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[{"key": "dc.contributor.advisor", "value": "J\u00e4\u00e4skel\u00e4inen, Jarmo", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Niskanen, Atte", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-12-08T07:25:00Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-12-08T07:25:00Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/78901", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa esitet\u00e4\u00e4n Buffonin neulaongelmalle kolme toisistaan hieman\npoikkeavaa todistusta ja k\u00e4yd\u00e4\u00e4n l\u00e4pi, kuinka piit\u00e4 voidaan arvioida t\u00e4m\u00e4n avulla.\nTutkielmassa todistetaan my\u00f6s kaksi Buffonin neulaongelman laajennusta ja tutustutaan\nBuffonin neulaongelmaan mittateoriassa. Lis\u00e4ksi pohditaan kuinka Buffonin neulaongelman\nvoi esitell\u00e4 lukion eri moduuleilla, ja esitell\u00e4\u00e4n GeoGebra-appletti, jonka\navulla neulaongelmaa voi simuloida.\nYksinkertaisuudessaan Buffonin neula tarkoittaa uhkapeli\u00e4, jossa pelaaja heitt\u00e4\u00e4\nneulan lattialle, joka sis\u00e4lt\u00e4\u00e4 yhdensuuntaisia suoria ja panostuksen kohteena on, osuuko\nneula suoraan. Tutkielman tavoitteena on laskea tarkka todenn\u00e4k\u00f6isyys tapahtumalle P(neula leikkaa suoran). Tarkka todenn\u00e4k\u00f6isyys voidaan n\u00e4ytt\u00e4\u00e4\npuhtaasti stokastisin menetelmin, geometriaa hyv\u00e4ksik\u00e4ytt\u00e4en sek\u00e4 odotusarvon\nlineaarisuuden avulla.\nOngelman laajennuksissa neulaa ei tiputeta lattialle, vaan laitetaan pallon sis\u00e4\u00e4n ja\npallo tiputetaan lattialle. T\u00e4ll\u00f6in todenn\u00e4k\u00f6isyys, ett\u00e4 neula ylitt\u00e4\u00e4 suoran muuttuu hieman\nja yll\u00e4tt\u00e4v\u00e4sti todenn\u00e4k\u00f6isyydess\u00e4 ei en\u00e4\u00e4 esiinny lukua \ud835\udf0b. Mittateoriassa Buffonin\nneula nousee esiin, kun tutkitaan Cantorin itsesimilaarisen joukon Favardin mittaa.\nT\u00e4ll\u00f6in neulana toimii suora, johon joukko projisoidaan.\nTutkielmassa k\u00e4yd\u00e4\u00e4n my\u00f6s l\u00e4pi, kuinka Buffonin neulaongelma voidaan pilkkoa\npienempiin palasiin siten, ett\u00e4 sit\u00e4 on mahdollista kuljettaa mukana usealla lukion pitk\u00e4n\nmatematiikan moduulilla. T\u00e4ll\u00e4 tavoin opettaja pystyy motivoimaan opiskelijoita palaamalla\njo aikaisemmilta kursseilta tuttuun aiheeseen, joka todenn\u00e4k\u00f6isesti uhkapelin\u00e4\nher\u00e4tt\u00e4\u00e4 my\u00f6s opiskelijoiden mielenkiinnon.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-12-08T07:25:00Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-12-08T07:25:00Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "41", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Buffonin neulat ja nuudelit", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202112085890", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "todenn\u00e4k\u00f6isyyslaskenta", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lukio", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "todenn\u00e4k\u00f6isyys", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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