{"id":780,"date":"2019-11-02T17:58:59","date_gmt":"2019-11-02T16:58:59","guid":{"rendered":"http:\/\/caneri.gmollet.free.fr\/?page_id=780"},"modified":"2023-09-13T15:50:34","modified_gmt":"2023-09-13T13:50:34","slug":"activite-xi","status":"publish","type":"page","link":"https:\/\/courscaneri.ovh\/?page_id=780","title":{"rendered":"Activit\u00e9 XI – Probabilit\u00e9s conditionnelles"},"content":{"rendered":"

Lucien, fumeur imp\u00e9nitent, d\u00e9cide d\u2019essayer de ne plus fumer.
\nS\u2019il ne fume pas un jour donn\u00e9, la probabilit\u00e9 qu\u2019il ne fume pas le lendemain est 0,76.
\nPar contre, s\u2019il fume un jour donn\u00e9, la probabilit\u00e9 qu\u2019il ne fume pas le lendemain est 0,36.<\/p>\n

On note \"F_n\"<\/code> l\u2019\u00e9v\u00e9nement \u00ab\u202fLucien fume\u202fle n-i\u00e8me jour \u00bb et \"\overline{F_n}\"<\/code> l\u2019\u00e9v\u00e9nement contraire.
\nOn pose \"\forall n \in \mathbb{N}, p_n=\mathbb{P}\left(F_n\right)\"<\/code>.<\/p>\n

    \n
  1. D\u00e9terminer \"p_0\"<\/code>.<\/li>\n
  2. D\u00e9terminer \"<code class="katex-inline">\mathbb{P}_{\overline{F_n}}\left(\overline{F_{n+1}}\right)\"<\/code> puis \"\mathbb{P}_{F_n}\left(\overline{F_{n+1}}\right)\"<\/code>.<\/li>\n
  3. Compl\u00e9ter cet arbre de probabilit\u00e9.\n

    \"\"<\/li>\n

  4. Calculer \"p_1\"<\/code>, \"p_2\"<\/code> puis \"p_3\"<\/code>.<\/li>\n
  5. Exprimer \"p_{n+1}\"<\/code> en fonction de \"p_n\"<\/code>.<\/li>\n
  6. On pose: \"\forall n \in \mathbb{N}, q_n=p_n-0,4\"<\/code>.
    \na. D\u00e9montrer que la suite \"\left(q_n\right)_{n\in\mathbb{N}}\"<\/code> est g\u00e9om\u00e9trique. On en pr\u00e9cisera la raison et le terme initial.
    \nb. En d\u00e9duire l’expression de \"q_n\"<\/code> en fonction de \"n\"<\/code>.
    \nc. En d\u00e9duire l’expression de \"p_n\"<\/code> en fonction de \"n\"<\/code>.
    \nd. D\u00e9terminer \"\lim\limits_{n\to + \infty}p_n\"<\/code>. Interpr\u00e9ter ce r\u00e9sultat.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    Lucien, fumeur imp\u00e9nitent, d\u00e9cide d\u2019essayer de ne plus fumer. S\u2019il ne fume pas un jour donn\u00e9, la probabilit\u00e9 qu\u2019il ne fume pas le lendemain est 0,76. Par contre, s\u2019il fume un jour donn\u00e9, la probabilit\u00e9 qu\u2019il ne fume pas le lendemain est 0,36. On note l\u2019\u00e9v\u00e9nement \u00ab\u202fLucien fume\u202fle n-i\u00e8me jour \u00bb et l\u2019\u00e9v\u00e9nement contraire. On […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-780","page","type-page","status-publish","czr-hentry"],"_links":{"self":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages\/780","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=780"}],"version-history":[{"count":2,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages\/780\/revisions"}],"predecessor-version":[{"id":3933,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages\/780\/revisions\/3933"}],"wp:attachment":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=780"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}