{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Blue Emp hasis" -1 262 "Times" 0 0 0 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 257 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 257 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 260 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 256 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 269 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 278 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 279 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 289 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 290 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 291 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 262 292 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple O utput" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } } {SECT 0 {PARA 3 "" 0 "" {TEXT -1 47 "Sets and the addition formula for probabilities" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo , B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 28.3.2007" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Sets" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 66 "Sets are designated in Maple by means of curly brackets \{ . . . \}." }}{PARA 0 "" 0 "" {TEXT -1 57 "The underlying concept of the notion of a set is that of \+ " }{TEXT 259 10 "membership" }{TEXT -1 177 ". A set is specified when \+ we know exactly which objects are members of the set. Thus a set can b e specified by listing its members, however the order of listing does \+ not matter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "A := \{6,8,5,3\};" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"AG<&\"\"$\"\"&\"\"'\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 58 "For printing purposes, a standard order i s used. Sets are " }{TEXT 259 5 "equal" }{TEXT -1 36 " when they conta in the same objects." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "if A=\{8,3,5,6\} then print('yes') end if ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$yesG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 6 "mem ber" }{TEXT -1 65 " can be used to test whether a given object is a me mber of a set." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "member(8,A);\nmember(9,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Sets can have any Maple objects as members, including sets." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "B := \{3,9 ,A\};\nmember(9,B);\nmember(A,B);\nmember(8,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG<%\"\"$\"\"*<&F&\"\"&\"\"'\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "A sequence can be made into a s et simply by enclosing the sequence in curly brackets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "\{seq(2 *i,i=0..10)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<-\"\"!\"\"#\"\"%\" \"'\"\")\"#5\"#7\"#9\"#;\"#=\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "If we construct the sequence backwards, w e still get the same set." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "seq(20-2*i,i=0..10);\n\{%\};" }} {PARA 11 "" 1 "" {XPPMATH 20 "6-\"#?\"#=\"#;\"#9\"#7\"#5\"\")\"\"'\"\" %\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<-\"\"!\"\"#\"\"%\"\"' \"\")\"#5\"#7\"#9\"#;\"#=\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 63 "The difference between a list and a set i s that for a list the " }{TEXT 259 13 "order matters" }{TEXT -1 217 ". Thus in a plot command you can specify which options are to be applie d to which graph by using lists. Notice that the graphs appear to be o n top of each other in the order of the listing with the first graph o n top." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot([x^2,x,sin(x)],x=-1..2,thickness=[3,5,2],color=[ green,blue,red]);" }}{PARA 13 "" 1 "" {GLPLOT2D 249 181 181 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$!\"\"\"\"!$\"\"\"F*7$$!1*****\\P&3Y$*!#;$\"1)))y O=J\\t)F07$$!1++Dcx6x()F0$\"1/zq5'zPq(F07$$!1,+]iTDP\")F0$\"1eB^I0\\@m F07$$!1****\\P\"\\J\\(F0$\"1)p\"o*RGZh&F07$$!1++DJa5_oF0$\"1vl4%)[8&p% F07$$!1,+DcexdiF0$\"1Vlqme(f\"RF07$$!1++D1?QUcF0$\"1x`Wquk$=$F07$$!1++ D13%f+&F0$\"1!*ycNV%f]#F07$$!1++D\"oS:P%F0$\"1sBy#zO5\">F07$$!1+++v@)* =PF0$\"1tnzTG3$Q\"F07$$!1++](G3U9$F0$\"1$o$=bd/'))*!#<7$$!1*****\\-\\r \\#F0$\"1[%eID`dB'F_o7$$!1+++vGVZ=F0$\"1m2jF#3IT$F_o7$$!1+++v4J@7F0$\" 1]al(\\+;\\\"F_o7$$!1,+]iIKFlF_o$\"1SWAOYfgU!#=7$$\"19+++DFOBFdp$\"1:j D%>p\"ea!#@7$$\"1,+++!R5'fF_o$\"1-@&ff)R`NFdp7$$\"1++vV!QBE\"F0$\"1w&) pOt\\$f\"F_o7$$\"1******\\\"o?&=F0$\"1?WCKk:IMF_o7$$\"1,+vVb4*\\#F0$\" 176zO&yaC'F_o7$$\"1,+DJ'=_6$F0$\"1Gr[?re/(*F_o7$$\"1,+]P%y!ePF0$\"1X-C a`J79F07$$\"1,+v=WU[VF0$\"1G\"eD\\z3*=F07$$\"1++]7B>&)\\F0$\"144ERU@&[ #F07$$\"1++v$>:mk&F0$\"1N)GYJE%)=$F07$$\"1++DcdQAiF0$\"1)Ge*\\%3=(QF07 $$\"1,+]PPBWoF0$\"1ALNXNN%o%F07$$\"1******\\Nm'[(F0$\"1e)*36J,0cF07$$ \"1****\\(yb^6)F0$\"1t%RbMvbe'F07$$\"1++vVVDB()F0$\"1C`dMm^4wF07$$\"1+ +]7TW)R*F0$\"1\"fyNk_)e:Fiu7$$\"1+] (=Yb;J\"Fiu$\"1?lm]+W?=+:Fiu$\"1h.7.ga]AFiu7$$\"1++DE&4Q c\"Fiu$\"1\\-RM-]XCFiu7$$\"1+]P%>5pi\"Fiu$\"1,8c!yOok#Fiu7$$\"1+++bJ*[ o\"Fiu$\"1aewV\\')QGFiu7$$\"1++Dr\"[8v\"Fiu$\"1?2%pT?s1$Fiu7$$\"1+++Ij y5=Fiu$\"1p[\"H8Z*yKFiu7$$\"1+]P/)fT(=Fiu$\"1Q\\Ls\\Z7NFiu7$$\"1+]i0j \"[$>Fiu$\"1uB^OT^VPFiu7$$\"\"#F*$\"\"%F*-%'COLOURG6&%$RGBGF*$\"*++++ \"!\")F*-%*THICKNESSG6#\"\"$-F$6%7S7$F(F(7$F.F.7$F4F47$F9F97$F>F>7$FCF C7$FHFH7$FMFM7$FRFR7$FWFW7$FfnFfn7$F[oF[o7$FaoFao7$FfoFfo7$F[pF[p7$F`p F`p7$FfpFfp7$F\\qF\\q7$FaqFaq7$FfqFfq7$F[rF[r7$F`rF`r7$FerFer7$FjrFjr7 $F_sF_s7$FdsFds7$FisFis7$F^tF^t7$FctFct7$FhtFht7$F]uF]u7$FbuFbu7$FguFg u7$F]vF]v7$FbvFbv7$FgvFgv7$F\\wF\\w7$FawFaw7$FfwFfw7$F[xF[x7$F`xF`x7$F exFex7$FjxFjx7$F_yF_y7$FdyFdy7$FiyFiy7$F^zF^z7$FczFcz7$FhzFhz-F][l6&F_ [lF*F*F`[l-Fd[l6#\"\"&-F$6%7S7$F($!1l*y![)4ZT)F07$F.$!1*GE#)elO/)F07$F 4$!1$*H5w\"*y#p(F07$F9$!19345\"3&osF07$F>$!1+'z6It8\"oF07$FC$!1Ds)4Yg$ GjF07$FH$!1Mia#*oFdeF07$FM$!1'RVk`AxM&F07$FR$!1)Q\"o(3o%*z%F07$FW$!1?& Q.#)GOB%F07$Ffn$!1*yP\\RXQj$F07$F[o$!1O!)o`ul#4$F07$Fao$!1sa%p9x7Z#F07 $Ffo$!1VTZa>%p$=F07$F[p$!1@I[eqF=7F07$F`p$!135`.!*oAlF_o7$Ffp$\"1'\\0Z PqiL#Fdp7$F\\q$\"1#HxDI4v&fF_o7$Faq$\"1$QG+;))*e7F07$Ffq$\"1YM`b\")\\T =F07$F[r$\"1v:`VK;tCF07$F`r$\"1\"*eeRg2lIF07$Fer$\"1Z9aU6CqOF07$Fjr$\" 1:4$R6vE@%F07$F_s$\"1ZaLm^D\"y%F07$Fds$\"1-Co%3*H^`F07$Fis$\"1/GV;jbGe F07$F^t$\"1*z)3gNEAjF07$Fct$\"1yuE4Mi1oF07$Fht$\"1I>28LJ`sF07$F]u$\"1 \"\\S)3PEewF07$Fbu$\"1'y#\\iKmu!)F07$Fgu$\"1TEE_;[<%)F07$F]v$\"1.Rh?XH \\()F07$Fbv$\"1nlFFhI=!*F07$Fgv$\"1W#H%oM$oF*F07$F\\w$\"1,]p@IB&[*F07$ Faw$\"1HOv$)[5m'*F07$Ffw$\"1^o0S5t0)*F07$F[x$\"1D\"[\"fm!>\"**F07$F`x$ \"1%='[N%y](**F07$Fex$\"1?O3Cfv****F07$Fjx$\"1w\"H+IgU)**F07$F_y$\"1T- 2^,)\\$**F07$Fdy$\"1Dl4luWP)*F07$Fiy$\"1l*y1^.Mr*F07$F^z$\"1IGj:6PV&*F 07$Fcz$\"1#p#)QIJZM*F07$Fhz$\"1 " 0 "" {MPLTEXT 1 0 32 "S1 := convert([3,8,10,2,7],s et);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S1G<'\"\"#\"\"$\"\"(\"\")\" #5" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 28 ". . or by using the command " }{TEXT 0 2 "op" }{TEXT -1 109 " to extrac t the members of a list as a sequence, and then enclosing the resultin g sequence in curly brackets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "S2 := \{op([3,8,10,2,7])\}; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S2G<'\"\"#\"\"$\"\"(\"\")\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "The fo llowing commands construct the set P of all prime numbers which are le ss than 100." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "seqn := NULL:\nfor i from 1 to 100 do\n if ispr ime(i) then seqn := seqn,i end if;\nend do;\nP := \{seqn\};\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG<;\"\"#\"\"$\"\"&\"\"(\"#6\"#8\" #<\"#>\"#B\"#H\"#J\"#P\"#T\"#V\"#Z\"#`\"#f\"#h\"#n\"#r\"#t\"#z\"#$)\"# *)\"#(*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "The union \+ and intersection of sets" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 5 "union" }{TEXT -1 35 " of two sets A and B is the set of " }{TEXT 291 57 "all objects which belong to either A or B or both A and B" }{TEXT -1 55 ".\nThe M aple command for computing the union of sets is " }{TEXT 0 5 "union" } {TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 12 "int ersection" }{TEXT -1 35 " of two sets A and B is the set of " }{TEXT 292 40 "all objects which belong to both A and B" }{TEXT -1 62 ".\nThe Maple command for computing the intersection of sets is " }{TEXT 0 9 "intersect" }{TEXT -1 1 "." }}{PARA 13 "" 1 "" {GLPLOT2D 297 172 172 {PLOTDATA 2 "6+-%)POLYGONSG6$7in7$$\"#;!\"\"\"\"!7$$\"+&*=_%f\"!\"*$\" +KYGX5!#57$$\"+,w9y:F/$\"+4p6z?F27$$\"+;l0^:F/$\"+X*p,4$F27$$\"+eaa8:F /$\"+IkOnSF27$$\"+/a-m9F/$\"+++++]F27$$\"+&*p,49F/$\"+?D&y(eF27$$\"+E[ 9V8F/$\"+jgI\"p'F27$$\"+118p7F/$\"+c#[9V(F27$$\"+__y(=\"F/$\"+X*p,4)F2 7$$\"+++++6F/$\"+SSDg')F27$$\"+Vmt15F/$\"+zXXN\"*F27$$\"+X*p,4*F2$\"+g ^c5&*F27$$\"+/p6z!)F2$\"+3gZ\"y*F27$$\"+HYGXqF2$\"+a*=_%**F27$$\"\"'F* $\"\"\"F+7$$\"+r`ra\\F2F\\p7$$\"+'4$)3#RF2Fgo7$$\"+b+$)4HF2Fbo7$$\"+vN jK>F2F]o7$$\"+++++5F2Fhn7$$\"*![Z@7F2FY7$$!*igI\"pF2FT7$$!+b#[9V\"F2FO 7$$!+X*p,4#F2FJ7$$!+SSDgEF2FE7$$!+xXXNJF2F@7$$!+g^c5NF2F;7$$!+2gZ\"y$F 2F67$$!+a*=_%RF2F07$$!\"%F*F+7$F[s$!+KYGX5F27$Fhr$!+4p6z?F27$Fer$!+X*p ,4$F27$Fbr$!+IkOnSF27$F_r$!+++++]F27$F\\r$!+?D&y(eF27$Fiq$!+jgI\"p'F27 $Ffq$!+c#[9V(F27$Fcq$!+X*p,4)F27$F`q$!+SSDg')F27$F]q$!+zXXN\"*F27$Fjp$ !+g^c5&*F27$Fgp$!+3gZ\"y*F27$Fdp$!+a*=_%**F27$F_p$F*F+7$FjoFhu7$FeoFeu 7$F`oFbu7$F[oF_u7$FfnF\\u7$FWFit7$FRFft7$FMFct7$FHF`t7$FCF]t7$F>Fjs7$F 9Fgs7$F4Fds7$F-FasF'-%'COLOURG6&%$RGBG$\")=THv!\")F^wF^w-F$6$7in7$$\" \"%F*F+7$$\"+a*=_%RF2F07$$\"+2gZ\"y$F2F67$$\"+g^c5NF2F;7$$\"+xXXNJF2F@ 7$$\"+SSDgEF2FE7$$\"+X*p,4#F2FJ7$$\"+b#[9V\"F2FO7$$\"*igI\"pF2FT7$$!*! [Z@7F2FY7$$!+++++5F2Fhn7$$!+vNjK>F2F]o7$$!+b+$)4HF2Fbo7$$!+'4$)3#RF2Fg o7$$!+r`ra\\F2F\\p7$$!\"'F*Fap7$$!+HYGXqF2F\\p7$$!+/p6z!)F2Fgo7$$!+X*p ,4*F2Fbo7$$!+Vmt15F/F]o7$$!+++++6F/Fhn7$$!+__y(=\"F/FY7$$!+118p7F/FT7$ $!+E[9V8F/FO7$$!+&*p,49F/FJ7$$!+/a-m9F/FE7$$!+eaa8:F/F@7$$!+;l0^:F/F;7 $$!+,w9y:F/F67$$!+&*=_%f\"F/F07$$!#;F*F+7$F\\]lFas7$Fi\\lFds7$Ff\\lFgs 7$Fc\\lFjs7$F`\\lF]t7$F]\\lF`t7$Fj[lFct7$Fg[lFft7$Fd[lFit7$Fa[lF\\u7$F ^[lF_u7$F[[lFbu7$FhzFeu7$FezFhu7$FbzF[v7$F_zFhu7$F\\zFeu7$FiyFbu7$FfyF _u7$FcyF\\u7$F`yFit7$F]yFft7$FjxFct7$FgxF`t7$FdxF]t7$FaxFjs7$F^xFgs7$F [xFds7$FhwFasFdwFjv-%'CURVESG6$7'7$$\"\"#F+$\"1++++++]9!#:7$$!\"#F+Fe_ l7$Fi_l$!1++++++]9Fg_l7$Fc_lF\\`lFb_l-F[w6&F]w$\"*++++\"F`wF+F+-%%TEXT G6$7$$!\"(F*$\"#7F*%\"AG-Fd`l6$7$$\"\"(F*Fi`l%\"BG-%(SCALINGG6#%,CONST RAINEDG-%+AXESLABELSG6$%!GFial-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAUL TGFabl" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}{PARA 0 "" 0 "" {TEXT -1 95 "In the Venn diagram shown, where the separate circular regions \+ represent the sets A and B, the " }{TEXT 263 5 "union" }{TEXT -1 27 " \+ of A and B corresponds to " }{TEXT 264 21 "the total shaded area" } {TEXT -1 12 ", while the " }{TEXT 265 12 "intersection" }{TEXT -1 27 " of A and B corresponds to " }{TEXT 266 40 "the region where the two c ircles overlap" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "A := \{2,5,7,8\};\nB := \{1, 5,6,8,9\};\n'A union B' = A union B;\n'A intersect B' = A intersect B; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG<&\"\"#\"\"&\"\"(\"\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG<'\"\"\"\"\"&\"\"'\"\")\"\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&unionG6$%\"AG%\"BG<)\"\"\"\"\"#\" \"&\"\"'\"\"(\"\")\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*interse ctG6$%\"AG%\"BG<$\"\"&\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The union and intersection of 3 or more sets ca n be found." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 115 "C := \{8,9,10,11\};\n'A union B union C' = A union B union C;\n'A intersect B intersect C' = A intersect B intersect C; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG<&\"\")\"\"*\"#5\"#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&unionG6$-F%6$%\"AG%\"BG%\"CG<+\"\" \"\"\"#\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*intersectG6$-F%6$%\"AG%\"BG%\"CG<#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "We can use the commands \+ " }{TEXT 0 5 "union" }{TEXT -1 5 " and " }{TEXT 0 9 "intersect" } {TEXT -1 43 " in a functional form if we use backquotes." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "`unio n`(A,B,C);\n`intersect`(A,B,C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<+ \"\"\"\"\"#\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 97 "Let S and T be respe ctively the sets of multiples of 3 and 5 which are less than or equal \+ to 100." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "S := \{seq(3*i,i=1..33)\};\nT := \{seq(5*i,i=1..20)\} ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG%\"TG<6\"\"&\"#5\"#:\"#?\"#D\"#I\"#N\"#S\"#X\"#]\"#b\" #g\"#l\"#q\"#v\"#!)\"#&)\"#!*\"#&*\"$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The intersection of S and T is the set of all multiples of 15 which are less than 100." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "'S interse ct T'=S intersect T;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*intersectG 6$%\"SG%\"TG<(\"#:\"#I\"#X\"#g\"#v\"#!*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 58 "Let \+ P be the set of prime numbers which are less than 100." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "seqn : = NULL:\nfor i from 1 to 100 do\n if isprime(i) then seqn := seqn,i \+ end if;\nend do:\nP := \{seqn\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"PG<;\"\"#\"\"$\"\"&\"\"(\"#6\"#8\"#<\"#>\"#B\"#H\"#J\"#P\"#T\"#V\"#Z \"#`\"#f\"#h\"#n\"#r\"#t\"#z\"#$)\"#*)\"#(*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Let Q be the set of all numbers less than 100 whose digits have a sum which is divisible by 4." }} {PARA 0 "" 0 "" {TEXT -1 1 "\004" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "seqn := NULL:\nfor i from 1 to 99 do\n units := i mod 10;\n tens := (i-units)/10;\n if (units + tens) mod 4 = 0 then seqn := \+ seqn,i end if;\nend do:\nQ := \{seqn\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG<:\"\"%\"\")\"#8\"#<\"#A\"#E\"#J\"#N\"#R\"#S\"#W\"#[\"#`\" #d\"#i\"#m\"#r\"#v\"#z\"#!)\"#%)\"#))\"#$*\"#(*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "Find the intersection of these sets, that is, the set of all prime numbers less than 100 whose digits have a sum which is divisible by 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "'P intersect Q'=P \+ intersect Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*intersectG6$%\"PG% \"QG<)\"#8\"#<\"#J\"#`\"#r\"#z\"#(*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT -1 26 " Let's se t up a procedure " }{TEXT 0 9 "getdigits" }{TEXT -1 58 " for obtaining the sequence of decimal digits of a number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "getdigits := proc (n::posint)\n local residual,temp,digit,seqn;\n residual := n;\n \+ seqn := NULL;\n while residual > 0 do\n temp := floor(residual /10);\n digit := residual - temp*10;\n residual := temp;\n \+ seqn := digit,seqn;\n end do;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "getdigits(25 678934);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*\"\"#\"\"&\"\"'\"\"(\"\") \"\"*\"\"$\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Let P be set of all the prime numbers between 500 and 600 . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "seqn := NULL:\nfor i from 500 to 600 do\n if ispri me(i) then seqn := seqn,i end if;\nend do:\nP := \{seqn\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG<0\"$.&\"$4&\"$@&\"$B&\"$T&\"$Z&\"$d& \"$j&\"$p&\"$r&\"$x&\"$(e\"$$f\"$*f" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 " . . and let Q be the set of all numbe rs between 500 and 600 which have a sum of digits which is divisible b y 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "seqn := NULL:\nfor i from 500 to 600 do\n dig := g etdigits(i);\n sumdigits := add(d,d=dig);\n if sumdigits mod 4 = 0 then seqn := seqn,i end if;\nend do:\nQ := \{seqn\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG<:\"$.&\"$2&\"$7&\"$;&\"$@&\"$D&\"$H&\"$I&\" $M&\"$Q&\"$V&\"$Z&\"$_&\"$c&\"$h&\"$l&\"$p&\"$q&\"$u&\"$y&\"$$e\"$(e\" $#f\"$'f" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "Now form the intersection of P and Q." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "'P intersect Q'=P \+ intersect Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*intersectG6$%\"PG% \"QG<'\"$.&\"$@&\"$Z&\"$p&\"$(e" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Complem entation for sets and complementary events " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Given sets A and B the " } {TEXT 259 29 "relative complement of B in A" }{TEXT -1 32 " is the set A \\ B consisting of " }{TEXT 290 35 "all objects of A which are not \+ in B" }{TEXT -1 52 ". To form A \\ B just remove any members of B from A." }}{PARA 0 "" 0 "" {TEXT -1 45 "The Maple command for relative com plement is " }{TEXT 0 5 "minus" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "A := \{3,5,6 ,8,10,13\};\nB := \{6,8,10,12,14\};\n'A minus B'=A minus B;\n'B minus \+ A'=B minus A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG<(\"\"$\"\"&\" \"'\"\")\"#5\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG<'\"\"'\"\") \"#5\"#7\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&minusG6$%\"AG%\"BG <%\"\"$\"\"&\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&minusG6$%\"BG% \"AG<$\"#7\"#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Let S and T be respectively the sets of multiples of 3 an d 5 which are less than or equal to 100." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "S := \{seq(3*i,i=1.. 33)\};\nT := \{seq(5*i,i=1..20)\};\n'S intersect T'=S intersect T;\nN \+ := rhs(%);\n'S minus T'=S minus T;\nM := rhs(%);\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"SG%\"TG <6\"\"&\"#5\"#:\"#?\"#D\"#I\"#N\"#S\"#X\"#]\"#b\"#g\"#l\"#q\"#v\"#!)\" #&)\"#!*\"#&*\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*intersectG6 $%\"SG%\"TG<(\"#:\"#I\"#X\"#g\"#v\"#!*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG<(\"#:\"#I\"#X\"#g\"#v\"#!*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&minusG6$%\"SG%\"TG<=\"\"$\"\"'\"\"*\"#7\"#=\"#@\"#C\"#F\"#L \"#O\"#R\"#U\"#[\"#^\"#a\"#d\"#j\"#m\"#p\"#s\"#y\"#\")\"#%)\"#()\"#$* \"#'*\"#**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG<=\"\"$\"\"'\"\"* \"#7\"#=\"#@\"#C\"#F\"#L\"#O\"#R\"#U\"#[\"#^\"#a\"#d\"#j\"#m\"#p\"#s\" #y\"#\")\"#%)\"#()\"#$*\"#'*\"#**" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 106 "The relative complement of T in S is the set of multiples of 3 less than 100 which are not multiples of 5." }} {PARA 0 "" 0 "" {TEXT -1 91 "This is the same as the set of multiples \+ of 3 less than 100 which are not multiples of 15.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "'S minus N'=S minus N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&minusG6$%\"SG%\"NG<=\"\"$\"\"'\"\"*\"#7\"#=\"#@\"#C \"#F\"#L\"#O\"#R\"#U\"#[\"#^\"#a\"#d\"#j\"#m\"#p\"#s\"#y\"#\")\"#%)\"# ()\"#$*\"#'*\"#**" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The complement of an event E in a sample space S of equa lly likely outcomes for an experiment is called the " }{TEXT 259 19 "c omplementary event" }{TEXT -1 25 " to E and denoted by E '." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "For example, t he complementary event to the event of obtaining a head when a coin is flipped is the event of obtaining a tail." }}{PARA 0 "" 0 "" {TEXT -1 155 "The complementary event of obtaining a total score greater tha n or equal to 9 when two dice are rolled is the event of obtaining a t otal score less than 9." }}{PARA 0 "" 0 "" {TEXT -1 55 "Note that, if \+ E and E ' are complementary events, then " }}{PARA 256 "" 0 "" {TEXT -1 20 " p(E) + p(E ') = 1." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 267 9 "_________" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Disjoint sets and mutually exclusive events" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedur e " }{TEXT 0 4 "nops" }{TEXT -1 54 " can be used to obtain the number \+ of objects in a set." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A := \{2,3,5,7,8,11\};\nnops(A);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG<(\"\"#\"\"$\"\"&\"\"(\"\")\"#6 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "seqn := NULL:\nfo r i from 1 to 100 do\n if isprime(i) then seqn := seqn,i end if;\nen d do:\nP := \{seqn\};\nnops(P);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"PG<;\"\"#\"\"$\"\"&\"\"(\"#6\"#8\"#<\"#>\"#B\"#H\"#J\"#P\"#T\"#V\"# Z\"#`\"#f\"#h\"#n\"#r\"#t\"#z\"#$)\"#*)\"#(*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 51 "There are 25 prime numbers which are less than 100." }} {PARA 0 "" 0 "" {TEXT -1 42 "Sets with no members in common are called " }{TEXT 259 8 "disjoint" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Let S be the set of square numbers between 1 and 100 inclusive." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "S := \{seq(i^2,i=1..10)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG<,\"\"\"\"\"%\"\"*\"#;\"#D\"#O \"#\\\"#k\"#\")\"$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The sets S and P, where P is the set of primes less \+ than 100, are disjoint." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "'S intersect P'=S intersect P;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%*intersectG6$%\"SG%\"PG<\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "If we for m the union of two disjoint sets, the number of objects in the union i s the " }{TEXT 259 3 "sum" }{TEXT -1 42 " of the number of objects in \+ the two sets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "nops(S);\nnops(P);\nnops(S union P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#D " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#N" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "In terms of probability theory, e vents which are disjoint subsets of a given sample space are called " }{TEXT 259 18 "mutually exclusive" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "For example, when rolling two dice, let E the event of obtaining a " }{TEXT 271 35 "total score less than or equal to 5" }{TEXT -1 36 ", and F be the event of obtain ing a " }{TEXT 268 38 "total score greater than or equal to 9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 319 "pts := [seq(seq([i,j],i=1..6),j=1..6)]:\nS := \{op(p ts)\};\nfav := NULL:\nfor i from 1 to nops(pts) do\n if pts[i,1]+pts [i,2] <=5 then\n fav := fav, pts[i];\n end if;\nend do:\nE := \+ \{fav\};\nfav := NULL:\nfor i from 1 to nops(pts) do\n if pts[i,1]+p ts[i,2] >=9 then\n fav := fav, pts[i];\n end if;\nend do:\nF := \{fav\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SG%\"EG<,7$\"\"\"\"\"%7$F' F'7$\"\"#F'7$\"\"$F'7$F(F'7$F'F+7$F+F+7$F-F+7$F'F-7$F+F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"FG<,7$\"\"'\"\"$7$\"\"&\"\"%7$F'F+7$F+F*7$F* F*7$F'F*7$F(F'7$F+F'7$F*F'7$F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 96 "We draw a picture which shows the outcome s in the events E and F in green and blue respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 371 "SM := \+ S minus (E union F):\nE := [op(E)]: F := [op(F)]: SM := [op(SM)]:\nall := [circle,cross,diamond]: \np1 := plot([SM$3],style=point,symbol=all ,color=COLOR(RGB,.6,.6,.6)):\np2 := plot([E$3],style=point,symbol=all, color=red):\np3 := plot([F$3],style=point,symbol=all,color=blue):\nplo ts[display]([p1,p2,p3],view=[0..6,0..6],\nlabels=[\"1st die\",\"2nd di e\"],scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6.-%'CURVESG6&727$$\"\"%\"\"!$\"\"$F*7$$\"\"#F*F(7$F+F(7$ $\"\"&F*F+7$F(F(7$$\"\"\"F*F27$F.F27$F+F27$F6$\"\"'F*7$F.F;7$F2F67$F;F 67$F(F.7$F2F.7$F;F.7$F+F+-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$RGBG$F " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Because \+ the events E and F are mutually exclusive, the probability of E or F o ccurring, that is, the probabilty of the union of E and F is" }}{PARA 256 "" 0 "" {TEXT -1 14 " p(E or F) = " }{XPPEDIT 18 0 "(n(E)+n(F))/n (S);" "6#*&,&-%\"nG6#%\"EG\"\"\"-F&6#%\"FGF)F)-F&6#%\"SG!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "n(E)/n(S)+n(F)/n(S);" "6#,&*&-%\"nG6#%\"EG \"\"\"-F&6#%\"SG!\"\"F)*&-F&6#%\"FGF)-F&6#F,F-F)" }{TEXT -1 15 " = p(E ) + p(F)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Thus the " }{TEXT 259 25 "theoretical probabilities" }{TEXT -1 26 " of the events E, F and E " }{TEXT 270 2 "or" }{TEXT -1 8 " F are: " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 8 " p(E) = " }{XPPEDIT 18 0 "10/36 = 5/18;" "6#/*&\"#5\"\"\"\"#O!\"\"*&\"\"&F&\" #=F(" }{TEXT -1 1 " " }{TEXT 273 1 "~" }{TEXT -1 16 " 0.2777777778, \+ " }}{PARA 256 "" 0 "" {TEXT -1 7 "p(F) = " }{XPPEDIT 18 0 "10/36 = 5/1 8" "6#/*&\"#5\"\"\"\"#O!\"\"*&\"\"&F&\"#=F(" }{TEXT -1 1 " " }{TEXT 274 1 "~" }{TEXT -1 16 " 0.2777777778, " }}{PARA 256 "" 0 "" {TEXT -1 13 " p(E or F) = " }{XPPEDIT 18 0 "20/36 = 5/9;" "6#/*&\"#?\"\"\"\" #O!\"\"*&\"\"&F&\"\"*F(" }{TEXT -1 1 " " }{TEXT 272 1 "~" }{TEXT -1 14 " 0.5555555556." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 269 12 " ____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "This addition rule can be applied to any pair o f mutually exclusive events." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "We can run a simulation of this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "randomi ze():\nrolldie := rand(1..6):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "The totals for the events E and F are acc umulated in the variables e and f." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "n := 50000;\ne := 0: f := 0 : t := 0:\nfor i from 1 to n do\n firstscore := rolldie();\n secon dscore := rolldie();\n s := firstscore+secondscore;\n if s<=5 then e := e+1 end if;\n if s>=9 then f := f+1 end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"&++&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " The" }{TEXT 259 27 " experimental pr obabilities" }{TEXT -1 36 " of the events E, F and E or F are: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "evalf(e/n,3);\nevalf(f/n,3);\nevalf((e+f)/n,3);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"$x#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$z#! \"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$c&!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "The addition formula for probabilities " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "Let A and B be respectively the sets of multiples of 3 and 5 which are less than or equal to 50." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A := \{seq(3*i,i=1..16)\};\nB := \{seq(5* i,i=1..10)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG<2\"\"$\"\"'\" \"*\"#7\"#:\"#=\"#@\"#C\"#F\"#I\"#L\"#O\"#R\"#U\"#X\"#[" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"BG<,\"\"&\"#5\"#:\"#?\"#D\"#I\"#N\"#S\"#X\"# ]" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 251 "Th e intersection of A and B is the set of all multiples of 15 which are \+ less than 50 and the union of A and B is the set of all numbers less t han or equal to 50 which are divisible by either 3 or 5 including the \+ possibilty of being divisible by both. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "N := A intersect B;\nU \+ := A union B;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG<%\"#:\"#I\"#X " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG<9\"\"$\"\"&\"\"'\"\"*\"#5\" #7\"#:\"#=\"#?\"#@\"#C\"#D\"#F\"#I\"#L\"#N\"#O\"#R\"#S\"#U\"#X\"#[\"#] " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "nops(A);\nnops(B);\nnops(N); \nnops(U);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#B" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "The sets A, B, N and U are all subset s of the set S of integers from 1 to 50." }}{PARA 0 "" 0 "" {TEXT -1 97 "In the following Venn diagram the number of the 50 members of the \+ set S in each region is given. " }}{PARA 13 "" 1 "" {GLPLOT2D 321 185 185 {PLOTDATA 2 "61-%'CURVESG6$7'7$$\"\"#\"\"!$\"1++++++]9!#:7$$!\"#F* F+7$F/$!1++++++]9F-7$F(F2F'-%'COLOURG6&%$RGBGF*F*F*-F$6$7in7$$\"1+++++ ++;F-F*7$$\"1+++&*=_%f\"F-$\"1+++KYGX5!#;7$$\"1+++,w9y:F-$\"1+++4p6z?F D7$$\"1+++;l0^:F-$\"1+++X*p,4$FD7$$\"1+++eaa8:F-$\"1+++IkOnSFD7$$\"1++ +/a-m9F-$\"1+++++++]FD7$$\"1+++&*p,49F-$\"1+++?D&y(eFD7$$\"1+++E[9V8F- $\"1+++jgI\"p'FD7$$\"1+++118p7F-$\"1*****fD[9V(FD7$$\"1+++__y(=\"F-$\" 1+++X*p,4)FD7$$\"1+++++++6F-$\"1,++SSDg')FD7$$\"1+++Vmt15F-$\"1+++zXXN \"*FD7$$\"1+++X*p,4*FD$\"1+++g^c5&*FD7$$\"1+++/p6z!)FD$\"1+++3gZ\"y*FD 7$$\"1+++HYGXqFD$\"1+++a*=_%**FD7$$\"1+++++++gFD$\"\"\"F*7$$\"1+++r`ra \\FDF^q7$$\"1+++'4$)3#RFDFip7$$\"1+++b+$)4HFDFdp7$$\"1+++vNjK>FDF_p7$$ \"1+++++++5FDFjo7$$\"1++++[Z@7!#FDF_p7$$!1+++b+$)4HFDFdp7$$!1+++'4$)3#RFDFip7$$!1+++r`ra\\FDF^q 7$$!1+++++++gFDFcq7$$!1+++HYGXqFDF^q7$$!1+++/p6z!)FDFip7$$!1+++X*p,4*F DFdp7$$!1+++Vmt15F-F_p7$$!1+++++++6F-Fjo7$$!1+++__y(=\"F-Feo7$$!1+++11 8p7F-F`o7$$!1+++E[9V8F-F[o7$$!1+++&*p,49F-Ffn7$$!1+++/a-m9F-FW7$$!1+++ eaa8:F-FR7$$!1+++;l0^:F-FM7$$!1+++,w9y:F-FH7$$!1+++&*=_%f\"F-FB7$$!1++ +++++;F-F*7$F^^lFdt7$F[^lFgt7$Fh]lFjt7$Fe]lF]u7$Fb]lF`u7$F_]lFcu7$F\\] lFfu7$Fi\\lFiu7$Ff\\lF\\v7$Fc\\lF_v7$F`\\lFbv7$F]\\lFev7$Fj[lFhv7$Fg[l F[w7$Fd[lF^w7$Fa[lF[w7$F^[lFhv7$F[[lFev7$FhzFbv7$FezF_v7$FbzF\\v7$F_zF iu7$F\\zFfu7$FiyFcu7$FfyF`u7$FcyF]u7$F`yFjt7$F]yFgt7$FjxFdtFfx-F66&F8F *F*F`x-%%TEXTG6%7$$!\"(F_w$\"#7F_w%\"AGF``l-Fc`l6%7$$\"\"(F_wFh`l%\"BG F^x-Fc`l6%7$F*$\"\"&F_w%\"NGF5-Fc`l6%7$F*$F_wF_w%\"3GF5-Fc`l6%7$FcqF*% \"7GF5-Fc`l6%7$F^wF*%#13GF5-Fc`l6%7$$!#=F_w$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "An example involving rolling two \+ dice" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 152 " When rolling two dice, let E be the event that the first die comes up \+ with a 4 and F be the event of obtaining a total score greater than or equal to 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "pts := [seq(seq([i,j],i=1..6),j=1..6)]:\nS := \{o p(pts)\};\nE := \{seq([4,j],j=1..6)\};\nfav := NULL:\nfor i from 1 to \+ nops(pts) do\n if pts[i,1]+pts[i,2] >=8 then\n fav := fav, pts[ i];\n end if;\nend do:\nF := \{fav\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SG%\"EG<(7$\"\"%\"\"$7$F'F'7$F'\"\"&7$F'\"\"'7$F'\"\"\"7$F'\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG<17$\"\"'\"\"$7$\"\"&F(7$\" \"%F,7$F*F,7$F'F,7$F(F*7$F,F*7$F*F*7$F'F*7$\"\"#F'7$F(F'7$F,F'7$F*F'7$ F'F'7$F'F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "In the following picture the members of the intersection of E \+ and F are coloured magenta, while the remaining members of E and F are coloured red and blue respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 504 "SM := S minus (E union F) :\nN := E intersect F:\nEM := E minus N:\nFM := F minus N:\nSM := [op( SM)]: N := [op(N)]: EM := [op(EM)]: FM := [op(FM)]:\nall := [circle,cr oss,diamond]: \np1 := plot([SM$3],style=point,symbol=all,color=COLOR(R GB,.7,.7,.7)):\np2 := plot([EM$3],style=point,symbol=all,color=red):\n p3 := plot([FM$3],style=point,symbol=all,color=blue):\np4 := plot([N$3 ],style=point,symbol=all,color=magenta):\nplots[display]([p1,p2,p3,p4] ,view=[0..6,0..6],\nlabels=[\"1st die\",\"2nd die\"],scaling=constrain ed);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "61-%'CURVE SG6&747$$\"\"\"\"\"!$\"\"%F*7$$\"\"#F*F+7$$\"\"$F*F+7$F($\"\"&F*7$F.F4 7$F($\"\"'F*7$F(F(7$F.F(7$F1F(7$F4F(7$F8F(7$F(F.7$F.F.7$F1F.7$F4F.7$F( F17$F.F17$F1F1-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$RGBG$\"\"(!\"\"FNFN-%& STYLEG6#%&POINTG-F$6&F&-FG6#%&CROSSGFJFQ-F$6&F&-FG6#%(DIAMONDGFJFQ-F$6 &7%7$F+F17$F+F(7$F+F.FF-%'COLOURG6&FM$\"*++++\"!\")$F*F*FeoFQ-F$6&F[oF WF_oFQ-F$6&F[oFfnF_oFQ-F$6&7.7$F8F17$F4F17$F4F+7$F8F+7$F1F47$F4F47$F8F 47$F.F87$F1F87$F4F87$F8F87$F8F.FF-F`o6&FMFeoFeoFboFQ-F$6&F\\pFWFipFQ-F $6&F\\pFfnFipFQ-F$6&7%7$F+F+7$F+F47$F+F8FF-F`o6&FMFboFeoFboFQ-F$6&FaqF WFeqFQ-F$6&FaqFfnFeqFQ-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q(1st ~die6\"Q(2nd~dieFcr%(DEFAULTG-%%VIEWG6$;FeoF8Fir" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Cu rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C urve 11" "Curve 12" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Thus the \+ " }{TEXT 259 25 "theoretical probabilities" }{TEXT -1 26 " of the even ts E, F and E " }{TEXT 282 3 "and" }{TEXT -1 8 " F are: " }}{PARA 256 "" 0 "" {TEXT -1 8 " p(E) = " }{XPPEDIT 18 0 "6/36 = 1/6;" "6#/*&\"\"' \"\"\"\"#O!\"\"*&F&F&F%F(" }{TEXT -1 1 " " }{TEXT 284 1 "~" }{TEXT -1 15 " 0.1666666667, " }}{PARA 256 "" 0 "" {TEXT -1 8 " p(F) = " } {XPPEDIT 18 0 "15/36 = 5/12;" "6#/*&\"#:\"\"\"\"#O!\"\"*&\"\"&F&\"#7F( " }{TEXT -1 1 " " }{TEXT 285 1 "~" }{TEXT -1 16 " 0.4166666667, " }} {PARA 256 "" 0 "" {TEXT -1 5 " p(E " }{TEXT 283 3 "and" }{TEXT -1 6 " \+ F) = " }{XPPEDIT 18 0 "3/36 = 1/12;" "6#/*&\"\"$\"\"\"\"#O!\"\"*&F&F& \"#7F(" }{TEXT -1 1 " " }{TEXT 286 1 "~" }{TEXT -1 16 " 0.08333333333. " }}{PARA 0 "" 0 "" {TEXT -1 6 " Then " }}{PARA 256 "" 0 "" {TEXT -1 6 " p(E " }{TEXT 287 2 "or" }{TEXT -1 24 " F) = p(E) + p(F) - p(E " } {TEXT 288 3 "and" }{TEXT -1 7 " F) = " }{XPPEDIT 18 0 "1/6+5/12-1/12 \+ = 1/2;" "6#/,(*&\"\"\"F&\"\"'!\"\"F&*&\"\"&F&\"#7F(F&*&F&F&F+F(F(*&F&F &\"\"#F(" }{TEXT -1 8 " = 0.5. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 289 25 "_________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "We can run a simulation of this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "randomize(): \nrolldie := rand(1..6):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 164 "The totals for the events E and F are accumulated in the variables e and f and those for the events (E and F) and (E or F) are accumulated in the variables t and u." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 335 "n := 50000; \ne := 0: f := 0: t := 0: u := 0:\nfor i from 1 to n do\n firstscore := rolldie();\n secondscore := rolldie();\n s := firstscore+secon dscore;\n if firstscore=4 then e := e+1 end if;\n if s>=8 then f : = f+1 end if; \n if firstscore=4 and s>=8 then t := t+1 end if;\n \+ if firstscore=4 or s>=8 then u := u+1 end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"&++&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "The" }{TEXT 259 27 " experimental probabil ities" }{TEXT -1 23 " of the events E, F, E " }{TEXT 280 3 "and" } {TEXT -1 9 " F and E " }{TEXT 281 2 "or" }{TEXT -1 8 " F are: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "evalf(e/n,3);\nevalf(f/n,3);\nevalf(t/n,3);\nevalf(u/n,3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$n\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$>%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$N)!\" %" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$-&!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 99 "Find the set of all \+ prime numbers less than 300, which have a sum of digits that is divisi ble by 7." }}{PARA 0 "" 0 "" {TEXT -1 32 "____________________________ ____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }} {PARA 0 "" 0 "" {TEXT -1 79 "(a) When two dice are rolled, find the th eoretical probabilities of obtaining: " }}{PARA 0 "" 0 "" {TEXT -1 63 " (i) E . . the same number turning up on both dice, " }} {PARA 0 "" 0 "" {TEXT -1 57 " (ii) F . . a total score less than or equal to 10," }}{PARA 0 "" 0 "" {TEXT -1 21 " (iii) E and F, " }}{PARA 0 "" 0 "" {TEXT -1 19 " (iv) E or F." }}{PARA 0 "" 0 " " {TEXT -1 110 "(b) Perform a simulation with a large number of trials , to obtain experimental values for these probabilities." }}{PARA 0 " " 0 "" {TEXT -1 32 "________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "____________________ ____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 20 "Code f or 1st picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 352 "with(plots):\nd := Pi/30: h := 'h':\nc1 := [seq (evalf([cos(d*i)+h,sin(d*i)]),i=0..60)]:\nh := 0.6:\np1 := polygonplot (c1,color=gray):\nh := -0.6:\np2 := polygonplot(c1,color=gray):\nh := \+ 2: k := 1.45:\np3 := plot([[h,k],[-h,k],[-h,-k],[h,-k],[h,k]],color=re d):\nt1 := textplot([[-.7,1.2,`A`],[.7,1.2,`B`]]):\ndisplay(\{p1,p2,p3 ,t1\},scaling=constrained,axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 " " 0 "" {TEXT -1 20 "Code for 2nd picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 528 "with(plots):\nA := ' A': B := 'B': N := 'N':S := 'S':\nd := Pi/30: h := 'h':\nc1 := [seq(ev alf([cos(d*i)+h,sin(d*i)]),i=0..60)]:\nh := 0.6:\np1 := plot(c1,color= red):\nh := -0.6:\np2 := plot(c1,color=blue):\nh := 2: k := 1.45:\np3 \+ := plot([[h,k],[-h,k],[-h,-k],[h,-k],[h,k]],color=black):\nt1 := textp lot([-.7,1.2,`A`],color=blue):\nt2 := textplot([.7,1.2,`B`],color=red) :\nt3 := textplot([[0,0.5,`N`],[0,-.1,`3`],\n[1,0,`7`],[-1,0,`13`],[-1 .8,1.7,`S`],[-1.7,-1,`27`]],color=black):\ndisplay(\{p1,p2,p3,t1,t2,t3 \},axes=none,scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }