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SaturnCloud Samples.html
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<!doctype html>
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<title>SaturnCloud Samples</title><link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; }
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<div id='write' class = 'is-node'><p> </p><p> </p><p><span>Greetings!</span></p><p><span>I do not have any professional publish work to showcase. However, I do have a few math and logic puzzles that I would like to share with you. I have a small little channel inside a big gaming clan online where I share math puzzles and problems to several clan members. The clan members vary greatly in their technical backgrounds. Some of these problems involve programming, while others are more logic based. I try to present several problems that would excite the members and lead to interesting discussions. A lot of these problems were modified from several sources, are hybridization of multiple items, or are unique problems from me. The solutions are all original from me. </span></p><p><span>I have attached a link on the </span><strong><span>last page</span></strong><span> with a Data Science Project I did that might be of interest to you. I also added several more questions in case you were curious of some of the problems I gave my clanmates. The problems were never meant to be shared beyond a small circle of friends, so they aren't the most polished. </span></p><p> </p><h4><a name="question-2" class="md-header-anchor"></a><span>Question 2</span></h4><p><span>Henrique owns a parking lot for vehicles. A vehicle is either a motorcycle with two wheels or a car with four wheels. There are 100 vehicles in the parking lot. The total number of wheels in Henrique's parking lot is 326. If Henrique charges </span><span>$</span><span>1.00 for each motorcycle and </span><span>$</span><span>2.00 for each car per day, what is the total revenue for today?</span></p><h6><a name="solution" class="md-header-anchor"></a><span>Solution</span></h6><p><span>There are a myriad of methods to solve this. The commonality in all of these methods is we need to solve for a system of equations. Let </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.039ex" height="1.41ex" viewBox="0 -504.6 878 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E9-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E9-MJMATHI-6D" x="0" y="0"></use></g></svg></span><script type="math/tex">m</script><span> be the number of motorcycles and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.006ex" height="1.41ex" viewBox="0 -504.6 433 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E10-MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E10-MJMATHI-63" x="0" y="0"></use></g></svg></span><script type="math/tex">c</script><span> the number of cars on the lot. We end with two equations based on the above scenario. 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This isn't particularly time saving for this particular question, but for future questions, this will likely be the preferred operation. We extract the coefficients from equations above and translate them into matrices. Don't worry if you aren't familiar with the techniques presented below. 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0 & 1 &|\space63}
\right\rgroup</script></div></div><p><span>So we are left with the number of cars </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.426ex" height="1.994ex" viewBox="0 -755.9 2766.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E11-MJMATHI-63" d="M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path><path stroke-width="0" id="E11-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E11-MJMAIN-36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z"></path><path stroke-width="0" id="E11-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E11-MJMATHI-63" x="0" y="0"></use><use xlink:href="#E11-MJMAIN-3D" x="710" y="0"></use><g transform="translate(1766,0)"><use xlink:href="#E11-MJMAIN-36"></use><use xlink:href="#E11-MJMAIN-33" x="500" y="0"></use></g></g></svg></span><script type="math/tex">c=63</script><span>, and </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="19.202ex" height="2.11ex" viewBox="0 -755.9 8267.6 908.7" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="0" id="E12-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E12-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E12-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E12-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E12-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E12-MJMAIN-36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z"></path><path stroke-width="0" id="E12-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="0" id="E12-MJMAIN-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E12-MJMATHI-6D" x="0" y="0"></use><use xlink:href="#E12-MJMAIN-3D" x="1155" y="0"></use><g transform="translate(2211,0)"><use xlink:href="#E12-MJMAIN-31"></use><use xlink:href="#E12-MJMAIN-30" x="500" y="0"></use><use xlink:href="#E12-MJMAIN-30" x="1000" y="0"></use></g><use xlink:href="#E12-MJMAIN-2212" x="3933" y="0"></use><g transform="translate(4934,0)"><use xlink:href="#E12-MJMAIN-36"></use><use xlink:href="#E12-MJMAIN-33" x="500" y="0"></use></g><use xlink:href="#E12-MJMAIN-3D" x="6211" y="0"></use><g transform="translate(7267,0)"><use xlink:href="#E12-MJMAIN-33"></use><use xlink:href="#E12-MJMAIN-37" x="500" y="0"></use></g></g></svg></span><script type="math/tex">m=100-63=37</script><span> motorcycles. So the amount of money Henrique makes is </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="36.983ex" height="2.227ex" viewBox="0 -806.1 15923 958.9" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="0" id="E13-MJMAIN-24" d="M162 187Q162 164 146 149T109 133H103V130Q108 115 115 105Q122 92 131 82T150 64T170 52T190 44T206 40T220 37L227 36V313Q190 320 162 335Q116 358 86 404T55 508Q55 567 85 614T165 685Q186 696 225 704H227V750H273V704L286 703Q369 690 413 631Q441 588 444 531Q444 514 443 509Q439 490 425 479T391 468Q368 468 353 483T337 522Q337 546 353 560T390 575L394 576V578Q386 599 372 614T342 637T314 649T288 656L273 658V408L288 405Q329 394 355 376Q396 348 420 300T444 199Q444 130 408 76T313 1Q286 -9 276 -9H273V-56H227V-10H221Q202 -6 193 -4T155 11T108 41T74 94T55 176V182Q55 227 95 238Q103 240 108 240Q129 240 145 226T162 187ZM225 657Q219 657 204 651T169 632T135 594T121 538Q121 512 131 491T156 457T187 435T213 423T227 420V539Q227 657 225 657ZM378 169Q378 230 339 265T274 301Q273 301 273 169V37Q324 50 351 87T378 169Z"></path><path stroke-width="0" id="E13-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E13-MJMAIN-2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 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209Q352 289 347 316T329 361Q302 397 257 397Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E13-MJMAIN-24" x="0" y="0"></use><use xlink:href="#E13-MJMAIN-31" x="500" y="0"></use><use xlink:href="#E13-MJMAIN-2217" x="1222" y="0"></use><use xlink:href="#E13-MJMATHI-6D" x="1944" y="0"></use><use xlink:href="#E13-MJMAIN-2B" x="3044" y="0"></use><use xlink:href="#E13-MJMAIN-24" x="4044" y="0"></use><use xlink:href="#E13-MJMAIN-32" x="4544" y="0"></use><use xlink:href="#E13-MJMAIN-2217" x="5267" y="0"></use><use xlink:href="#E13-MJMATHI-63" x="5989" y="0"></use><use xlink:href="#E13-MJMAIN-3D" x="6700" y="0"></use><use xlink:href="#E13-MJMAIN-24" x="7755" y="0"></use><use xlink:href="#E13-MJMAIN-31" x="8255" y="0"></use><use xlink:href="#E13-MJMAIN-2217" x="8978" y="0"></use><g transform="translate(9700,0)"><use xlink:href="#E13-MJMAIN-33"></use><use xlink:href="#E13-MJMAIN-37" x="500" y="0"></use></g><use xlink:href="#E13-MJMAIN-2B" x="10922" y="0"></use><use xlink:href="#E13-MJMAIN-24" x="11922" y="0"></use><use xlink:href="#E13-MJMAIN-32" x="12422" y="0"></use><use xlink:href="#E13-MJMAIN-2217" x="13145" y="0"></use><g transform="translate(13867,0)"><use xlink:href="#E13-MJMAIN-36"></use><use xlink:href="#E13-MJMAIN-33" x="500" y="0"></use></g><use xlink:href="#E13-MJMAIN-3D" x="15145" y="0"></use></g></svg></span><script type="math/tex">\$1*m+\$2*c =\$1*37+\$2*63=</script><span>$</span><span>163$</span></p><p> </p><h4><a name="question-8" class="md-header-anchor"></a><span>Question 8</span></h4><p><img src="C:\Users\aldoc\Documents\Daily Math Questions\daily math questions.assets\1571338639533.png" referrerpolicy="no-referrer" alt="1571338639533"></p><p><span>The shaded region is defined by the circle with origin </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.162ex" height="2.577ex" viewBox="0 -806.1 2222.7 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E14-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E14-MJMAIN-36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z"></path><path 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xlink:href="#E17-MJMAIN-30" x="1000" y="0"></use></g></g></svg></span><script type="math/tex">Area_{circle}\le100</script><span>, what is the approximate area of the shaded region?</span></p><ul><li class='md-task-list-item task-list-item task-list-not-done' ><input type='checkbox' disabled='disabled' /><p><span>less than 75</span></p></li><li class='md-task-list-item task-list-item task-list-not-done' ><input type='checkbox' disabled='disabled' /><p><span>between 75 and 125</span></p></li><li class='md-task-list-item task-list-item task-list-not-done' ><input type='checkbox' disabled='disabled' /><p><span>between 125 and 175</span></p></li><li class='md-task-list-item task-list-item task-list-not-done' ><input type='checkbox' disabled='disabled' /><p><span>between 175 and 225</span></p></li><li class='md-task-list-item task-list-item task-list-not-done' ><input type='checkbox' disabled='disabled' /><p><span>between 225 and 275</span></p></li><li class='md-task-list-item task-list-item task-list-not-done' ><input type='checkbox' disabled='disabled' /><p><span>between 275 and 325</span></p></li><li class='md-task-list-item task-list-item task-list-not-done' ><input type='checkbox' disabled='disabled' /><p><span>more than 325</span></p></li></ul><h6><a name="solution-n36" class="md-header-anchor"></a><span>Solution</span></h6><p><img src="C:\Users\aldoc\Documents\Daily Math Questions\daily math questions.assets\1571338659988.png" referrerpolicy="no-referrer" alt="1571338659988"></p><p><span>We can break the shape of the shaded region approximately into the sum of 2 rectangles and one-quarter of a circle. We can create an upper boundary and a lower boundary by looking at the above figure. Clearly the figure on the left will have a larger area than the figure on the right since the rectangles extend pass the radius of the circle at certain points. </span></p><p><strong><span>The radius of the circle</span></strong><span> </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n40" cid="n40" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-3-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="25.431ex" height="2.577ex" viewBox="0 -755.9 10949.4 1109.7" role="img" focusable="false" style="vertical-align: -0.822ex; max-width: 100%;"><defs><path stroke-width="0" id="E26-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 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xlink:href="#E28-MJMAIN-34" x="1778" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-5">\frac{1}{4}Area_{circle}=\pi r^2 \longrightarrow \frac{1}{4}\pi (10)^2 =25\pi
\\
\array{
Area_{rectangle_1}=base*height = (6-0)(12-0)=72
\cr
Area_{rectangle_2}=base*height = (15.1-6)(4-0)\approx36.4
}
\\
\frac{1}{4}Area_{circle} +Area_{rectangle_1} +Area_{rectangle_2}
\\
25\pi+72+36.4
\\
25\pi+108.4</script></div></div><p><span>To find the base of the horizontal rectangle, we need to find the x-intercept of the circle. 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y="0"></use><g transform="translate(4859,0)"><use xlink:href="#E29-MJMAIN-31"></use><use xlink:href="#E29-MJMAIN-30" x="500" y="0"></use><use xlink:href="#E29-MJMAIN-30" x="1000" y="0"></use></g><use xlink:href="#E29-MJMAIN-2212" x="6581" y="0"></use><g transform="translate(7581,0)"><use xlink:href="#E29-MJMAIN-31"></use><use xlink:href="#E29-MJMAIN-36" x="500" y="0"></use></g></g><g transform="translate(16232,-5063)"><use xlink:href="#E29-MJSZ2-221A" x="0" y="43"></use><rect stroke="none" width="3525" height="60" x="1000" y="1133"></rect><g transform="translate(1000,0)"><use xlink:href="#E29-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E29-MJMATHI-78" x="389" y="0"></use><use xlink:href="#E29-MJMAIN-2212" x="1183" y="0"></use><use xlink:href="#E29-MJMAIN-36" x="2183" y="0"></use><g transform="translate(2683,0)"><use xlink:href="#E29-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E29-MJMAIN-32" x="550" y="583"></use></g></g><use xlink:href="#E29-MJMAIN-2264" x="4803" y="0"></use><g transform="translate(5859,0)"><use xlink:href="#E29-MJMAIN-221A" x="0" y="3"></use><rect stroke="none" width="1000" height="60" x="833" y="743"></rect><g transform="translate(833,0)"><use xlink:href="#E29-MJMAIN-38"></use><use xlink:href="#E29-MJMAIN-34" x="500" y="0"></use></g></g></g><g transform="translate(16509,-7353)"><use xlink:href="#E29-MJSZ2-221A" x="0" y="43"></use><rect stroke="none" width="3525" height="60" x="1000" y="1133"></rect><g transform="translate(1000,0)"><use xlink:href="#E29-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E29-MJMATHI-78" x="389" y="0"></use><use xlink:href="#E29-MJMAIN-2212" x="1183" y="0"></use><use xlink:href="#E29-MJMAIN-36" x="2183" y="0"></use><g transform="translate(2683,0)"><use xlink:href="#E29-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E29-MJMAIN-32" x="550" y="583"></use></g></g><use xlink:href="#E29-MJAMS-2A85" x="4803" y="0"></use><g transform="translate(5859,0)"><use xlink:href="#E29-MJMAIN-39"></use><use xlink:href="#E29-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E29-MJMAIN-31" x="778" y="0"></use></g></g><g transform="translate(15184,-9141)"><g transform="translate(-15,0)"><g transform="translate(0,-50)"><use xlink:href="#E29-MJMATHI-78" x="0" y="0"></use><use xlink:href="#E29-MJMAIN-2248" x="849" y="0"></use><g transform="translate(1905,0)"><use xlink:href="#E29-MJMAIN-31"></use><use xlink:href="#E29-MJMAIN-35" x="500" y="0"></use><use xlink:href="#E29-MJMAIN-2E" x="1000" y="0"></use><use xlink:href="#E29-MJMAIN-31" x="1278" y="0"></use></g></g></g><g transform="translate(4668,0)"><g transform="translate(0,-50)"><use xlink:href="#E29-MJMATHI-6F" x="0" y="0"></use><use xlink:href="#E29-MJMATHI-72" x="485" y="0"></use></g></g><g transform="translate(6604,0)"><g transform="translate(0,-50)"><use xlink:href="#E29-MJMATHI-78" x="0" y="0"></use><use xlink:href="#E29-MJMAIN-2248" x="849" y="0"></use><g transform="translate(1905,0)"><use xlink:href="#E29-MJMAIN-33"></use><use xlink:href="#E29-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E29-MJMAIN-31" x="778" y="0"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-6">(x-6)^2+(y-4)^2\le100
\\
(x-6)^2+(0-4)^2\le100
\\
(x-6)^2 \le 100-16
\\
\sqrt{(x-6)^2}\le\sqrt{84}
\\
\sqrt{(x-6)^2}\lessapprox 9.1
\\
\array{
x \approx 15.1 & or& x\approx3.1
}</script></div></div><p><span>We can approximate </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.257ex" height="2.811ex" viewBox="0 -1007.2 1833 1210.2" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E19-MJMAIN-38" d="M70 417T70 494T124 618T248 666Q319 666 374 624T429 515Q429 485 418 459T392 417T361 389T335 371T324 363L338 354Q352 344 366 334T382 323Q457 264 457 174Q457 95 399 37T249 -22Q159 -22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368Q70 417 70 494ZM286 386L292 390Q298 394 301 396T311 403T323 413T334 425T345 438T355 454T364 471T369 491T371 513Q371 556 342 586T275 624Q268 625 242 625Q201 625 165 599T128 534Q128 511 141 492T167 463T217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194T353 216T330 234T301 253T274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z"></path><path stroke-width="0" id="E19-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path><path stroke-width="0" id="E19-MJMAIN-221A" d="M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 178Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E19-MJMAIN-221A" x="0" y="49"></use><rect stroke="none" width="1000" height="60" x="833" y="789"></rect><g transform="translate(833,0)"><use xlink:href="#E19-MJMAIN-38"></use><use xlink:href="#E19-MJMAIN-34" x="500" y="0"></use></g></g></svg></span><script type="math/tex">\sqrt{84}</script><span> as </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.179ex" height="2.811ex" viewBox="0 -956.9 3521.4 1210.2" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E20-MJMAIN-38" d="M70 417T70 494T124 618T248 666Q319 666 374 624T429 515Q429 485 418 459T392 417T361 389T335 371T324 363L338 354Q352 344 366 334T382 323Q457 264 457 174Q457 95 399 37T249 -22Q159 -22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368Q70 417 70 494ZM286 386L292 390Q298 394 301 396T311 403T323 413T334 425T345 438T355 454T364 471T369 491T371 513Q371 556 342 586T275 624Q268 625 242 625Q201 625 165 599T128 534Q128 511 141 492T167 463T217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194T353 216T330 234T301 253T274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z"></path><path stroke-width="0" id="E20-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E20-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E20-MJMATHI-65" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"></path><path stroke-width="0" id="E20-MJMAIN-221A" d="M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 178Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E20-MJMAIN-221A" x="0" y="14"></use><rect stroke="none" width="2688" height="60" x="833" y="754"></rect><g transform="translate(833,0)"><use xlink:href="#E20-MJMAIN-38"></use><use xlink:href="#E20-MJMAIN-31" x="500" y="0"></use><use xlink:href="#E20-MJMAIN-2B" x="1222" y="0"></use><use xlink:href="#E20-MJMATHI-65" x="2222" y="0"></use></g></g></svg></span><script type="math/tex">\sqrt{81+e}</script><span>, where </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.341ex" height="1.994ex" viewBox="0 -755.9 2299.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E21-MJMATHI-65" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"></path><path stroke-width="0" id="E21-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path stroke-width="0" id="E21-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E21-MJMATHI-65" x="0" y="0"></use><use xlink:href="#E21-MJMAIN-3E" x="743" y="0"></use><use xlink:href="#E21-MJMAIN-30" x="1799" y="0"></use></g></svg></span><script type="math/tex">e>0</script><span>, so we get 9+a small fraction. </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.968ex" height="1.994ex" viewBox="0 -755.9 1278 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E22-MJMAIN-39" d="M352 287Q304 211 232 211Q154 211 104 270T44 396Q42 412 42 436V444Q42 537 111 606Q171 666 243 666Q245 666 249 666T257 665H261Q273 665 286 663T323 651T370 619T413 560Q456 472 456 334Q456 194 396 97Q361 41 312 10T208 -22Q147 -22 108 7T68 93T121 149Q143 149 158 135T173 96Q173 78 164 65T148 49T135 44L131 43Q131 41 138 37T164 27T206 22H212Q272 22 313 86Q352 142 352 280V287ZM244 248Q292 248 321 297T351 430Q351 508 343 542Q341 552 337 562T323 588T293 615T246 625Q208 625 181 598Q160 576 154 546T147 441Q147 358 152 329T172 282Q197 248 244 248Z"></path><path stroke-width="0" id="E22-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="0" id="E22-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E22-MJMAIN-39"></use><use xlink:href="#E22-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E22-MJMAIN-31" x="778" y="0"></use></g></svg></span><script type="math/tex">9.1</script><span> is close enough for our purposes. </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.394ex" height="1.994ex" viewBox="0 -755.9 3183.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E23-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E23-MJMAIN-2248" d="M55 319Q55 360 72 393T114 444T163 472T205 482Q207 482 213 482T223 483Q262 483 296 468T393 413L443 381Q502 346 553 346Q609 346 649 375T694 454Q694 465 698 474T708 483Q722 483 722 452Q722 386 675 338T555 289Q514 289 468 310T388 357T308 404T224 426Q164 426 125 393T83 318Q81 289 69 289Q55 289 55 319ZM55 85Q55 126 72 159T114 210T163 238T205 248Q207 248 213 248T223 249Q262 249 296 234T393 179L443 147Q502 112 553 112Q609 112 649 141T694 220Q694 249 708 249T722 217Q722 153 675 104T555 55Q514 55 468 76T388 123T308 170T224 192Q164 192 125 159T83 84Q80 55 69 55Q55 55 55 85Z"></path><path stroke-width="0" id="E23-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="0" id="E23-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="0" id="E23-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E23-MJMATHI-78" x="0" y="0"></use><use xlink:href="#E23-MJMAIN-2248" x="849" y="0"></use><g transform="translate(1905,0)"><use xlink:href="#E23-MJMAIN-33"></use><use xlink:href="#E23-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E23-MJMAIN-31" x="778" y="0"></use></g></g></svg></span><script type="math/tex">x\approx3.1</script><span> does not make sense in this context so we only need </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.555ex" height="1.994ex" viewBox="0 -755.9 3683.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E24-MJMATHI-78" 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xlink:href="#E30-MJMAIN-39" x="1000" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-7">25\pi+108.4<{shaded\ region}<25\pi+124
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\\183.4<{shaded\ region}<199</script></div></div><p> </p><h4><a name="question-12" class="md-header-anchor"></a><span>Question 12</span></h4><p><span>Tired of losing to that MMO boss purely by chance? Why not embrace randomness, and play a game where our score is determined by a random number generator — all hail the RNG god!</span></p><p><span>Our random number generator produces integers from 0 to 9 inclusively. The first number generated gets added to the decimal place of our score. So if we roll a 5, our current score becomes 0.5. Each time we generate a random number, if the number generated is less than or equal to the previous number, we append it to our score. If the number generated is 0, we end our score. Anything else gets skipped. </span></p><p><span>For example, suppose we generated the following sequence: 7, 5, 4, 2, 5, 6, 1, 1, 0. Our score for that round would be 0.754211. </span></p><p><span>What will be the average final score produced?</span></p><h6><a name="solution-n56" class="md-header-anchor"></a><span>Solution</span></h6><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang=""><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 36px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 28px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -28px; width: 28px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 19px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">0.4737855368144297</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 23px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 27px;"></div></div></div></div></pre><p><span>I find working on these sorts of problem easier to handle computationally rather than constructing a pure mathematical proof. For this game, I construct a simple Python script. First, we create a function that represents the random number generator. It is a good idea to separate this component, since we can more easily modify the script in the future by keeping functions modular.</span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 36px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 28px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>2</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -28px; width: 28px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 19px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">generate_int</span>():</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 19px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable">random</span>.<span class="cm-property">randint</span>(<span class="cm-number">0</span>,<span class="cm-number">9</span>)</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 46px;"></div><div class="CodeMirror-gutters" style="height: 46px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 27px;"></div></div></div></div></pre><p><span>Our function </span><code>generate_int()</code><span> outputs a random integer from 0 to 9 inclusively and uniformly. In the future, we can modify the random number generator to follow a Gaussian normal distribution or Chi-Squared distribution. However, for our purposes, a good old uniform distribution is acceptable. </span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>13</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">roll</span>():</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">running_score</span> = []</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">is_True</span> = <span class="cm-keyword">True</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">while</span> <span class="cm-variable">is_True</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">digit</span> = <span class="cm-variable">generate_int</span>()</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">if</span> <span class="cm-variable">digit</span> == <span class="cm-number">0</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">is_True</span> = <span class="cm-keyword">False</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">elif</span> (<span class="cm-variable">running_score</span> == []) <span class="cm-keyword">or</span> (<span class="cm-variable">digit</span> <span class="cm-operator"><</span>= <span class="cm-variable">running_score</span>[<span class="cm-operator">-</span><span class="cm-number">1</span>]):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">running_score</span>.<span class="cm-property">append</span>(<span class="cm-variable">digit</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">digits_to_string</span> = [<span class="cm-builtin">str</span>(<span class="cm-variable">i</span>) <span class="cm-keyword">for</span> <span class="cm-variable">i</span> <span class="cm-keyword">in</span> <span class="cm-variable">running_score</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-builtin">float</span>(<span class="cm-string">"0."</span> <span class="cm-operator">+</span> <span class="cm-string">""</span>.<span class="cm-property">join</span>(<span class="cm-variable">digits_to_string</span>))</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 299px;"></div><div class="CodeMirror-gutters" style="height: 299px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p><span>The function </span><code>roll()</code><span> simulates each round of the game. The crux of the function is a </span><strong><span>while loop</span></strong><span>. Because we don't know how long any particular game will run, a while loop gives us the flexibility that we need to have games of undetermined length. We will store our temporary results in a list called </span><code>running_score</code><span> to make our lives easier. </span></p><p><span>Inside the loop, we have a sequence of conditions. These are the conditions that represent the rules of our game. The </span><code>if</code><span> statement is our stop condition. Recall that our game is over at any point that the RNG generates a 0, so this is why this condition is first. The following conditions adjusts our </span><code>running_score</code><span>. The first part of the </span><code>elif</code><span> statement checks if our </span><code>running_score</code><span> is empty. This should only occur once when we initialize the score with a valid digit. The second part of the </span><code>elif</code><span> statements checks to see if the digit generated is less than or equal to the previously appended digit. If that condition is true, we append the current generated digit to our </span><code>running_score</code><span>. </span></p><p><span>After the while loop, we format the integers in our </span><code>running_score</code><span> to a string using a </span><strong><span>list comprehension</span></strong><span>. We then join those elements together and add a "0." in front to return our final float. This gives us our final score for that round. </span></p><p><span>There are a few idiosyncratic things to note. In the second part of the </span><code>elif</code><span> statement we used </span><code>==</code><span> as the comparison operator as opposed to </span><code>is</code><span>. Can you explain the reason why? Try replacing </span><code>==</code><span> with </span><code>is</code><span> and see what the script outputs. </span></p><p><span>After simulating one round of our game, why not simulate another? What about simulating 10,000 games? The beauty of a digital simulation, is that we can easily run multiple subsequent scenarios. The </span><code>test_rolls()</code><span> function repeats the </span><code>roll()</code><span> function simulating thousands of scenarios and returns an average score of each round. Try a few simulations and see what average scores you obtain. Please note that your average score might differ slightly from the answer given in this solution, but it should be fairly close. </span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang=""><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 36px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 28px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>1</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -28px; width: 28px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 19px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">0.4737855368144297</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="height: 23px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 27px;"></div></div></div></div></pre><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><span><span></span>x</span></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">random</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">generate_int</span>():</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable">random</span>.<span class="cm-property">randint</span>(<span class="cm-number">0</span>,<span class="cm-number">9</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">roll</span>():</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">digits</span> = []</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">is_True</span> = <span class="cm-keyword">True</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">while</span> <span class="cm-variable">is_True</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">digit</span> = <span class="cm-variable">generate_int</span>()</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">if</span> <span class="cm-variable">digit</span> == <span class="cm-number">0</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">is_True</span> = <span class="cm-keyword">False</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">elif</span> (<span class="cm-variable">digits</span> == []) <span class="cm-keyword">or</span> (<span class="cm-variable">digit</span> <span class="cm-operator"><</span>= <span class="cm-variable">digits</span>[<span class="cm-operator">-</span><span class="cm-number">1</span>]):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">digits</span>.<span class="cm-property">append</span>(<span class="cm-variable">digit</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">digits_to_string</span> = [<span class="cm-builtin">str</span>(<span class="cm-variable">i</span>) <span class="cm-keyword">for</span> <span class="cm-variable">i</span> <span class="cm-keyword">in</span> <span class="cm-variable">digits</span>]</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-builtin">float</span>(<span class="cm-string">"0."</span> <span class="cm-operator">+</span> <span class="cm-string">""</span>.<span class="cm-property">join</span>(<span class="cm-variable">digits_to_string</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">test_rolls</span>(<span class="cm-variable">n</span>=<span class="cm-number">100000</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">temp</span> = []</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">for</span> <span class="cm-variable">i</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable">n</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">if</span> <span class="cm-variable">i</span><span class="cm-operator">%</span><span class="cm-number">10000</span> ==<span class="cm-number">0</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-builtin">print</span>(<span class="cm-string">"Progress: {:0.2f}"</span>.<span class="cm-property">format</span>(<span class="cm-variable">i</span><span class="cm-operator">/</span><span class="cm-variable">n</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">x</span> = <span class="cm-variable">roll</span>()</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">temp</span>.<span class="cm-property">append</span>(<span class="cm-variable">x</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">average</span> = <span class="cm-builtin">sum</span>(<span class="cm-variable">temp</span>)<span class="cm-operator">/</span> <span class="cm-builtin">len</span>(<span class="cm-variable">temp</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable">average</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-operator">%</span><span class="cm-variable">time</span> <span class="cm-variable">test_rolls</span>()</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 690px;"></div><div class="CodeMirror-gutters" style="height: 690px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><h4><a name="question-4" class="md-header-anchor"></a><span>Question 4</span></h4><p><span>The Battle of Winterfell has come and gone. Regardless of what your feelings are for the final season of Game of Thrones (I'm still not ready to come to grips with it), there is a lot of inspiration we can take from it. Lets take the Night King vs Jon Snow in the Battle of Winterfell. The army of the undead bombarded Jon's rag tag team of the obstinate living. Let's come up with a simulation to see how likely Jon's army would come out as triumphant. </span></p><p><span>The undead may have supernatural abilities, but Jon's army is full of battle-hardened experienced soldiers. Let's give them a 50-50 chance of beating the undead via a one-on-one battle. If the undead kills the living, not only does the living lose a member, but the undead army gains a new member as the Night King is continuously raising the dead. If the living defeats the undead, that undead is permanently out of commission. Lets assume that all members of the living are carrying dragonglass. What distribution of living vs undead yields yields what probability of either side coming out as triumphant?</span></p><h6><a name="solution-n72" class="md-header-anchor"></a><span>Solution</span></h6><p><span>There are multiple ways to attack this problem. I want to tackle this from a bottom-up approach. The following function will focus on the living. </span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 36px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 28px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>7</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -28px; width: 28px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 19px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">one_one_one</span>(<span class="cm-variable">living</span>, <span class="cm-variable">undead</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">if</span> <span class="cm-variable">random</span>.<span class="cm-property">random</span>()<span class="cm-operator">></span><span class="cm-number">.5</span>: <span class="cm-comment"># living wins</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">undead</span> -= <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">else</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">undead</span> += <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 19px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">living</span> -= <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -28px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 19px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> (<span class="cm-variable">living</span>, <span class="cm-variable">undead</span>)</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 161px;"></div><div class="CodeMirror-gutters" style="height: 161px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 27px;"></div></div></div></div></pre><p><span>The function </span><code>one_on_one()</code><span> takes in the living and undead and generates a random number. If the random number is greater than 0.5, the living soldier has vanquished the undead. If the random number is less than 0.5, the undead kills the living and recruits the newly risen dead into the army of the Night King. </span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>11</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">battle_of_winterfell</span>(<span class="cm-variable">living</span>, <span class="cm-variable">undead</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">long_night</span> = <span class="cm-keyword">True</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">while</span> <span class="cm-variable">long_night</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">if</span> <span class="cm-variable">living</span> <span class="cm-keyword">and</span> <span class="cm-variable">undead</span> <span class="cm-operator">></span> <span class="cm-number">0</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable cm-error">living</span>, <span class="cm-variable">undead</span> = <span class="cm-variable">one_one_one</span>(<span class="cm-variable">living</span>,<span class="cm-variable">undead</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword cm-error">else</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">long_night</span> = <span class="cm-keyword">False</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">if</span> <span class="cm-variable">living</span><span class="cm-operator">></span> <span class="cm-variable">undead</span>: <span class="cm-comment">#returns 1 if Living wins</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">else</span>: </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-number">0</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 253px;"></div><div class="CodeMirror-gutters" style="height: 253px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p><span>We continue these one-on-one battles as long as both the living and the undead have at least 1 member. </span></p><p><span>I decided to make a heatmap showing what is the probability of either side winning given the starting numbers for both sides. </span></p><pre spellcheck="false" class="md-fences md-end-block md-fences-with-lineno ty-contain-cm modeLoaded" lang="python" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 44px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 36px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre><div class="CodeMirror-linenumber CodeMirror-gutter-elt"><div>56</div></div></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: -36px; width: 36px;"></div><div class="CodeMirror-gutter-wrapper CodeMirror-activeline-gutter" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">1</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">random</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">2</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">seaborn</span> <span class="cm-keyword">as</span> <span class="cm-variable">sns</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">3</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">pandas</span> <span class="cm-keyword">as</span> <span class="cm-variable">pd</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">4</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">matplotlib</span>.<span class="cm-property">pyplot</span> <span class="cm-keyword">as</span> <span class="cm-variable">plt</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">5</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">6</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">7</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">living</span>=<span class="cm-number">1000</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">8</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">undead</span>=<span class="cm-number">20</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">9</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">10</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">11</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">one_one_one</span>(<span class="cm-variable">living</span>, <span class="cm-variable">undead</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">12</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">if</span> <span class="cm-variable">random</span>.<span class="cm-property">random</span>()<span class="cm-operator">></span><span class="cm-number">.5</span>: <span class="cm-comment">#living wins</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">13</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">undead</span> -= <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">14</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">else</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">15</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">undead</span> += <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">16</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">living</span> -= <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">17</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> (<span class="cm-variable">living</span>,<span class="cm-variable">undead</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">18</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">19</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">20</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">battle_of_winterfell</span>(<span class="cm-variable">living</span>, <span class="cm-variable">undead</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">21</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">long_night</span> = <span class="cm-keyword">True</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">22</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">while</span> <span class="cm-variable">long_night</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">23</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">if</span> <span class="cm-variable">living</span> <span class="cm-keyword">and</span> <span class="cm-variable">undead</span> <span class="cm-operator">></span> <span class="cm-number">0</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">24</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable cm-error">living</span>, <span class="cm-variable">undead</span> = <span class="cm-variable">one_one_one</span>(<span class="cm-variable">living</span>,<span class="cm-variable">undead</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">25</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword cm-error">else</span>:</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">26</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">long_night</span> = <span class="cm-keyword">False</span> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">27</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">if</span> <span class="cm-variable">living</span><span class="cm-operator">></span> <span class="cm-variable">undead</span>: <span class="cm-comment">#returns 1 if Living wins</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">28</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-number">1</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">29</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">else</span>: </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">30</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-number">0</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">31</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">32</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">33</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">survival_rate</span>(<span class="cm-variable">living</span>, <span class="cm-variable">undead</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">34</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">n</span>=<span class="cm-number">50</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">35</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">rate</span> = []</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">36</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">for</span> <span class="cm-variable">trial</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable">n</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">37</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">rate</span>.<span class="cm-property">append</span>(<span class="cm-variable">battle_of_winterfell</span>(<span class="cm-variable">living</span>, <span class="cm-variable">undead</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">38</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-builtin">sum</span>(<span class="cm-variable">rate</span>)<span class="cm-operator">/</span><span class="cm-builtin">len</span>(<span class="cm-variable">rate</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">39</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">40</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">41</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">df</span> = <span class="cm-variable">pd</span>.<span class="cm-property">DataFrame</span>(<span class="cm-number">0.0</span>, <span class="cm-variable">index</span>=<span class="cm-builtin">range</span>(<span class="cm-variable">undead</span>), <span class="cm-variable">columns</span>=<span class="cm-builtin">range</span>(<span class="cm-variable">living</span>))</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">42</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">43</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">for</span> <span class="cm-variable">x</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable">living</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">44</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">for</span> <span class="cm-variable">y</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable">undead</span>):</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">45</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">df</span>[<span class="cm-variable">x</span>][<span class="cm-variable">y</span>] = <span class="cm-variable">survival_rate</span>(<span class="cm-variable">x</span>, <span class="cm-variable">y</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">46</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;" class=""><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">47</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">p1</span> = <span class="cm-variable">sns</span>.<span class="cm-property">heatmap</span>(<span class="cm-variable">df</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">48</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">title</span>(<span class="cm-string">"Battle of Winterfell"</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">49</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">ylabel</span>(<span class="cm-string">'# of White Walkers'</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">50</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">xlabel</span>(<span class="cm-string">'# of Winterfell soldiers'</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">51</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#plt.figure(figsize=(4,3))</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">52</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">savefig</span>(<span class="cm-string">'BattleOfWinterfell.png'</span>, <span class="cm-variable">dpi</span>=<span class="cm-number">600</span>)</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">53</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#plt.figure(dpi=600)</span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">54</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">show</span>()</span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt" style="left: 0px; width: 27px;">55</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div><div style="position: relative;"><div class="CodeMirror-gutter-wrapper" style="left: -36px;"><div class="CodeMirror-linenumber CodeMirror-gutter-elt CodeMirror-linenumber-show" style="left: 0px; width: 27px;">56</div></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#df.to_csv()</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 1288px;"></div><div class="CodeMirror-gutters" style="height: 1288px;"><div class="CodeMirror-gutter CodeMirror-linenumbers" style="width: 35px;"></div></div></div></div></pre><p><span>This is a heatmap using the data as from above. </span></p><p><img src="C:\Users\aldoc\Documents\Real Python\Real Python Writing Samples.assets\image-20200511125723895.png" referrerpolicy="no-referrer" alt="image-20200511125723895"></p><p><span>We can get a much smoother graph if we increase the values. Do note, that this will exponentially increase the time it takes to run. Can you think of ways to improve the efficiency of the code?</span><img src="C:\Users\aldoc\Downloads\BattleOfWinterfell - high res (1).png" referrerpolicy="no-referrer" alt="BattleOfWinterfell - high res (C:\Users\aldoc\Documents\Real Python\Real Python Writing Samples.assets\BattleOfWinterfell - high res (1).png)"></p><h4><a name="question-19" class="md-header-anchor"></a><span>Question 19</span></h4><p><span>Hello fellow Pokémon trainers! </span></p><p><span>I recently learned about Pokémon base stats. This got me thinking. The motto for Pokémon is </span><em><span>Gotta Catch 'Em All!</span></em><span> but just how many can you catch? How many Pokémon can Game Freak make that is competition valid?</span></p><p><span>There are 6 base stats categories ranging from 1 to 255 points–HP, Attack, Defense, Special Attack, Special Defense, and Speed. A simple way to gauge a Pokémon is to add up all 6 categories. Lets assume that every Pokémon is unique in its points distribution. The legendary status cutoff for base stats is 600 points; this means that if you want a Pokémon that is valid for competitive play, the total amount of points in all 6 categories must add up to 600 points or less. </span></p><p><span>How many ways can you distribute the points such that the Pokémon is valid for competitive play? You can arrange the points to be jack of all trades or really good in a couple of categories. </span></p><p><span>—</span><em><span>Aspiring Pokémon trainer</span></em></p><h6><a name="solution-n89" class="md-header-anchor"></a><span>Solution</span></h6><p><span>I haven't typed up the solution yet, but feel free to contact me for a sneak preview.</span></p><p> </p><p> </p><p> </p><h4><a name="project" class="md-header-anchor"></a><span>Project</span></h4><p><span>Sample Data Science Project1A.html</span></p><p><img src="C:\Users\aldoc\Documents\Real Python\Real Python Writing Samples.assets\image-20200511131127798.png" referrerpolicy="no-referrer" alt="image-20200511131127798"></p></div>
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