{"id":68,"date":"2022-01-11T14:10:00","date_gmt":"2022-01-11T13:10:00","guid":{"rendered":""},"modified":"2023-08-04T22:47:18","modified_gmt":"2023-08-04T20:47:18","slug":"buffonova-uloha-o-ihle","status":"publish","type":"post","link":"https:\/\/www.jankafialka.sk\/?p=68","title":{"rendered":"Buffonova \u00faloha o ihle"},"content":{"rendered":"\n

V\u010dera som h\u013eadala ak\u00e9si papiere. Ako to tak b\u00fdva, nena\u0161la som to, \u010do som h\u013eadala, ale nie\u010do in\u00e9. Na\u0161la som jeden \u0161an\u00f3n, ktor\u00fd som si odlo\u017eila z \u010dias, ke\u010f som chodila na vysok\u00fa. Boli v \u0148om okop\u00edrovan\u00e9 \u010dasopisy DM (didaktika matematiky), ktor\u00e9 vytv\u00e1ral a s\u00e1m aj tla\u010dil n\u00e1\u0161 nezabudnute\u013en\u00fd u\u010dite\u013e Du\u0161an Jedin\u00e1k. V jednom z nich je \u00faloha, ktor\u00e1 ma in\u0161pirovala k nap\u00edsaniu tohto pr\u00edspevku – Buffonova \u00faloha o ihle.<\/p>\n\n\n\n

Buffonovu \u00falohu o ihle vytvoril Georges Louis Leclerc de Buffon v roku 1777. Bol to franc\u00fazsky matematik, pr\u00edrodovedec a spisovate\u013e (v\u00fdborn\u00e1 kombin\u00e1cia \ud83d\ude09). V \u00falohe ide o to, vypo\u010d\u00edta\u0165, ak\u00e1 je pravdepodobnos\u0165, \u017ee ke\u010f hod\u00edm ihlu na zem, na ktorej s\u00fa vyzna\u010den\u00e9 rovnobe\u017en\u00e9 priamky v\u0161etky rovnako od seba vzdialen\u00e9, \u017ee t\u00e1to ihla pretne niektor\u00fa z t\u00fdch priamok.<\/p>\n\n\n\n

Sk\u00f4r ne\u017e sa pust\u00edm do rie\u0161enia, neodpust\u00edm si zamyslenie nad t\u00fdm, ako t\u00e1to \u00faloha asi vznikla. Predstavujem si p\u00e1na Buffona, ako si za\u0161\u00edva pono\u017eky v miestnosti s podlahou z dreven\u00fdch dos\u00e1k. V\u0161etky dosky s\u00fa rovnako \u0161irok\u00e9 a dos\u0165 dlh\u00e9 na to, aby vysta\u010dili na cel\u00fa \u0161\u00edrku miestnosti. Nitka sa minula, treba navliec\u0165 nov\u00fa a \u0161up. Ihla spadla na zem tak, \u017ee pr\u00e1ve v jej strede pretla \u0161k\u00e1ru medzi doskami. To ho zaujalo. Ak\u00e1 je pravdepodobnos\u0165, \u017ee sa toto stane? Sk\u00fasi hodi\u0165 ihlu e\u0161te p\u00e1r kr\u00e1t. Ihla niekedy padne na \u0161k\u00e1ru, niekedy nie. A tak vymysl\u00ed \u00falohu, ktor\u00e1 ponesie jeho meno.<\/p>\n\n\n\n

\"\"
Ihla na drevenej podlahe<\/figcaption><\/figure>\n\n\n\n

Mo\u017eno to tak bolo, mo\u017eno nie. Ke\u010f\u017ee som si to pr\u00e1ve teraz vymyslela, tak sk\u00f4r nie. \ud83d\ude01 Newtonovi tie\u017e jablko na hlavu pravdepodobne nepadlo.<\/p>\n\n\n\n

Ja by som ale zadanie roz\u0161\u00edrila o \u010fal\u0161iu ot\u00e1zku. Spravila by som z toho s\u00fa\u0165a\u017e. Dvaja h\u00e1d\u017eu na zem ihlu (naz\u00fdvam ju ihla, ale z bezpe\u010dnostn\u00fdch d\u00f4vodov by som pri praktickej realiz\u00e1cii pou\u017eila nejak\u00fa neostr\u00fa pali\u010dku). Na zemi s\u00fa rovnobe\u017en\u00e9 \u010diary v\u017edy 10 cm od seba. Ke\u010f hr\u00e1\u010d traf\u00ed \u010diaru, m\u00e1 bod. Je to f\u00e9rov\u00e1 s\u00fa\u0165a\u017e – obaja maj\u00fa rovnak\u00e9 \u0161ance. V tom ale jeden z hr\u00e1\u010dov navrhne, \u017ee si to s\u0165a\u017e\u00ed. Bude h\u00e1dza\u0165 polovi\u010dnou ihlou, ale za ka\u017ed\u00e9 trafenie dostane dva body. Je jeho n\u00e1vrh f\u00e9rov\u00fd?<\/p>\n\n\n\n

Rie\u0161enie<\/h2>\n\n\n\n

Teraz m\u00f4\u017eeme prejs\u0165 k samotn\u00e9mu v\u00fdpo\u010dtu. M\u00e1me tu nieko\u013eko premenn\u00fdch. Nevieme, kam ihla dopadne – budeme uva\u017eova\u0165 o vzdialenosti stredu ihly od najbli\u017e\u0161ej \u010diary. T\u00fato premenn\u00fa nazveme d<\/i> a je jasn\u00e9, \u017ee m\u00f4\u017ee ma\u0165 hodnoty od nuly do 5 cm, ke\u010f\u017ee \u010diary s\u00fa od seba vzdialen\u00e9 10 cm. Druhou premennou bude uhol, ktor\u00fd bude ihla zviera\u0165 s \u010diarami. Tento m\u00f4\u017ee by\u0165 medzi nulou a prav\u00fdm uhlom, \u010di\u017ee \u03c0\/2. Pri h\u00e1dzan\u00ed s\u00fa tieto dve premenn\u00e9 nez\u00e1visl\u00e9 – neovplyv\u0148uj\u00fa sa. To, kam padne ihla nem\u00e1 vplyv na uhol, ani naopak. V pr\u00edpade, \u017ee ihla pretne \u010diaru, plat\u00ed vz\u0165ah medzi t\u00fdmito dvoma premenn\u00fdmi. Ten je odvoden\u00fd na obr\u00e1zku. Aha, e\u0161te je tu h<\/i> – d\u013a\u017eka ihly. T\u00fa si predstav\u00edme ako nejak\u00e9 konkr\u00e9tne \u010d\u00edslo, zatia\u013e nevieme, ak\u00e9.<\/p>\n\n\n\n

\"Buffonova
Buffonova \u00faloha – rie\u0161enie<\/figcaption><\/figure>\n\n\n\n

No a ke\u010f chceme vypo\u010d\u00edta\u0165 pravdepodobnos\u0165, mus\u00edme ur\u010di\u0165 po\u010det priazniv\u00fdch pr\u00edpadov a po\u010det v\u0161etk\u00fdch mo\u017en\u00fdch pr\u00edpadov. V tejto \u00falohe s\u00fa oba nekone\u010dn\u00e9 – ihla m\u00f4\u017ee dopadn\u00fa\u0165 kamko\u013evek a aj pretn\u00fa\u0165 \u010diaru kdeko\u013evek.<\/p>\n\n\n\n

V tak\u00fdchto pr\u00edpadoch n\u00e1m skvele pom\u00f4\u017ee geometrick\u00e1<\/b> pravdepodobnos\u0165<\/b>. Premenn\u00e9 zmen\u00edme na geometrick\u00e9 objekty, a tak ur\u010d\u00edme ve\u013ekos\u0165 mno\u017einy priazniv\u00fdch udalost\u00ed aj mno\u017einy v\u0161etk\u00fdch mo\u017en\u00fdch udalost\u00ed. Ke\u010f\u017ee m\u00e1me dve premenn\u00e9, p\u00f4jde o plochy. Ve\u013ekos\u0165 mno\u017einy v\u0161etk\u00fdch mo\u017en\u00fdch udalost\u00ed je obsahom obd\u013a\u017enika so stranami 5 cm za premenn\u00fa d<\/i> a \u03c0\/2 za uhol. Mno\u017einu priazniv\u00fdch udalost\u00ed ur\u010d\u00edme ako ve\u013ekos\u0165 plochy pod \u010diarou, ktor\u00e1 ur\u010duje vypo\u010d\u00edtan\u00fd vz\u0165ah medzi t\u00fdmito dvoma premenn\u00fdmi. Spomenieme si, ako sa po\u010d\u00edta ur\u010dit\u00fd integr\u00e1l a v\u00fdsledok je za chv\u00ed\u013eu na svete.<\/p>\n\n\n\n

Pravdepodobnos\u0165 je teda h<\/i>\/5\u03c0. Rastie priamo\u00famerne s d\u013a\u017ekou ihly, a\u017e k\u00fdm ihla nebude dlh\u0161ia, ne\u017e je vzdialenos\u0165 \u010diar. (Tam by sa musel upravi\u0165 v\u00fdpo\u010det geometrickej pravdepodobnosti.) S polovi\u010dnou ihlou bude teda naozaj polovi\u010dn\u00e1 pravdepodobnos\u0165 a teda n\u00e1vrh je f\u00e9rov\u00fd. Zauj\u00edmav\u00e9 je, \u017ee ak by sme zvolili ihlu d\u013a\u017eky 5 cm – polovica vzdialenosti rovnobe\u017eiek, tak pravdepodobnos\u0165 je 1\/\u03c0. T\u00fdm dost\u00e1vame pekn\u00fd sp\u00f4sob, ako pribli\u017ene ur\u010di\u0165 hodnotu \u010d\u00edsla \u03c0. Sta\u010d\u00ed ve\u013emi ve\u013emi ve\u013ea kr\u00e1t h\u00e1dza\u0165 ihlu na zem s rovnobe\u017ekami a zapisova\u0165, ko\u013ekokr\u00e1t ihla pretla nejak\u00fa \u010diaru. \ud83d\ude09<\/p>\n","protected":false},"excerpt":{"rendered":"

V\u010dera som h\u013eadala ak\u00e9si papiere. Ako to tak b\u00fdva, nena\u0161la som to, \u010do som h\u013eadala, ale nie\u010do in\u00e9. Na\u0161la som jeden \u0161an\u00f3n, ktor\u00fd som si odlo\u017eila z \u010dias, ke\u010f som chodila na vysok\u00fa. Boli v \u0148om okop\u00edrovan\u00e9 \u010dasopisy DM (didaktika matematiky), ktor\u00e9 vytv\u00e1ral a s\u00e1m aj tla\u010dil n\u00e1\u0161 nezabudnute\u013en\u00fd u\u010dite\u013e Du\u0161an Jedin\u00e1k. V jednom z […]<\/p>\n","protected":false},"author":2,"featured_media":1945,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[27,28,29],"class_list":["post-68","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-nezaradene","tag-buffonova-ihla","tag-geometricka-pravdepodobnost","tag-slavne-ulohy"],"yoast_head":"\nBuffonova \u00faloha o ihle - jankafialka<\/title>\n<meta name=\"description\" content=\"Buffonova \u00faloha o ihle obsahuje zadanie a rie\u0161enie zn\u00e1mej \u00falohy - ak\u00e1 je pravdepodobnos\u0165, \u017ee ihla spadnut\u00e1 na dl\u00e1\u017eku padne na \u0161k\u00e1ru.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.jankafialka.sk\/?p=68\" \/>\n<meta property=\"og:locale\" content=\"sk_SK\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Buffonova \u00faloha o ihle - 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