{"id":2850,"date":"2023-12-06T16:48:01","date_gmt":"2023-12-06T15:48:01","guid":{"rendered":"https:\/\/math.univ-lyon1.fr\/dream\/?p=2850"},"modified":"2026-01-24T17:40:23","modified_gmt":"2026-01-24T16:40:23","slug":"le-probleme-de-l-echange","status":"publish","type":"post","link":"https:\/\/math.univ-lyon1.fr\/dream\/?p=2850","title":{"rendered":"Le probl\u00e8me de l&rsquo;\u00e9change"},"content":{"rendered":"\n<h2>\n\t\t\t\u00c9nonc\u00e9 du probl\u00e8me\t<\/h2>\n\tOn dispose de 4 jetons de forme et de poids identiques mais de couleurs diff\u00e9rentes (rouge, vert, noir et rouge) et un sac opaque. L&rsquo;un de ces jetons est dit \u00ab gagnant \u00bb (le vert) et les 3 autres sont dits \u00ab perdants \u00bb.<br \/>\nAvec ces jetons nous allons proc\u00e9der \u00e0 des \u00e9changes.<br \/>\nPour commencer, on d\u00e9pose les 3 jetons \u00ab perdants \u00bb dans le sac et on garde en main le jeton \u00ab gagnant \u00bb.<br \/>\nPour le premier \u00e9change, on tire au hasard un jeton du sac et on d\u00e9pose \u00e0 la place le jeton qu&rsquo;on avait pr\u00e9alablement en main (ici le jeton \u00ab gagnant \u00bb)<br \/>\nPour les \u00e9changes suivants, le principe est le m\u00eame, on tire au hasard un jeton du sac et on d\u00e9pose \u00e0 la place le jeton d\u00e9tenu dans la main.<br \/>\n<strong>Dans la premi\u00e8re partie du probl\u00e8me, on s&rsquo;int\u00e9resse \u00e0 la probabilit\u00e9 de r\u00e9cup\u00e9rer le jeton \u00ab gagnant \u00bb au bout de 3 \u00e9changes maximum. Cette probabilit\u00e9 est-elle sup\u00e9rieure ou inf\u00e9rieure \u00e0 1\/2 ?<\/strong>\n\t<p><iframe loading=\"lazy\" title=\"Vid\u00e9oDREAMaths_PbEchange\" width=\"560\" height=\"315\" src=\"https:\/\/tube-sciences-technologies.apps.education.fr\/videos\/embed\/3To9kgDri35VhmbzMdZ9gH\" allow=\"fullscreen\" sandbox=\"allow-same-origin allow-scripts allow-popups allow-forms\"><\/iframe><\/p>\n\t\t\t<a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?page_id=79\" target=\"_self\">\n\t\t\t\t\t\tRetour Panier \u00e0 probl\u00e8mes\n\t\t\t\t\t<\/a>\n\t<h3>Connaissances math\u00e9matiques<\/h3>\nLe th\u00e8me abord\u00e9 concerne les probabilit\u00e9s du cycle 4 et du lyc\u00e9e. Plus pr\u00e9cis\u00e9ment :<br \/>\n\u2666\u00a0exp\u00e9rience al\u00e9atoire, \u00e9preuve, univers, issues. . .vocabulaire des probabilit\u00e9s<br \/>\n\u2666\u00a0mod\u00e8le d&rsquo;\u00e9quiprobabilit\u00e9<br \/>\n\u2666\u00a0arbre de d\u00e9nombrement<br \/>\n\u2666\u00a0exprimer et calculer sur des probabilit\u00e9s sous la forme de fractions, de pourcentages, de<br \/>\nd\u00e9cimaux<br \/>\n\u2666 probabilit\u00e9s conditionnelles<br \/>\n\u2666 syst\u00e8me complet d&rsquo;\u00e9v\u00e8nements<br \/>\n\u2666\u00a0simulation et \u00e9chantillonnage\n\t\t\t<a href=\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2022\/03\/Le-probleme-de-lechange.pdf\" target=\"_blank\" rel=\"noopener\">\n\t\t\t\t\t\t\tT\u00e9l\u00e9charcher le document complet\n\t\t\t<\/a>\n\n","protected":false},"excerpt":{"rendered":"<p>Quatre jetons dont un \u00ab\u202fgagnant\u202f\u00bb vert et trois perdants. Apr\u00e8s trois \u00e9changes al\u00e9atoires dans un sac, quelle est la probabilit\u00e9 de r\u00e9cup\u00e9rer le jeton gagnant\u202f?<\/p>\n","protected":false},"author":3,"featured_media":2853,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[107],"tags":[],"niveau":[113,114],"type_situation":[116],"theme_maths":[118,119],"probleme_associe":[],"type-expe":[],"class_list":{"0":"post-2850","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-situation-probleme","8":"niveau-cycle-4","9":"niveau-lycee","10":"type_situation-situation-probleme","11":"theme_maths-nombres-et-calculs","12":"theme_maths-ogd-fonctions","13":"czr-hentry"},"rttpg_featured_image_url":{"full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE.png",651,301,false],"landscape":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE.png",651,301,false],"portraits":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE.png",651,301,false],"thumbnail":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE-150x150.png",150,150,true],"medium":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE-300x139.png",300,139,true],"large":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE.png",651,301,false],"1536x1536":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE.png",651,301,false],"2048x2048":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE.png",651,301,false],"tc-grid-full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE.png",651,301,false],"tc-grid":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE-570x301.png",570,301,true],"tc-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE-270x250.png",270,250,true],"slider-full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE.png",651,301,false],"slider":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE.png",651,301,false],"tc-sq-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE-510x301.png",510,301,true],"tc-ws-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE.png",651,301,false],"tc-ws-small-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE-528x297.png",528,297,true],"tc-slider-small":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/12\/PROBLEMEECHANGE-517x141.png",517,141,true]},"rttpg_author":{"display_name":"St\u00e9phanie CROQUELOIS","author_link":"https:\/\/math.univ-lyon1.fr\/dream\/?author=3"},"rttpg_comment":0,"rttpg_category":"<a href=\"https:\/\/math.univ-lyon1.fr\/dream\/?cat=107\" rel=\"category\">Situation Probl\u00e8me<\/a>","rttpg_excerpt":"Quatre jetons dont un \u00ab\u202fgagnant\u202f\u00bb vert et trois perdants. Apr\u00e8s trois \u00e9changes al\u00e9atoires dans un sac, quelle est la probabilit\u00e9 de r\u00e9cup\u00e9rer le jeton gagnant\u202f?","_links":{"self":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/2850","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2850"}],"version-history":[{"count":10,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/2850\/revisions"}],"predecessor-version":[{"id":3927,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/2850\/revisions\/3927"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/media\/2853"}],"wp:attachment":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2850"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2850"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2850"},{"taxonomy":"niveau","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fniveau&post=2850"},{"taxonomy":"type_situation","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype_situation&post=2850"},{"taxonomy":"theme_maths","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftheme_maths&post=2850"},{"taxonomy":"probleme_associe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fprobleme_associe&post=2850"},{"taxonomy":"type-expe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype-expe&post=2850"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}