t'4O\"F17$F($!1BmIq&o?5%F--%'COLOURG6&%$RG
BG$\"#5F)F*F*-F$6#7$7$$\"\"\"F*F*7$$\"+8YSYn!#5$!+\\\\Y\"Q(Fi[l-F$6#7$
Fc[l7$$\"+L#oP$*)Fi[l$!+B>3$\\%Fi[l-F$6#7$Fc[l7$$\"+^>Id**Fi[l$!+h88J#
*!#6-F$6#7$Fc[l7$$\"+/h!*e'*Fi[l$!+qV]*e#Fi[l-F$6#7$Fc[l7$$!+Ne&=Z\"Fi
[l$\"+R*)3\"*)*Fi[l-F$6#7$Fc[l7$$!+j!otE#Fi[l$\"+r1cR(*Fi[l-F$6#7$Fc[l
7$$!+3>pXuFi[l$!+WHXvmFi[l-F$6#7$Fc[l7$$\"+aM!z_(Fi[l$!+dNg#e'Fi[l-F$6
#7$Fc[l7$$!+BfKZsFi[l$\"+'Q-.*oFi[l-F$6#7$Fc[l7$$!+i$*)o*GFi[l$\"+%\\3
7d*Fi[l-F$6#7$Fc[l7$$!+SOc*G(Fi[l$!+];gXoFi[l-F$6#7$Fc[l7$$\"+0'Qbc*Fi
[l$\"+anb:HFi[l-F$6#7$Fc[l7$$\"+]<\\7\"*Fi[l$\"+c[b=TFi[l-F$6#7$Fc[l7$
$\"+(p![@*)Fi[l$!+F4V<XFi[l-F$6#7$Fc[l7$$!+N&=\\!**Fi[l$!+IRrv8Fi[l-F$
6#7$Fc[l7$$!+?bGImFi[l$!+&3Tf[(Fi[l-F$6#7$Fc[l7$$\"+mY(QJ*Fi[l$!+lPHSO
Fi[l-F$6#7$Fc[l7$$\"+YNL$>)Fi[l$\"+8T<LdFi[l-F$6#7$Fc[l7$$!+0xbnrFi[l$
!++HCtpFi[l-F$6#7$Fc[l7$$\"+]fIuHF\\]l$!+tdd&***Fi[l-F$6#7$Fc[l7$$!+IW
D;[Fi[l$\"+M;xj()Fi[l-F$6#7$Fc[l7$$!+M$*HK$)Fi[l$\"+![m#HbFi[l-F$6#7$F
c[l7$$\"+1^`+**Fi[l$!+[J\"pS\"Fi[l-F$6#7$Fc[l7$$!+(GFU,*Fi[l$!+Q-SHVFi
[l-F$6#7$Fc[l7$$!+tl!>,)Fi[l$!+m;4%)fFi[l-F$6#7$Fc[l7$$!+Q:V-**Fi[l$\"
+]E]$R\"Fi[l-F$6#7$Fc[l7$$\"+?+P0(*F\\]l$!+k9z_**Fi[l-F$6#7$Fc[l7$$!+x
S8JvFi[l$!+nr!*ylFi[l-F$6#7$Fc[l7$$\"+6D_NDFi[l$\"+@p@t'*Fi[l-F$6#7$Fc
[l7$$\"+5Ndg()F\\]l$\"+kAbh**Fi[l-F$6#7$Fc[l7$$\"+\\yzS**Fi[l$\"+7d_'3
\"Fi[l-F$6#7$Fc[l7$$!+.emQhFi[l$!+Kz4%*yFi[l-F$6#7$Fc[l7$$!+oyH#***Fi[
l$!+t32CRF\\]l-F$6#7$Fc[l7$$!+W?([!yFi[l$!+bvr^iFi[l-F$6#7$Fc[l7$$!+N.
W2PFi[l$!+p3N(G*Fi[l-F$6#7$Fc[l7$$!+Bc'o$)*Fi[l$!+6$4*)z\"Fi[l-F$6#7$F
c[l7$$!+:](\\p)Fi[l$!+!Rt$R\\Fi[l-F$6#7$Fc[l7$$!+t/@ogFi[l$\"+Y[Q[zFi[
l-F$6#7$Fc[l7$$!+P(GfN'Fi[l$!+gTC?xFi[l-F$6#7$Fc[l7$$\"+AXo*4\"Fi[l$!+
\"y]$R**Fi[l-F$6#7$Fc[l7$$!+'G&*pS$Fi[l$!+\\Bs,%*Fi[l-F$6#7$Fc[l7$$!+@
%4r%zFi[l$\"+62))pgFi[l-F$6#7$Fc[l7$$!+\")QDgxFi[l$\"+/n,2jFi[l-F$6#7$
Fc[l7$$\"+ecqA_Fi[l$\"+/%)zF&)Fi[l-F$6#7$Fc[l7$$!+<N;/!*Fi[l$\"+r=H]VF
i[l-F$6#7$Fc[l7$$\"+GEcVRFi[l$!+njd*=*Fi[l-F$6#7$Fc[l7$$\"+FxF([$!#7$
\"+%>R*****Fi[l-F$6#7$Fc[l7$$!+qt-Z)*Fi[l$\"+MqUU<Fi[l-F$6#7$Fc[l7$$!+
Z&Gba*Fi[l$\"+woT!)HFi[l-F$6#7$Fc[l7$$\"+eWBG!*Fi[l$\"+^U6+VFi[l-%(SCA
LINGG6#%,CONSTRAINEDG" 2 254 254 254 2 0 1 0 2 9 0 4 1 1.000000 
45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 
0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "BertrandsParado
x(50,4);" }}{PARA 6 "" 1 "" {TEXT -1 39 ".2400000000   .5800000000   .
3000000000" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0
 4 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 }