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Geometry and Probability Simulation

Geometry and Probability Simulation
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2014-08-09 20:09
VectorTrigonometryCoordinate GeometryProbability & StatisticsGeometric Construction
In collections Statistics Simulations Statistics Simulations
This example demonstrates the intersection of geometry and probability through interactive constructions. It involves core concepts like probability distributions, geometric constructions, and stochastic processes. Key GeoGebra commands used include Polygon, Segment, Intersect, If, Sum, DotPlot, Mean, and Vector. Suitable for high school students and math enthusiasts exploring interdisciplinary applications.

GeoGebra Commands & Parameters

  1. Polygon((0, 0), (1, 0), 4) → poly1
  2. Segment(B, (0, 0)) → a
  3. Segment(A, (1, 0)) → b
  4. Intersect(b, a) → C
  5. Segment((0, 0), (random(), random())) → A1
  6. Segment((1, 0), (random(), random())) → B1
  7. Intersect(A1, B1) → C1
  8. If((x(C1) * y(C1)) > 0, 1, 0) → D1
  9. Segment((0, 0), (random(), random())) → A2
  10. Segment((1, 0), (random(), random())) → B2
  11. Intersect(A2, B2) → C2
  12. If((x(C2) * y(C2)) > 0, 1, 0) → D2
  13. Segment((0, 0), (random(), random())) → A3
  14. Segment((1, 0), (random(), random())) → B3
  15. Intersect(A3, B3) → C3
  16. If((x(C3) * y(C3)) > 0, 1, 0) → D3
  17. Segment((0, 0), (random(), random())) → A4
  18. Segment((1, 0), (random(), random())) → B4
  19. Intersect(A4, B4) → C4
  20. If((x(C4) * y(C4)) > 0, 1, 0) → D4
  21. Segment((0, 0), (random(), random())) → A5
  22. Segment((1, 0), (random(), random())) → B5
  23. Intersect(A5, B5) → C5
  24. If((x(C5) * y(C5)) > 0, 1, 0) → D5
  25. Segment((0, 0), (random(), random())) → A6
  26. Segment((1, 0), (random(), random())) → B6
  27. Intersect(A6, B6) → C6
  28. If((x(C6) * y(C6)) > 0, 1, 0) → D6
  29. Segment((0, 0), (random(), random())) → A7
  30. Segment((1, 0), (random(), random())) → B7
  31. Intersect(A7, B7) → C7
  32. If((x(C7) * y(C7)) > 0, 1, 0) → D7
  33. Segment((0, 0), (random(), random())) → A8
  34. Segment((1, 0), (random(), random())) → B8
  35. Intersect(A8, B8) → C8
  36. If((x(C8) * y(C8)) > 0, 1, 0) → D8
  37. Segment((0, 0), (random(), random())) → A9
  38. Segment((1, 0), (random(), random())) → B9
  39. Intersect(A9, B9) → C9
  40. If((x(C9) * y(C9)) > 0, 1, 0) → D9
  41. Segment((0, 0), (random(), random())) → A10
  42. Segment((1, 0), (random(), random())) → B10
  43. Intersect(A10, B10) → C10
  44. If((x(C10) * y(C10)) > 0, 1, 0) → D10
  45. Segment((0, 0), (random(), random())) → A11
  46. Segment((1, 0), (random(), random())) → B11
  47. Intersect(A11, B11) → C11
  48. If((x(C11) * y(C11)) > 0, 1, 0) → D11
  49. Segment((0, 0), (random(), random())) → A12
  50. Segment((1, 0), (random(), random())) → B12
  51. Intersect(A12, B12) → C12
  52. If((x(C12) * y(C12)) > 0, 1, 0) → D12
  53. Segment((0, 0), (random(), random())) → A13
  54. Segment((1, 0), (random(), random())) → B13
  55. Intersect(A13, B13) → C13
  56. If((x(C13) * y(C13)) > 0, 1, 0) → D13
  57. Segment((0, 0), (random(), random())) → A14
  58. Segment((1, 0), (random(), random())) → B14
  59. Intersect(A14, B14) → C14
  60. If((x(C14) * y(C14)) > 0, 1, 0) → D14
  61. Segment((0, 0), (random(), random())) → A15
  62. Segment((1, 0), (random(), random())) → B15
  63. Intersect(A15, B15) → C15
  64. If((x(C15) * y(C15)) > 0, 1, 0) → D15
  65. Segment((0, 0), (random(), random())) → A16
  66. Segment((1, 0), (random(), random())) → B16
  67. Intersect(A16, B16) → C16
  68. If((x(C16) * y(C16)) > 0, 1, 0) → D16
  69. Segment((0, 0), (random(), random())) → A17
  70. Segment((1, 0), (random(), random())) → B17
  71. Intersect(A17, B17) → C17
  72. If((x(C17) * y(C17)) > 0, 1, 0) → D17
  73. Segment((0, 0), (random(), random())) → A18
  74. Segment((1, 0), (random(), random())) → B18
  75. Intersect(A18, B18) → C18
  76. If((x(C18) * y(C18)) > 0, 1, 0) → D18
  77. Segment((0, 0), (random(), random())) → A19
  78. Segment((1, 0), (random(), random())) → B19
  79. Intersect(A19, B19) → C19
  80. If((x(C19) * y(C19)) > 0, 1, 0) → D19
  81. Segment((0, 0), (random(), random())) → A20
  82. Segment((1, 0), (random(), random())) → B20
  83. Intersect(A20, B20) → C20
  84. If((x(C20) * y(C20)) > 0, 1, 0) → D20
  85. Segment((0, 0), (random(), random())) → A21
  86. Segment((1, 0), (random(), random())) → B21
  87. Intersect(A21, B21) → C21
  88. If((x(C21) * y(C21)) > 0, 1, 0) → D21
  89. Segment((0, 0), (random(), random())) → A22
  90. Segment((1, 0), (random(), random())) → B22
  91. Intersect(A22, B22) → C22
  92. If((x(C22) * y(C22)) > 0, 1, 0) → D22
  93. Segment((0, 0), (random(), random())) → A23
  94. Segment((1, 0), (random(), random())) → B23
  95. Intersect(A23, B23) → C23
  96. If((x(C23) * y(C23)) > 0, 1, 0) → D23
  97. Segment((0, 0), (random(), random())) → A24
  98. Segment((1, 0), (random(), random())) → B24
  99. Intersect(A24, B24) → C24
  100. If((x(C24) * y(C24)) > 0, 1, 0) → D24
  101. Segment((0, 0), (random(), random())) → A25
  102. Segment((1, 0), (random(), random())) → B25
  103. Intersect(A25, B25) → C25
  104. If((x(C25) * y(C25)) > 0, 1, 0) → D25
  105. Segment((0, 0), (random(), random())) → A26
  106. Segment((1, 0), (random(), random())) → B26
  107. Intersect(A26, B26) → C26
  108. If((x(C26) * y(C26)) > 0, 1, 0) → D26
  109. Segment((0, 0), (random(), random())) → A27
  110. Segment((1, 0), (random(), random())) → B27
  111. Intersect(A27, B27) → C27
  112. If((x(C27) * y(C27)) > 0, 1, 0) → D27
  113. Segment((0, 0), (random(), random())) → A28
  114. Segment((1, 0), (random(), random())) → B28
  115. Intersect(A28, B28) → C28
  116. If((x(C28) * y(C28)) > 0, 1, 0) → D28
  117. Segment((0, 0), (random(), random())) → A29
  118. Segment((1, 0), (random(), random())) → B29
  119. Intersect(A29, B29) → C29
  120. If((x(C29) * y(C29)) > 0, 1, 0) → D29
  121. Segment((0, 0), (random(), random())) → A30
  122. Segment((1, 0), (random(), random())) → B30
  123. Intersect(A30, B30) → C30
  124. If((x(C30) * y(C30)) > 0, 1, 0) → D30
  125. Segment((0, 0), (random(), random())) → A31
  126. Segment((1, 0), (random(), random())) → B31
  127. Intersect(A31, B31) → C31
  128. If((x(C31) * y(C31)) > 0, 1, 0) → D31
  129. Segment((0, 0), (random(), random())) → A32
  130. Segment((1, 0), (random(), random())) → B32
  131. Intersect(A32, B32) → C32
  132. If((x(C32) * y(C32)) > 0, 1, 0) → D32
  133. Segment((0, 0), (random(), random())) → A33
  134. Segment((1, 0), (random(), random())) → B33
  135. Intersect(A33, B33) → C33
  136. If((x(C33) * y(C33)) > 0, 1, 0) → D33
  137. Segment((0, 0), (random(), random())) → A34
  138. Segment((1, 0), (random(), random())) → B34
  139. Intersect(A34, B34) → C34
  140. If((x(C34) * y(C34)) > 0, 1, 0) → D34
  141. Segment((0, 0), (random(), random())) → A35
  142. Segment((1, 0), (random(), random())) → B35
  143. Intersect(A35, B35) → C35
  144. If((x(C35) * y(C35)) > 0, 1, 0) → D35
  145. Segment((0, 0), (random(), random())) → A36
  146. Segment((1, 0), (random(), random())) → B36
  147. Intersect(A36, B36) → C36
  148. If((x(C36) * y(C36)) > 0, 1, 0) → D36
  149. Segment((0, 0), (random(), random())) → A37
  150. Segment((1, 0), (random(), random())) → B37
  151. Intersect(A37, B37) → C37
  152. If((x(C37) * y(C37)) > 0, 1, 0) → D37
  153. Segment((0, 0), (random(), random())) → A38
  154. Segment((1, 0), (random(), random())) → B38
  155. Intersect(A38, B38) → C38
  156. If((x(C38) * y(C38)) > 0, 1, 0) → D38
  157. Segment((0, 0), (random(), random())) → A39
  158. Segment((1, 0), (random(), random())) → B39
  159. Intersect(A39, B39) → C39
  160. If((x(C39) * y(C39)) > 0, 1, 0) → D39
  161. Segment((0, 0), (random(), random())) → A40
  162. Segment((1, 0), (random(), random())) → B40
  163. Intersect(A40, B40) → C40
  164. If((x(C40) * y(C40)) > 0, 1, 0) → D40
  165. Segment((0, 0), (random(), random())) → A41
  166. Segment((1, 0), (random(), random())) → B41
  167. Intersect(A41, B41) → C41
  168. If((x(C41) * y(C41)) > 0, 1, 0) → D41
  169. Segment((0, 0), (random(), random())) → A42
  170. Segment((1, 0), (random(), random())) → B42
  171. Intersect(A42, B42) → C42
  172. If((x(C42) * y(C42)) > 0, 1, 0) → D42
  173. Segment((0, 0), (random(), random())) → A43
  174. Segment((1, 0), (random(), random())) → B43
  175. Intersect(A43, B43) → C43
  176. If((x(C43) * y(C43)) > 0, 1, 0) → D43
  177. Segment((0, 0), (random(), random())) → A44
  178. Segment((1, 0), (random(), random())) → B44
  179. Intersect(A44, B44) → C44
  180. If((x(C44) * y(C44)) > 0, 1, 0) → D44
  181. Segment((0, 0), (random(), random())) → A45
  182. Segment((1, 0), (random(), random())) → B45
  183. Intersect(A45, B45) → C45
  184. If((x(C45) * y(C45)) > 0, 1, 0) → D45
  185. Segment((0, 0), (random(), random())) → A46
  186. Segment((1, 0), (random(), random())) → B46
  187. Intersect(A46, B46) → C46
  188. If((x(C46) * y(C46)) > 0, 1, 0) → D46
  189. Segment((0, 0), (random(), random())) → A47
  190. Segment((1, 0), (random(), random())) → B47
  191. Intersect(A47, B47) → C47
  192. If((x(C47) * y(C47)) > 0, 1, 0) → D47
  193. Segment((0, 0), (random(), random())) → A48
  194. Segment((1, 0), (random(), random())) → B48
  195. Intersect(A48, B48) → C48
  196. If((x(C48) * y(C48)) > 0, 1, 0) → D48
  197. Segment((0, 0), (random(), random())) → A49
  198. Segment((1, 0), (random(), random())) → B49
  199. Intersect(A49, B49) → C49
  200. If((x(C49) * y(C49)) > 0, 1, 0) → D49
  201. Segment((0, 0), (random(), random())) → A50
  202. Segment((1, 0), (random(), random())) → B50
  203. Intersect(A50, B50) → C50
  204. If((x(C50) * y(C50)) > 0, 1, 0) → D50
  205. Segment((0, 0), (random(), random())) → A51
  206. Segment((1, 0), (random(), random())) → B51
  207. Intersect(A51, B51) → C51
  208. If((x(C51) * y(C51)) > 0, 1, 0) → D51
  209. Segment((0, 0), (random(), random())) → A52
  210. Segment((1, 0), (random(), random())) → B52
  211. Intersect(A52, B52) → C52
  212. If((x(C52) * y(C52)) > 0, 1, 0) → D52
  213. Segment((0, 0), (random(), random())) → A53
  214. Segment((1, 0), (random(), random())) → B53
  215. Intersect(A53, B53) → C53
  216. If((x(C53) * y(C53)) > 0, 1, 0) → D53
  217. Segment((0, 0), (random(), random())) → A54
  218. Segment((1, 0), (random(), random())) → B54
  219. Intersect(A54, B54) → C54
  220. If((x(C54) * y(C54)) > 0, 1, 0) → D54
  221. Segment((0, 0), (random(), random())) → A55
  222. Segment((1, 0), (random(), random())) → B55
  223. Intersect(A55, B55) → C55
  224. If((x(C55) * y(C55)) > 0, 1, 0) → D55
  225. Segment((0, 0), (random(), random())) → A56
  226. Segment((1, 0), (random(), random())) → B56
  227. Intersect(A56, B56) → C56
  228. If((x(C56) * y(C56)) > 0, 1, 0) → D56
  229. Segment((0, 0), (random(), random())) → A57
  230. Segment((1, 0), (random(), random())) → B57
  231. Intersect(A57, B57) → C57
  232. If((x(C57) * y(C57)) > 0, 1, 0) → D57
  233. Segment((0, 0), (random(), random())) → A58
  234. Segment((1, 0), (random(), random())) → B58
  235. Intersect(A58, B58) → C58
  236. If((x(C58) * y(C58)) > 0, 1, 0) → D58
  237. Segment((0, 0), (random(), random())) → A59
  238. Segment((1, 0), (random(), random())) → B59
  239. Intersect(A59, B59) → C59
  240. If((x(C59) * y(C59)) > 0, 1, 0) → D59
  241. Segment((0, 0), (random(), random())) → A60
  242. Segment((1, 0), (random(), random())) → B60
  243. Intersect(A60, B60) → C60
  244. If((x(C60) * y(C60)) > 0, 1, 0) → D60
  245. Segment((0, 0), (random(), random())) → A61
  246. Segment((1, 0), (random(), random())) → B61
  247. Intersect(A61, B61) → C61
  248. If((x(C61) * y(C61)) > 0, 1, 0) → D61
  249. Segment((0, 0), (random(), random())) → A62
  250. Segment((1, 0), (random(), random())) → B62
  251. Intersect(A62, B62) → C62
  252. If((x(C62) * y(C62)) > 0, 1, 0) → D62
  253. Segment((0, 0), (random(), random())) → A63
  254. Segment((1, 0), (random(), random())) → B63
  255. Intersect(A63, B63) → C63
  256. If((x(C63) * y(C63)) > 0, 1, 0) → D63
  257. Segment((0, 0), (random(), random())) → A64
  258. Segment((1, 0), (random(), random())) → B64
  259. Intersect(A64, B64) → C64
  260. If((x(C64) * y(C64)) > 0, 1, 0) → D64
  261. Segment((0, 0), (random(), random())) → A65
  262. Segment((1, 0), (random(), random())) → B65
  263. Intersect(A65, B65) → C65
  264. If((x(C65) * y(C65)) > 0, 1, 0) → D65
  265. Segment((0, 0), (random(), random())) → A66
  266. Segment((1, 0), (random(), random())) → B66
  267. Intersect(A66, B66) → C66
  268. If((x(C66) * y(C66)) > 0, 1, 0) → D66
  269. Segment((0, 0), (random(), random())) → A67
  270. Segment((1, 0), (random(), random())) → B67
  271. Intersect(A67, B67) → C67
  272. If((x(C67) * y(C67)) > 0, 1, 0) → D67
  273. Segment((0, 0), (random(), random())) → A68
  274. Segment((1, 0), (random(), random())) → B68
  275. Intersect(A68, B68) → C68
  276. If((x(C68) * y(C68)) > 0, 1, 0) → D68
  277. Segment((0, 0), (random(), random())) → A69
  278. Segment((1, 0), (random(), random())) → B69
  279. Intersect(A69, B69) → C69
  280. If((x(C69) * y(C69)) > 0, 1, 0) → D69
  281. Segment((0, 0), (random(), random())) → A70
  282. Segment((1, 0), (random(), random())) → B70
  283. Intersect(A70, B70) → C70
  284. If((x(C70) * y(C70)) > 0, 1, 0) → D70
  285. Segment((0, 0), (random(), random())) → A71
  286. Segment((1, 0), (random(), random())) → B71
  287. Intersect(A71, B71) → C71
  288. If((x(C71) * y(C71)) > 0, 1, 0) → D71
  289. Segment((0, 0), (random(), random())) → A72
  290. Segment((1, 0), (random(), random())) → B72
  291. Intersect(A72, B72) → C72
  292. If((x(C72) * y(C72)) > 0, 1, 0) → D72
  293. Segment((0, 0), (random(), random())) → A73
  294. Segment((1, 0), (random(), random())) → B73
  295. Intersect(A73, B73) → C73
  296. If((x(C73) * y(C73)) > 0, 1, 0) → D73
  297. Segment((0, 0), (random(), random())) → A74
  298. Segment((1, 0), (random(), random())) → B74
  299. Intersect(A74, B74) → C74
  300. If((x(C74) * y(C74)) > 0, 1, 0) → D74
  301. Segment((0, 0), (random(), random())) → A75
  302. Segment((1, 0), (random(), random())) → B75
  303. Intersect(A75, B75) → C75
  304. If((x(C75) * y(C75)) > 0, 1, 0) → D75
  305. Segment((0, 0), (random(), random())) → A76
  306. Segment((1, 0), (random(), random())) → B76
  307. Intersect(A76, B76) → C76
  308. If((x(C76) * y(C76)) > 0, 1, 0) → D76
  309. Segment((0, 0), (random(), random())) → A77
  310. Segment((1, 0), (random(), random())) → B77
  311. Intersect(A77, B77) → C77
  312. If((x(C77) * y(C77)) > 0, 1, 0) → D77
  313. Segment((0, 0), (random(), random())) → A78
  314. Segment((1, 0), (random(), random())) → B78
  315. Intersect(A78, B78) → C78
  316. If((x(C78) * y(C78)) > 0, 1, 0) → D78
  317. Segment((0, 0), (random(), random())) → A79
  318. Segment((1, 0), (random(), random())) → B79
  319. Intersect(A79, B79) → C79
  320. If((x(C79) * y(C79)) > 0, 1, 0) → D79
  321. Segment((0, 0), (random(), random())) → A80
  322. Segment((1, 0), (random(), random())) → B80
  323. Intersect(A80, B80) → C80
  324. If((x(C80) * y(C80)) > 0, 1, 0) → D80
  325. Segment((0, 0), (random(), random())) → A81
  326. Segment((1, 0), (random(), random())) → B81
  327. Intersect(A81, B81) → C81
  328. If((x(C81) * y(C81)) > 0, 1, 0) → D81
  329. Segment((0, 0), (random(), random())) → A82
  330. Segment((1, 0), (random(), random())) → B82
  331. Intersect(A82, B82) → C82
  332. If((x(C82) * y(C82)) > 0, 1, 0) → D82
  333. Segment((0, 0), (random(), random())) → A83
  334. Segment((1, 0), (random(), random())) → B83
  335. Intersect(A83, B83) → C83
  336. If((x(C83) * y(C83)) > 0, 1, 0) → D83
  337. Segment((0, 0), (random(), random())) → A84
  338. Segment((1, 0), (random(), random())) → B84
  339. Intersect(A84, B84) → C84
  340. If((x(C84) * y(C84)) > 0, 1, 0) → D84
  341. Segment((0, 0), (random(), random())) → A85
  342. Segment((1, 0), (random(), random())) → B85
  343. Intersect(A85, B85) → C85
  344. If((x(C85) * y(C85)) > 0, 1, 0) → D85
  345. Segment((0, 0), (random(), random())) → A86
  346. Segment((1, 0), (random(), random())) → B86
  347. Intersect(A86, B86) → C86
  348. If((x(C86) * y(C86)) > 0, 1, 0) → D86
  349. Segment((0, 0), (random(), random())) → A87
  350. Segment((1, 0), (random(), random())) → B87
  351. Intersect(A87, B87) → C87
  352. If((x(C87) * y(C87)) > 0, 1, 0) → D87
  353. Segment((0, 0), (random(), random())) → A88
  354. Segment((1, 0), (random(), random())) → B88
  355. Intersect(A88, B88) → C88
  356. If((x(C88) * y(C88)) > 0, 1, 0) → D88
  357. Segment((0, 0), (random(), random())) → A89
  358. Segment((1, 0), (random(), random())) → B89
  359. Intersect(A89, B89) → C89
  360. If((x(C89) * y(C89)) > 0, 1, 0) → D89
  361. Segment((0, 0), (random(), random())) → A90
  362. Segment((1, 0), (random(), random())) → B90
  363. Intersect(A90, B90) → C90
  364. If((x(C90) * y(C90)) > 0, 1, 0) → D90
  365. Segment((0, 0), (random(), random())) → A91
  366. Segment((1, 0), (random(), random())) → B91
  367. Intersect(A91, B91) → C91
  368. If((x(C91) * y(C91)) > 0, 1, 0) → D91
  369. Segment((0, 0), (random(), random())) → A92
  370. Segment((1, 0), (random(), random())) → B92
  371. Intersect(A92, B92) → C92
  372. If((x(C92) * y(C92)) > 0, 1, 0) → D92
  373. Segment((0, 0), (random(), random())) → A93
  374. Segment((1, 0), (random(), random())) → B93
  375. Intersect(A93, B93) → C93
  376. If((x(C93) * y(C93)) > 0, 1, 0) → D93
  377. Segment((0, 0), (random(), random())) → A94
  378. Segment((1, 0), (random(), random())) → B94
  379. Intersect(A94, B94) → C94
  380. If((x(C94) * y(C94)) > 0, 1, 0) → D94
  381. Segment((0, 0), (random(), random())) → A95
  382. Segment((1, 0), (random(), random())) → B95
  383. Intersect(A95, B95) → C95
  384. If((x(C95) * y(C95)) > 0, 1, 0) → D95
  385. Segment((0, 0), (random(), random())) → A96
  386. Segment((1, 0), (random(), random())) → B96
  387. Intersect(A96, B96) → C96
  388. If((x(C96) * y(C96)) > 0, 1, 0) → D96
  389. Segment((0, 0), (random(), random())) → A97
  390. Segment((1, 0), (random(), random())) → B97
  391. Intersect(A97, B97) → C97
  392. If((x(C97) * y(C97)) > 0, 1, 0) → D97
  393. Segment((0, 0), (random(), random())) → A98
  394. Segment((1, 0), (random(), random())) → B98
  395. Intersect(A98, B98) → C98
  396. If((x(C98) * y(C98)) > 0, 1, 0) → D98
  397. Segment((0, 0), (random(), random())) → A99
  398. Segment((1, 0), (random(), random())) → B99
  399. Intersect(A99, B99) → C99
  400. If((x(C99) * y(C99)) > 0, 1, 0) → D99
  401. Segment((0, 0), (random(), random())) → A100
  402. Segment((1, 0), (random(), random())) → B100
  403. Intersect(A100, B100) → C100
  404. If((x(C100) * y(C100)) > 0, 1, 0) → D100
  405. Sum(D1:D100) → E1
  406. DotPlot(data) → c
  407. Mean(data) → d
  408. Vector(D, E) → u
  409. Vector(F, G) → v

AI Drawing Prompt Reference

Draw a square with sides starting from the origin, then draw random line segments from each vertex inside the square, mark their intersections with circles, color them differently based on the sign of the product of their coordinates, and display the distribution of these random points with a histogram, showcasing the intersection of geometry and probability.
The prompt above was compiled from the original drawing commands and is for reference only.