{"id":5756,"date":"2019-05-23T06:31:50","date_gmt":"2019-05-23T05:31:50","guid":{"rendered":"http:\/\/mathsguyon.fr\/?page_id=5756"},"modified":"2019-06-05T21:16:40","modified_gmt":"2019-06-05T20:16:40","slug":"loi-normale-loi-uniforme","status":"publish","type":"page","link":"https:\/\/mathsguyon.fr\/?page_id=5756","title":{"rendered":"Loi normale, loi uniforme"},"content":{"rendered":"

Les fondamentaux pour le Bac :<\/span><\/h1>\n

 <\/p>\n

\n

Exercice 1 :<\/strong><\/p>\n

En suivant la loi uniforme, on choisit un nombre au hasard dans l\u2019intervalle [4 ;11]. La probabilit\u00e9 que ce nombre soit inf\u00e9rieur \u00e0 10 est :<\/p>\n

a) $$\\dfrac{6}{11}$$ \u00a0\u00a0\u00a0 b) $$\\dfrac{10}{7}$$ \u00a0\u00a0\u00a0 c) $$\\dfrac{10}{11}$$ \u00a0\u00a0\u00a0 d) $$\\dfrac{6}{7}$$<\/strong><\/p>\n

Correction en vid\u00e9o<\/strong><\/span><\/a><\/p>\n

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Exercice 2 :<\/strong><\/p>\n

On choisit au hasard un nombre r\u00e9el dans l\u2019intervalle [10;50]. La probabilit\u00e9 que ce nombre appartienne \u00e0 l\u2019intervalle [15; 20] est :<\/p>\n

a)<\/strong> $$\\dfrac{5}{50}$$ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 b)<\/strong> $$\\dfrac{1}{8}$$ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 c)<\/strong> $$\\dfrac{1}{40}$$ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 d)<\/strong> $$\\dfrac{1}{5}$$<\/p>\n

Correction en vid\u00e9o<\/strong><\/span><\/a><\/p>\n

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Exercice 3 :<\/strong><\/p>\n

$$X$$ est une variable al\u00e9atoire qui suit la loi normale de moyenne 3 et d\u2019\u00e9cart-type 2 alors une valeur approch\u00e9e au centi\u00e8me de la probabilit\u00e9 $$p(X\\geqslant5)$$ est<\/p>\n

a)<\/strong> \u00a0\u00a0 0,14\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0\u00a0 b)\u00a0\u00a0<\/strong> \u00a0 0,16 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 c)\u00a0<\/strong>\u00a0 0,32 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 d)\u00a0<\/strong>\u00a0 0,84<\/p>\n

Correction en vid\u00e9o<\/strong><\/span><\/a><\/p>\n

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Exercice 4 :<\/strong><\/p>\n

La variable al\u00e9atoire $$X$$ suit une loi normale d\u2019esp\u00e9rance $$\\mu=0$$\u00a0 et d\u2019\u00e9cart type $$\\sigma$$ inconnu mais on sait que $$P(-10\u00a0 \\leqslant X \\leqslant 10)=0,8$$.<\/p>\n

On peut en d\u00e9duire :<\/p>\n

a)<\/strong>\u00a0 $$ P(X\\leqslant10)=0,1$$ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0\u00a0\u00a0 b)\u00a0<\/strong> $$ P(X\\leqslant10)=0,2\u00a0\u00a0\u00a0 $$ \u00a0
\n<\/strong><\/p>\n

c)<\/strong>\u00a0 $$ P(X\\leqslant10)=0,5$$ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0\u00a0\u00a0 d)\u00a0<\/strong>\u00a0 $$P(X\\leqslant10)=0,9$$<\/p>\n

Correction en vid\u00e9o<\/strong><\/span><\/a><\/p>\n

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Exercice 5 :<\/strong><\/p>\n

Une variable al\u00e9atoire $$X$$ suit la loi uniforme sur l\u2019intervalle [0; 5] dont la fonction de densit\u00e9 est repr\u00e9sent\u00e9e ci-dessous. D\u00e9terminer la bonne \u00e9galit\u00e9 :<\/p>\n

\"\"<\/p>\n

a)<\/strong> \u00a0 $$P(X\\geqslant3)=P(X\\leqslant3)$$ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 b)\u00a0\u00a0<\/strong> $$P(1\\leqslant X \\leqslant 4)=\\dfrac{1}{3}$$\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 <\/strong><\/p>\n

c)<\/strong> \u00a0 $$E(X)=\\dfrac{5}{2}$$ \u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 d)<\/strong> \u00a0 $$E(X)=\\dfrac{1}{5}$$<\/p>\n

Correction en vid\u00e9o<\/strong><\/span><\/a><\/p>\n

\n

Exercice 6 :<\/strong><\/p>\n

On mod\u00e9lise le nombre de parties jou\u00e9es par jour \u00e0 une\u00a0 loterie par une variable al\u00e9atoire $$X$$ qui suit une loi normale d\u2019esp\u00e9rance $$\\mu=150$$ et d\u2019\u00e9cart-type $$\\sigma=10$$.<\/p>\n

Une valeur approch\u00e9e \u00e0 $$10^{-3}$$ pr\u00e8s de $$P(140 \\leqslant X \\leqslant 160)$$ est<\/p>\n

a)\u00a0<\/strong>\u00a0\u00a0\u00a0 0,954\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 b)\u00a0<\/strong>\u00a0\u00a0\u00a0 0,683\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 c)<\/strong>\u00a0\u00a0 0,997\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 d)<\/strong>\u00a0\u00a0 0,841<\/p>\n

Correction en vid\u00e9o<\/strong><\/span><\/a><\/p>\n

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Exercice 7 :<\/strong><\/p>\n

Cette \u00e9preuve permet de d\u00e9velopper sa VMA (vitesse maximale a\u00e9robie) qui correspond \u00e0 une vitesse de course rapide. L\u2019unit\u00e9 de mesure de la VMA est le km\/h.On choisit un \u00e9l\u00e8ve au hasard parmi les 120 \u00e9l\u00e8ves.<\/p>\n

On admet que la VMA d\u2019un \u00e9l\u00e8ve pris au hasard est mod\u00e9lis\u00e9e par une variable al\u00e9atoire $$Y$$ qui suit la loi normale d\u2019esp\u00e9rance $$\\mu=11,8$$ et d\u2019\u00e9cart type $$\\sigma=1,2$$.<\/p>\n

    \n
  1. Quelle est la probabilit\u00e9 arrondie \u00e0 $$10^{-3}$$ , qu\u2019un \u00e9l\u00e8ve de terminale de ce lyc\u00e9e ait une VMA comprise entre 10 et 13 km\/h?<\/li>\n
  2. D\u00e9terminer la valeur arrondie au dixi\u00e8me de $$\\alpha$$ tel que $$P(Y \\leqslant \\alpha)=0,8$$. Interpr\u00e9ter cette valeur dans le contexte de l\u2019exercice.<\/li>\n<\/ol>\n

    Correction<\/a><\/strong><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

    Les fondamentaux pour le Bac :   Exercice 1 : En suivant la loi uniforme, on choisit un nombre au hasard dans l\u2019intervalle [4 ;11]. La probabilit\u00e9 que ce nombre soit inf\u00e9rieur \u00e0 10 est : a) $$\\dfrac{6}{11}$$ \u00a0\u00a0\u00a0 b) … Continuer la lecture →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":4521,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-5756","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/5756","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5756"}],"version-history":[{"count":30,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/5756\/revisions"}],"predecessor-version":[{"id":5832,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/5756\/revisions\/5832"}],"up":[{"embeddable":true,"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=\/wp\/v2\/pages\/4521"}],"wp:attachment":[{"href":"https:\/\/mathsguyon.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5756"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}