{"id":747,"date":"2019-04-10T09:09:49","date_gmt":"2019-04-10T07:09:49","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/lehning\/?p=747"},"modified":"2019-04-23T12:19:24","modified_gmt":"2019-04-23T10:19:24","slug":"galilee-cosme-ii-de-medicis-et-le-paradoxe-de-toscane","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/04\/10\/galilee-cosme-ii-de-medicis-et-le-paradoxe-de-toscane\/","title":{"rendered":"Galil\u00e9e, Cosme II de M\u00e9dicis et le paradoxe de Toscane"},"content":{"rendered":"

Cosme II de M\u00e9dicis, Grand-duc de Toscane, avait remarqu\u00e9 qu\u2019en jetant trois d\u00e9s, le total dix sortait plus souvent que le neuf. Pourtant, il existait autant de fa\u00e7ons de d\u00e9composer neuf et dix en somme de trois nombres entre un et six, ce qui lui semblait contradictoire.<\/p>\n

\"\"
Une fa\u00e7on de d\u00e9composer dix en somme de trois nombres.<\/figcaption><\/figure>\n

Ce paradoxe est connu sous le nom de paradoxe de Toscane.<\/p>\n

La solution de Galil\u00e9e<\/h2>\n

Galil\u00e9e (1564 – 1642), qui fut le pr\u00e9cepteur de Cosme II, trouva la raison de cette bizarrerie. On peut comprendre son m\u00e9canisme en consid\u00e9rant le jeu de pile ou face. Si la pi\u00e8ce n\u2019est pas pip\u00e9e, la probabilit\u00e9 d\u2019obtenir pile est \u00e9gale \u00e0 \u00bd et de m\u00eame celle d\u2019obtenir face. Si on joue deux fois de suite, chacune des possibilit\u00e9s PP, PF, FP et FF est \u00e9quiprobable donc leurs probabilit\u00e9s sont toutes \u00e9gales \u00e0 \u00bc. Si on jette les deux pi\u00e8ces \u00e0 la fois, les probabilit\u00e9s d\u2019avoir deux piles ou deux faces sont \u00e9gales \u00e0 \u00bc mais celle d\u2019avoir un pile et un face est \u00e9gale \u00e0 \u00bd car elle regroupe les deux cas PF et FP. Il en va exactement de m\u00eame dans le paradoxe de Toscane. Les d\u00e9compositions de neuf et dix ne sont pas \u00e9quivalentes de ce point de vue. La diff\u00e9rence tient en la d\u00e9composition de neuf en trois fois le m\u00eame nombre, ce qui est impossible pour dix. Le calcul permet d\u2019\u00e9tablir que la probabilit\u00e9 d\u2019obtenir neuf est \u00e9gale \u00e0 25\/216 alors que celle d\u2019obtenir dix est \u00e9gale \u00e0 27\u00a0\/\u00a0216 soit 1\/8. Ces deux nombres montrent que Cosme \u00e9tait fin observateur, et vraiment tr\u00e8s grand joueur, car les probabilit\u00e9s ne diff\u00e8rent que de 1\u00a0%.<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Cosme II de M\u00e9dicis, Grand-duc de Toscane, avait remarqu\u00e9 qu\u2019en jetant trois d\u00e9s, le total dix sortait plus souvent que le neuf. Pourtant, il existait autant de fa\u00e7ons de d\u00e9composer neuf et dix en somme de trois nombres entre un et six, ce qui lui semblait contradictoire. Ce paradoxe est connu sous le nom de … Continuer la lecture de Galil\u00e9e, Cosme II de M\u00e9dicis et le paradoxe de Toscane<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":12,"featured_media":748,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[63,29],"tags":[285,281,287,286,283,284,282],"class_list":["post-747","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-histoire","category-maths-2","tag-cosme-ii","tag-galilee","tag-jeu-de-des","tag-medicis","tag-paradoxe","tag-probabilite","tag-toscane"],"yoast_head":"\nGalil\u00e9e, Cosme II de M\u00e9dicis et le paradoxe de Toscane, par Herv\u00e9 Lehning<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/04\/10\/galilee-cosme-ii-de-medicis-et-le-paradoxe-de-toscane\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Galil\u00e9e, Cosme II de M\u00e9dicis et le paradoxe de Toscane, par Herv\u00e9 Lehning\" \/>\n<meta property=\"og:description\" content=\"Cosme II de M\u00e9dicis, Grand-duc de Toscane, avait remarqu\u00e9 qu\u2019en jetant trois d\u00e9s, le total dix sortait plus souvent que le neuf. 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