{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }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 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Headi ng 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 " " 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 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 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 259 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 262 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 263 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 264 1 {CSTYLE " " -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 265 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 266 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 267 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 268 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 269 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 270 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 271 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 15 272 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 273 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 274 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 276 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 278 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 279 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 280 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 281 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 282 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 283 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 284 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 285 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 288 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 289 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 290 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 291 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 292 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 293 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 294 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 295 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 15 296 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 297 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 298 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 299 1 {CSTYLE " " -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 300 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 302 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 303 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 304 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 306 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 307 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 309 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 310 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 312 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 313 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 314 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 315 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 316 1 {CSTYLE " " -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 317 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 318 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 319 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 320 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 321 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 323 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 324 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 326 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 328 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 330 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 331 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 334 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 335 1 {CSTYLE " " -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 336 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 337 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 338 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 339 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 340 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 341 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 342 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 343 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 344 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 345 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 346 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 347 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 348 1 {CSTYLE " " -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 349 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 350 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 351 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 352 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 353 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 354 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 355 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 356 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 357 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 358 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 359 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 360 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 361 1 {CSTYLE " " -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 362 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 363 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 364 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 365 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 366 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 367 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 368 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 369 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 370 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 372 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 373 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 375 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 377 1 {CSTYLE " " -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 379 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 15 380 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT 256 8 "Kent Ngo" }}{PARA 256 " " 0 "" {TEXT -1 13 "Final Project" }}{PARA 257 "" 0 "" {TEXT -1 8 "Mat h 183" }}{PARA 258 "" 0 "" {TEXT -1 12 "Prof. Wavrik" }}{PARA 259 "" 0 "" {TEXT -1 7 "6/13/02" }}{PARA 261 "" 0 "" {TEXT 259 28 "Winning Pr obability of Craps" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 "Introducti on" }}{PARA 265 "" 0 "" {TEXT -1 188 "The game is played using 2 6-sid ed dice. The shooter throws the dice across the craps table, and the d ice have to bounce against the wall on the opposite side to make the t hrow to be fair." }}{PARA 266 "" 0 "" {TEXT -1 302 "There are 2 basic \+ ways the shooter (the person rolling the dice) wins:\n1. Throwing eith er a 7 or 11 on his first roll, this is called a \"natural\".\n2. Thro wing either a 4, 5, 6, 8, 9, 10 on his first roll and then rolling tha t number again before he rolls a 7 (craps), this is called \"making hi s point\"." }}{PARA 267 "" 0 "" {TEXT -1 64 "If the the first roll is \+ a 2, 3 or a 12, then the shooter loses." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Objectives" }}{PARA 268 "" 0 "" {TEXT -1 79 "Calculate th e winning probability of the shooter using probability mathematics." } }{PARA 269 "" 0 "" {TEXT -1 88 "Create a simulation of the game of cra ps, so that we can play it just like in real life." }}{PARA 270 "" 0 " " {TEXT -1 156 "Run the simulation and then find the winning probabili ty of the shooter from the simulation by dividing the number of wins b y the number of games we played." }}{PARA 271 "" 0 "" {TEXT -1 82 "The n compare that with the true probability. It should be close to the tr ue value." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 34 "Winning probability of the shooter" }{TEXT 258 0 "" }}{EXCHG {PARA 272 "" 0 "" {TEXT -1 103 "The probability of winning in the first throw (throwing a \"natur al\") is : P(7) + P(11) = 6/36 + 2/36 = " }{TEXT 260 4 "8/36" }}{PARA 273 "" 0 "" {TEXT -1 481 "To calculate the probability of winning in t he second throw, we assume that x is the \"point\" with probability p. The probability of not winning in any given throw is given by the pro bability of not x and not 7 = r = 1 - p - 1/6 . The conditional probab ility is :\n\n P\{win | x in first throw\} = p + r*p + (r^2) * p + \+ \205 an infinite geometric series. \n \+ = p / (1- r) , and\n P \{win in second or subsequent throws\} \+ = p*p / (1 - r) = p^2 / (1 - r)" }}{PARA 274 "" 0 "" {TEXT -1 15 "So w hen x = 4, " }}{PARA 275 "" 0 "" {TEXT -1 81 " P ( win after \+ first with x = 4) = (3/36)^2 / (1 - (1 - 3/36 - 1/6 )) = " }{TEXT 261 4 "1/36" }}{PARA 276 "" 0 "" {TEXT -1 6 "x = 5," }}{PARA 277 "" 0 "" {TEXT -1 80 " P (win after first with x = 5) = (4/36)^2 / (1 \+ - (1 - 4/36 - 1/6 )) = " }{TEXT 262 6 "16/360" }}{PARA 278 "" 0 "" {TEXT -1 6 "x = 6," }}{PARA 279 "" 0 "" {TEXT -1 80 " P (win \+ after first with x = 6) = (5/36)^2 / (1 - (1 - 5/36 - 1/6 )) = " } {TEXT 263 6 "25/396" }}{PARA 280 "" 0 "" {TEXT -1 224 "same probabilit ies for 8, 9, 10, since the probability of throwing a 4 is the same as the probability of throwing a 10, and same with 5 and 9, and 6 and 8. \n\nThus, P (shooter wins ) = 8/36 + 2*(1/36) + 2*(16/360) + 2*(25/396 )" }}{PARA 281 "" 0 "" {TEXT -1 38 " \+ = " }{TEXT 257 5 "0.493" }}{PARA 282 "" 0 "" {TEXT -1 0 "" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "Graph of probabilities" }}{PARA 283 "" 0 "" {TEXT -1 118 "We're going to graph the probabilities of al l possible ways to win, e.g. throwing a \"natural\" and \"making the p oint\".\n" }}{EXCHG {PARA 284 "> " 0 "" {MPLTEXT 1 0 151 "with(plottoo ls):\nL := curve([[0,8/36], [1,1/36],[2,16/360],[3,25/396],[4,25/396], [5,16/360],[6,1/36]], color=green, thickness = 2):\nplots[display](L); \n" }}{PARA 285 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6#-%'CURVESG6%7)7$$\"\"!F)$\"+AAAAA!#57$$\"\" \"F)$\"+yxxxF!#67$$\"\"#F)$\"+WWWWWF27$$\"\"$F)$\"+8888jF27$$\"\"%F)F; 7$$\"\"&F)F67$$\"\"'F)F0-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-%*THICKNE SSG6#F5" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" }}}}{EXCHG {PARA 288 "" 0 "" {TEXT -1 83 " In the graphs abov e, at x = 0 is the winning probability by throwing a natural" }}{PARA 289 "" 0 "" {TEXT -1 96 " x = 1 \+ is the winning probability by making the point of 4" }}{PARA 290 "" 0 "" {TEXT -1 96 " x = 2 is the wi nning probability by making the point of 5" }}{PARA 291 "" 0 "" {TEXT -1 96 " x = 3 is the winning pro bability by making the point of 6" }}{PARA 292 "" 0 "" {TEXT -1 96 " \+ x = 4 is the winning probability \+ by making the point of 8" }}{PARA 293 "" 0 "" {TEXT -1 96 " \+ x = 5 is the winning probability by making the point of 9" }}{PARA 294 "" 0 "" {TEXT -1 97 " \+ x = 6 is the winning probability by making the poin t of 10" }}}{PARA 295 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 19 "Simulation of craps" }}{PARA 296 "" 0 "" {TEXT -1 159 " This program simulates the game of craps. It will play the game as man y times as you want, and will keep track of the number of wins and los ses of the shooter." }}{PARA 297 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 298 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 299 "> " 0 "" {MPLTEXT 1 0 18 "die := rand(1..6):" }}}{EXCHG {PARA 300 "> " 0 "" {MPLTEXT 1 0 4 "die;" }}{PARA 2 "" 1 "" {TEXT -1 50 " \+ die" }}}{EXCHG {PARA 302 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 303 "" 0 "" {TEXT -1 44 "The program below simulates throwing 2 dice." }}}{EXCHG {PARA 304 "> " 0 "" {MPLTEXT 1 0 131 "dice := proc()local d,i;\n\n#simulates throwing 2 di ce\n d:=0;\n for i from 1 to 2 do\n d := d + die();\n \+ od;\n d;\nend:" }}}{EXCHG {PARA 306 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 307 "> " 0 "" {MPLTEXT 1 0 7 "dice();" }}{PARA 2 "" 1 "" {TEXT -1 49 " 7" }}} {EXCHG {PARA 309 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 310 "> " 0 "" {MPLTEXT 1 0 7 "dice();" }}{PARA 2 "" 1 "" {TEXT -1 49 " \+ 10" }}}{EXCHG {PARA 312 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 313 "> " 0 "" {MPLTEXT 1 0 121 "te st := proc (n)local List,i,d;\n\n#Generates a list containing the numb er of times a sum of 2, 3 ... 12\n#of 2 dice appears" }}{PARA 314 "> \+ " 0 "" {MPLTEXT 1 0 223 " List := [seq(0,i = 2..12)];\n\n #Creat es a list having a total of 11 elements\n #with the first element s toring the number of times a sum of 2\n #appears and the next eleme nt for the sum of 3, and so on, up to 12.\n" }}{PARA 315 "> " 0 "" {MPLTEXT 1 0 24 " for i from 1 to n do" }}{PARA 316 "> " 0 "" {MPLTEXT 1 0 20 " d := dice();" }}{PARA 317 "> " 0 "" {MPLTEXT 1 0 51 " List := subsop( d-1 = List[d-1] + 1, List);" }}{PARA 318 "> " 0 "" {MPLTEXT 1 0 7 " od;" }}{PARA 319 "> " 0 "" {MPLTEXT 1 0 9 " List;" }}{PARA 320 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 321 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 323 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 324 "> " 0 "" {MPLTEXT 1 0 14 "test( 10000 );" }}{PARA 2 "" 1 "" {TEXT -1 78 " [281, 548, 875, 1099, 1398, 1709, 1386, 1042, 839, 562, 261]" }}}{EXCHG {PARA 326 "> " 0 "" {MPLTEXT 1 0 15 "evalf(%/10000);" }}{PARA 2 "" 1 "" {TEXT -1 160 " [.02810000000, .05480000000, .08750000000, .1099000000 , .1398000000, .1709000000, .1386000000,\n\n .1042000000, .0839 0000000, .05620000000, .02610000000]" }}}{EXCHG {PARA 328 "> " 0 "" {MPLTEXT 1 0 10 "List := %;" }}{PARA 2 "" 1 "" {TEXT -1 168 " List := [.02810000000, .05480000000, .08750000000, .1099000000, .1398000000, \+ .1709000000,\n\n .1386000000, .1042000000, .08390000000, .05620 000000, .02610000000]" }}}{EXCHG {PARA 330 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 331 "> " 0 "" {MPLTEXT 1 0 226 "with(plottools):\nL \+ := curve([[0, List[1]],[1, List[2]],[2, List[3]],[3, List[4]],[4, List [5]],\n[5, List[6]],[6, List[7]],[7, List[8]],[8, List[9]],[9, List[10 ]],[10, List[11]]],\ncolor=green, thickness = 2):\nplots[display](L); \n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6#-%'CURVESG 6%7-7$$\"\"!F)$\"++++5G!#67$$\"\"\"F)$\"++++![&F,7$$\"\"#F)$\"++++]()F ,7$$\"\"$F)$\"++++*4\"!#57$$\"\"%F)$\"++++)R\"F<7$$\"\"&F)$\"++++4 " 0 "" {MPLTEXT 1 0 133 "craps := proc (n)local sh oot_win,shoot_loss,house_win,house_loss,i,j,toss,prev_toss,List;\n\n#T his program simulates the game of craps\n" }}{PARA 335 "> " 0 "" {MPLTEXT 1 0 19 " shoot_win := 0;" }}{PARA 336 "> " 0 "" {MPLTEXT 1 0 20 " shoot_loss := 0;" }}{PARA 337 "> " 0 "" {MPLTEXT 1 0 19 " \+ house_win := 0;" }}{PARA 338 "> " 0 "" {MPLTEXT 1 0 20 " house_l oss := 0;" }}{PARA 339 "> " 0 "" {MPLTEXT 1 0 30 " List := [seq(0,i = 1..2)];" }}{PARA 340 "> " 0 "" {MPLTEXT 1 0 4 " " }{TEXT -1 0 " " }{MPLTEXT 1 0 25 "for i from 1 by 1 to n do" }}{PARA 341 "> " 0 "" {MPLTEXT 1 0 82 " toss := dice();\n \n #if it's a natural, then the shooter wins" }}{PARA 342 "> " 0 "" {MPLTEXT 1 0 39 " if (toss = 7 or toss = 11) then" }}{PARA 343 "> " 0 "" {MPLTEXT 1 0 39 " shoot_win := shoot_win + 1;" }}{PARA 344 "> " 0 "" {MPLTEXT 1 0 94 " house_loss := house_loss + 1;\n \n #if the first throw is 4, 5, 6, 8, 9, or 10" }}{PARA 345 "> \+ " 0 "" {MPLTEXT 1 0 89 " elif (toss = 4 or toss = 5 or toss = 6 or toss = 8 or toss = 9 or toss = 10) then" }}{PARA 346 "> " 0 "" {MPLTEXT 1 0 30 " prev_toss := toss;" }}{PARA 347 "> " 0 " " {MPLTEXT 1 0 27 " toss := dice();" }}{PARA 348 "> " 0 "" {MPLTEXT 1 0 70 " for j from 1 by 1 while toss <> prev_toss and toss <> 7 do" }}{PARA 349 "> " 0 "" {MPLTEXT 1 0 34 " \+ prev_toss := toss;" }}{PARA 350 "> " 0 "" {MPLTEXT 1 0 31 " \+ toss := dice();" }}{PARA 351 "> " 0 "" {MPLTEXT 1 0 76 " \+ od;\n \n #the shooter gets that number ag ain" }}{PARA 352 "> " 0 "" {MPLTEXT 1 0 38 " if (toss = pre v_toss) then" }}{PARA 353 "> " 0 "" {MPLTEXT 1 0 43 " s hoot_win := shoot_win + 1;" }}{PARA 354 "> " 0 "" {MPLTEXT 1 0 80 " \+ house_loss := house_loss + 1;\n\n #the shooter gets a 7" }}{PARA 355 "> " 0 "" {MPLTEXT 1 0 16 " else" }} {PARA 356 "> " 0 "" {MPLTEXT 1 0 43 " house_win := hous e_win + 1;" }}{PARA 357 "> " 0 "" {MPLTEXT 1 0 44 " sho ot_loss := shoot_loss +1;" }}{PARA 358 "> " 0 "" {MPLTEXT 1 0 80 " \+ fi;\n\n #the first throw is 2, 3, or 12, then the shoot er loses" }}{PARA 359 "> " 0 "" {MPLTEXT 1 0 12 " else" }} {PARA 360 "> " 0 "" {MPLTEXT 1 0 39 " house_win := house_wi n + 1;" }}{PARA 361 "> " 0 "" {MPLTEXT 1 0 41 " shoot_loss \+ := shoot_loss + 1;" }}{PARA 362 "> " 0 "" {MPLTEXT 1 0 11 " fi; " }}{PARA 363 "> " 0 "" {MPLTEXT 1 0 7 " od;" }}{PARA 364 "> " 0 " " {MPLTEXT 1 0 3 " " }}{PARA 365 "> " 0 "" {MPLTEXT 1 0 45 " \+ List := subsop( 1 = shoot_win, List);" }}{PARA 366 "> " 0 "" {MPLTEXT 1 0 46 " List := subsop( 2 = shoot_loss, List);" }}{PARA 367 "> " 0 "" {MPLTEXT 1 0 9 " List;" }}{PARA 368 "" 0 "" {MPLTEXT 0 21 149 " #the first element of list is the number of wins of the sho oter\n #the second element of list is the number of losses of the shooter\n " }}{PARA 369 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 370 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 372 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 373 "> " 0 "" {MPLTEXT 1 0 14 "craps (10000);" }}{PARA 2 "" 1 "" {TEXT -1 54 " \+ [4794, 5206]" }}}{EXCHG {PARA 375 "> " 0 "" {MPLTEXT 1 0 15 "evalf(%/10000);" }}{PARA 2 "" 1 "" {TEXT -1 61 " \+ [.4794000000, .5206000000]" }}}{EXCHG {PARA 377 "> \+ " 0 "" {MPLTEXT 1 0 5 "%[1];" }}{PARA 2 "" 1 "" {TEXT -1 54 " \+ .4794000000" }}}{EXCHG {PARA 379 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 380 "" 0 "" {TEXT -1 357 "The above value is the win ning probability of the shooter that we got from the simulation. It's \+ close to the true probability that we calculated earlier. We only play the game 10000, and only got the probability of winning correct in th e tenth decimal. If we want it to be correct in the thousandth decimal , then we have to play a lot more than 10000 times." }}}}}{MARK "0 0 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }