{"id":1563,"date":"2026-09-08T08:06:57","date_gmt":"2026-09-08T08:06:57","guid":{"rendered":"https:\/\/maroczain.com\/scolaire.maroczain.com\/?page_id=1563"},"modified":"2026-09-08T08:06:57","modified_gmt":"2026-09-08T08:06:57","slug":"modeliser-conditionner-prevoir","status":"publish","type":"page","link":"https:\/\/maroczain.com\/scolaire.maroczain.com\/modeliser-conditionner-prevoir\/","title":{"rendered":"Mod\u00e9liser. Conditionner. Pr\u00e9voir."},"content":{"rendered":"\n&#8220;`html id=&#8221;mza-probabilites-17&#8243;\n<div id=\"mza-probabilites-cpge\">\n\n<style>\n#mza-probabilites-cpge,#mza-probabilites-cpge *{box-sizing:border-box}\n#mza-probabilites-cpge{\n --navy:#06172c;--navy2:#0d416d;--gold:#c99a36;--gold2:#efd68c;\n --paper:#f5f6f8;--white:#fff;--text:#18283a;--muted:#687789;--line:#dfe5eb;\n font-family:Arial,Helvetica,sans-serif;background:var(--paper);color:var(--text);overflow:hidden\n}\n#mza-probabilites-cpge a{text-decoration:none;color:inherit}\n#mza-probabilites-cpge .wrap{max-width:1200px;margin:auto;padding:0 24px}\n#mza-probabilites-cpge section{padding:68px 0}\n#mza-probabilites-cpge .hero{\n color:#fff;padding:72px 0;\n background:\n radial-gradient(circle at 84% 15%,rgba(213,169,68,.28),transparent 28%),\n linear-gradient(135deg,#041224,#082b4c 65%,#10517e)\n}\n#mza-probabilites-cpge .badge{\n display:inline-block;padding:8px 13px;border-radius:999px;\n border:1px solid 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.grid,#mza-probabilites-cpge .method{grid-template-columns:1fr}\n}\n<\/style>\n\n<header class=\"hero\">\n<div class=\"wrap\">\n\n<span class=\"badge\">COURS 17 \u2022 PROBABILIT\u00c9S<\/span>\n\n<h1>Mod\u00e9liser.<br>Conditionner. Pr\u00e9voir.<\/h1>\n\n<p class=\"lead\">\nLes probabilit\u00e9s permettent de mesurer l&#8217;incertitude, raisonner avec des \u00e9v\u00e9nements,\nmettre \u00e0 jour une information gr\u00e2ce au conditionnement et mod\u00e9liser des exp\u00e9riences\nal\u00e9atoires \u00e0 l&#8217;aide de variables al\u00e9atoires.\n<\/p>\n\n<div class=\"actions\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda COMMENCER<\/a>\n<a href=\"#methode\" class=\"btn glass\">\ud83e\udde0 M\u00c9THODE<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 QCM<\/a>\n<\/div>\n\n<\/div>\n<\/header>\n\n<section id=\"cours\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">1. UNIVERS &#038; \u00c9V\u00c9NEMENTS<\/span>\n<h2>D\u00e9crire une exp\u00e9rience al\u00e9atoire.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Univers \u03a9<\/h3>\n<p>\n\u03a9 contient toutes les issues possibles d&#8217;une exp\u00e9rience al\u00e9atoire.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>\u00c9v\u00e9nement<\/h3>\n<p>\nUn \u00e9v\u00e9nement A est un sous-ensemble de \u03a9.\n<\/p>\n\n<div class=\"formula\">A \u2282 \u03a9<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>\u00c9v\u00e9nement contraire<\/h3>\n\n<div class=\"formula\">\nP(A\u0304)=1-P(A)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>\u00c9v\u00e9nement impossible<\/h3>\n\n<div class=\"formula\">\nP(\u2205)=0\n<\/div>\n\n<p>et P(\u03a9)=1.<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">2. UNION &#038; INTERSECTION<\/span>\n<h2>Combiner des \u00e9v\u00e9nements.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Intersection<\/h3>\n\n<div class=\"formula\">\nA \u2229 B\n<\/div>\n\n<p>A et B se r\u00e9alisent simultan\u00e9ment.<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Union<\/h3>\n\n<div class=\"formula\">\nA \u222a B\n<\/div>\n\n<p>Au moins l&#8217;un des deux \u00e9v\u00e9nements se r\u00e9alise.<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Formule<\/h3>\n\n<div class=\"formula\">\nP(A\u222aB)=P(A)+P(B)-P(A\u2229B)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>\u00c9v\u00e9nements incompatibles<\/h3>\n\n<div class=\"formula\">\nA\u2229B=\u2205\n<\/div>\n\n<p>Alors P(A\u222aB)=P(A)+P(B).<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">3. PROBABILIT\u00c9 CONDITIONNELLE<\/span>\n<h2>Calculer sous une information nouvelle.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>\nSi P(B)&gt;0, la probabilit\u00e9 de A sachant B est :\n<\/p>\n\n<div class=\"formula\">\nP(A|B)=P(A\u2229B)\/P(B)\n<\/div>\n\n<\/div>\n\n<div class=\"grid\" style=\"margin-top:16px\">\n\n<div class=\"card\">\n<h3>Produit<\/h3>\n\n<div class=\"formula\">\nP(A\u2229B)=P(B)P(A|B)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Autre \u00e9criture<\/h3>\n\n<div class=\"formula\">\nP(A\u2229B)=P(A)P(B|A)\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"methode\" class=\"dark\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\" style=\"color:#efd68c\">MZA PROBABILITY ENGINE<\/span>\n<h2>Six r\u00e9flexes.<\/h2>\n<\/div>\n\n<div class=\"method\">\n<div><b>01<\/b><span>UNIVERS<\/span><\/div>\n<div><b>02<\/b><span>\u00c9V\u00c9NEMENTS<\/span><\/div>\n<div><b>03<\/b><span>ARBRE<\/span><\/div>\n<div><b>04<\/b><span>CONDITIONNER<\/span><\/div>\n<div><b>05<\/b><span>TOTALISER<\/span><\/div>\n<div><b>06<\/b><span>V\u00c9RIFIER<\/span><\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">4. IND\u00c9PENDANCE<\/span>\n<h2>Quand une information ne change rien.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>D\u00e9finition<\/h3>\n\n<div class=\"formula\">\nP(A\u2229B)=P(A)P(B)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Conditionnement<\/h3>\n\n<p>Si P(B)&gt;0 :<\/p>\n\n<div class=\"formula\">\nP(A|B)=P(A)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Attention<\/h3>\n\n<p>\nInd\u00e9pendance et incompatibilit\u00e9 sont deux notions diff\u00e9rentes.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Pi\u00e8ge CPGE<\/h3>\n\n<div class=\"tip\">\nDeux \u00e9v\u00e9nements incompatibles de probabilit\u00e9 non nulle ne sont pas ind\u00e9pendants.\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">5. FORMULE DES PROBABILIT\u00c9S TOTALES<\/span>\n<h2>D\u00e9composer selon plusieurs sc\u00e9narios.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>\nSi B\u2081,&#8230;,B\u2099 forment un syst\u00e8me complet d&#8217;\u00e9v\u00e9nements :\n<\/p>\n\n<div class=\"formula\">\nP(A)=\u03a3 P(B\u1d62)P(A|B\u1d62)\n<\/div>\n\n<\/div>\n\n<div class=\"tip\">\nTr\u00e8s utile dans les arbres de probabilit\u00e9s : on additionne les chemins menant au m\u00eame \u00e9v\u00e9nement.\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">6. FORMULE DE BAYES<\/span>\n<h2>Remonter de l&#8217;effet vers la cause.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<div class=\"formula\">\nP(B|A)=P(B)P(A|B)\/P(A)\n<\/div>\n\n<p>\nBayes permet de r\u00e9viser la probabilit\u00e9 d&#8217;une hypoth\u00e8se apr\u00e8s l&#8217;observation d&#8217;un \u00e9v\u00e9nement.\n<\/p>\n\n<\/div>\n\n<div class=\"tip\">\nR\u00e9flexe : calculer d&#8217;abord P(A) avec la formule des probabilit\u00e9s totales.\n<\/div>\n\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">7. VARIABLE AL\u00c9ATOIRE<\/span>\n<h2>Transformer les issues en valeurs num\u00e9riques.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>D\u00e9finition<\/h3>\n\n<p>\nUne variable al\u00e9atoire X associe une valeur r\u00e9elle \u00e0 chaque issue de \u03a9.\n<\/p>\n\n<div class=\"formula\">\nX : \u03a9 \u2192 \u211d\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Loi<\/h3>\n\n<p>\nLa loi de X donne les probabilit\u00e9s des diff\u00e9rentes valeurs possibles.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Variable discr\u00e8te<\/h3>\n\n<div class=\"formula\">\nP(X=x\u1d62)=p\u1d62\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Contr\u00f4le<\/h3>\n\n<div class=\"formula\">\n\u03a3 p\u1d62 = 1\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">8. ESP\u00c9RANCE<\/span>\n<h2>La valeur moyenne th\u00e9orique.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>D\u00e9finition<\/h3>\n\n<div class=\"formula\">\nE(X)=\u03a3 x\u1d62P(X=x\u1d62)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Lin\u00e9arit\u00e9<\/h3>\n\n<div class=\"formula\">\nE(aX+b)=aE(X)+b\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Somme<\/h3>\n\n<div class=\"formula\">\nE(X+Y)=E(X)+E(Y)\n<\/div>\n\n<p>sans hypoth\u00e8se d&#8217;ind\u00e9pendance.<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Interpr\u00e9tation<\/h3>\n\n<p>\nSur un tr\u00e8s grand nombre d&#8217;exp\u00e9riences, la moyenne observ\u00e9e tend \u00e0 se rapprocher de l&#8217;esp\u00e9rance dans les cadres usuels.\n<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">9. VARIANCE &#038; \u00c9CART-TYPE<\/span>\n<h2>Mesurer la dispersion.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Variance<\/h3>\n\n<div class=\"formula\">\nVar(X)=E[(X-E(X))\u00b2]\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Formule de K\u00f6nig-Huygens<\/h3>\n\n<div class=\"formula\">\nVar(X)=E(X\u00b2)-E(X)\u00b2\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>\u00c9cart-type<\/h3>\n\n<div class=\"formula\">\n\u03c3(X)=\u221aVar(X)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Transformation affine<\/h3>\n\n<div class=\"formula\">\nVar(aX+b)=a\u00b2Var(X)\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">10. LOI DE BERNOULLI<\/span>\n<h2>Succ\u00e8s ou \u00e9chec.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>D\u00e9finition<\/h3>\n\n<p>\nX suit une loi de Bernoulli de param\u00e8tre p si :\n<\/p>\n\n<div class=\"formula\">\nP(X=1)=p<br>\nP(X=0)=1-p\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Esp\u00e9rance<\/h3>\n\n<div class=\"formula\">\nE(X)=p\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Variance<\/h3>\n\n<div class=\"formula\">\nVar(X)=p(1-p)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Interpr\u00e9tation<\/h3>\n\n<p>\n1 repr\u00e9sente g\u00e9n\u00e9ralement un succ\u00e8s, 0 un \u00e9chec.\n<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">11. LOI BINOMIALE<\/span>\n<h2>Compter les succ\u00e8s dans des r\u00e9p\u00e9titions ind\u00e9pendantes.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>\nSi X compte le nombre de succ\u00e8s obtenus au cours de n \u00e9preuves de Bernoulli ind\u00e9pendantes de m\u00eame param\u00e8tre p :\n<\/p>\n\n<div class=\"formula\">\nX ~ B(n,p)\n<\/div>\n\n<div class=\"formula\">\nP(X=k)=C(n,k)p\u1d4f(1-p)\u207f\u207b\u1d4f\n<\/div>\n\n<\/div>\n\n<div class=\"grid\" style=\"margin-top:16px\">\n\n<div class=\"card\">\n<h3>Esp\u00e9rance<\/h3>\n\n<div class=\"formula\">\nE(X)=np\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Variance<\/h3>\n\n<div class=\"formula\">\nVar(X)=np(1-p)\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"exercices\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">12. EXERCICES PROGRESSIFS<\/span>\n<h2>Du conditionnement aux variables al\u00e9atoires.<\/h2>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 1<\/span>\n\n<h3>Exercice 1 \u2014 Compl\u00e9mentaire<\/h3>\n\n<p>\nUne exp\u00e9rience donne P(A)=0,72. Calculer P(A\u0304).\n<\/p>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaProbaHint('proHint1')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"proHint1\" class=\"hint\">\nUtilisez P(A\u0304)=1-P(A).\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 2<\/span>\n\n<h3>Exercice 2 \u2014 Conditionnement<\/h3>\n\n<p>\nOn conna\u00eet :\n<\/p>\n\n<div class=\"formula\">\nP(A\u2229B)=0,18<br>\nP(B)=0,30\n<\/div>\n\n<p>D\u00e9terminer P(A|B).<\/p>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaProbaHint('proHint2')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"proHint2\" class=\"hint\">\nDivisez la probabilit\u00e9 de l&#8217;intersection par P(B).\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 3<\/span>\n\n<h3>Exercice 3 \u2014 Ind\u00e9pendance<\/h3>\n\n<p>\nOn donne P(A)=0,4, P(B)=0,5 et P(A\u2229B)=0,2.\n\u00c9tudier l&#8217;ind\u00e9pendance de A et B.\n<\/p>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaProbaHint('proHint3')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"proHint3\" class=\"hint\">\nComparez P(A\u2229B) et P(A)P(B).\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU PR\u00c9PA<\/span>\n\n<h3>Exercice 4 \u2014 Binomiale<\/h3>\n\n<p>\nUne pi\u00e8ce donne pile avec probabilit\u00e9 0,6.\nOn effectue 10 lancers ind\u00e9pendants.\nCalculer la probabilit\u00e9 d&#8217;obtenir exactement 7 piles.\n<\/p>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaProbaHint('proHint4')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"proHint4\" class=\"hint\">\nReconnaissez une loi binomiale de param\u00e8tres n=10 et p=0,6.\n<\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">13. PROBLEM LAB<\/span>\n<h2>Mission : diagnostic m\u00e9dical probabiliste.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<h3>Mission CPGE \u2014 Bayes<\/h3>\n\n<p>\nDans une population, une maladie touche 2% des individus.\nUn test est positif chez 95% des personnes malades et chez 4% des personnes non malades.\n<\/p>\n\n<p>Votre mission :<\/p>\n\n<ol>\n<li>d\u00e9finir les \u00e9v\u00e9nements M et T ;<\/li>\n<li>\u00e9crire P(M), P(T|M) et P(T|M\u0304) ;<\/li>\n<li>construire un arbre de probabilit\u00e9s ;<\/li>\n<li>calculer P(T) ;<\/li>\n<li>calculer P(M|T) par Bayes ;<\/li>\n<li>interpr\u00e9ter correctement le r\u00e9sultat ;<\/li>\n<li>expliquer pourquoi un test tr\u00e8s sensible peut n\u00e9anmoins produire des faux positifs significatifs dans une population o\u00f9 la maladie est rare.<\/li>\n<\/ol>\n\n<div class=\"actions\">\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaProbaHint('proMission1')\">\ud83d\udca1 INDICE<\/button>\n<button type=\"button\" class=\"btn\" style=\"background:#06172c;color:#fff\" onclick=\"mzaProbaHint('proMission2')\">\ud83e\udde0 M\u00c9THODE<\/button>\n<\/div>\n\n<div id=\"proMission1\" class=\"hint\">\nCommencez par calculer P(T) en additionnant les deux chemins menant \u00e0 un test positif.\n<\/div>\n\n<div id=\"proMission2\" class=\"hint\">\nUtilisez ensuite P(M|T)=P(M)P(T|M)\/P(T). Ne confondez pas P(T|M) avec P(M|T).\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section class=\"dark\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\" style=\"color:#efd68c\">MZA PROBABILITY LAB<\/span>\n<h2>Traduire avant de calculer.<\/h2>\n\n<p>\nUne grande partie des erreurs en probabilit\u00e9s vient d&#8217;une mauvaise d\u00e9finition des \u00e9v\u00e9nements ou d&#8217;une confusion entre P(A|B) et P(B|A).\n<\/p>\n<\/div>\n\n<div class=\"method\">\n<div><b>A<\/b><span>D\u00c9FINIR<\/span><\/div>\n<div><b>B<\/b><span>ARBRE<\/span><\/div>\n<div><b>C<\/b><span>INTERSECTION<\/span><\/div>\n<div><b>D<\/b><span>TOTAL<\/span><\/div>\n<div><b>E<\/b><span>BAYES<\/span><\/div>\n<div><b>F<\/b><span>INTERPR\u00c9TER<\/span><\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section id=\"qcm\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">14. QCM DE VALIDATION<\/span>\n<h2>Diagnostic Probabilit\u00e9s CPGE.<\/h2>\n\n<p>Le score est affich\u00e9 sans r\u00e9v\u00e9ler les r\u00e9ponses correctes.<\/p>\n<\/div>\n\n<div class=\"qcm\">\n\n<form id=\"probaQuiz\">\n\n<div class=\"question\">\n<strong>1. Si P(A)=0,35, alors P(A\u0304) vaut :<\/strong>\n<label><input type=\"radio\" name=\"p1\" value=\"1\"> 0,65<\/label>\n<label><input type=\"radio\" name=\"p1\" value=\"0\"> 0,35<\/label>\n<label><input type=\"radio\" name=\"p1\" value=\"0\"> 1,35<\/label>\n<label><input type=\"radio\" name=\"p1\" value=\"0\"> 0<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>2. P(A|B) est \u00e9gale \u00e0 :<\/strong>\n<label><input type=\"radio\" name=\"p2\" value=\"1\"> P(A\u2229B)\/P(B)<\/label>\n<label><input type=\"radio\" name=\"p2\" value=\"0\"> P(A)+P(B)<\/label>\n<label><input type=\"radio\" name=\"p2\" value=\"0\"> P(A)\/P(A\u2229B)<\/label>\n<label><input type=\"radio\" name=\"p2\" value=\"0\"> 1-P(B)<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>3. A et B sont ind\u00e9pendants lorsque :<\/strong>\n<label><input type=\"radio\" name=\"p3\" value=\"1\"> P(A\u2229B)=P(A)P(B)<\/label>\n<label><input type=\"radio\" name=\"p3\" value=\"0\"> A\u2229B=\u2205 toujours<\/label>\n<label><input type=\"radio\" name=\"p3\" value=\"0\"> P(A)=P(B)<\/label>\n<label><input type=\"radio\" name=\"p3\" value=\"0\"> P(A\u222aB)=0<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>4. L&#8217;esp\u00e9rance d&#8217;une variable discr\u00e8te X est :<\/strong>\n<label><input type=\"radio\" name=\"p4\" value=\"1\"> \u03a3 x\u1d62P(X=x\u1d62)<\/label>\n<label><input type=\"radio\" name=\"p4\" value=\"0\"> \u03a3 P(X=x\u1d62)\u00b2<\/label>\n<label><input type=\"radio\" name=\"p4\" value=\"0\"> toujours \u00e9gale \u00e0 1<\/label>\n<label><input type=\"radio\" name=\"p4\" value=\"0\"> toujours \u00e9gale \u00e0 0<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>5. La variance peut se calculer par :<\/strong>\n<label><input type=\"radio\" name=\"p5\" value=\"1\"> E(X\u00b2)-E(X)\u00b2<\/label>\n<label><input type=\"radio\" name=\"p5\" value=\"0\"> E(X\u00b2)+E(X)\u00b2<\/label>\n<label><input type=\"radio\" name=\"p5\" value=\"0\"> E(X)<\/label>\n<label><input type=\"radio\" name=\"p5\" value=\"0\"> 1\/E(X)<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>6. Si X suit B(n,p), son esp\u00e9rance vaut :<\/strong>\n<label><input type=\"radio\" name=\"p6\" value=\"1\"> np<\/label>\n<label><input type=\"radio\" name=\"p6\" value=\"0\"> n+p<\/label>\n<label><input type=\"radio\" name=\"p6\" value=\"0\"> p\/n<\/label>\n<label><input type=\"radio\" name=\"p6\" value=\"0\"> n(1-p)<\/label>\n<\/div>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaProbaScore()\">\nVALIDER MON QCM\n<\/button>\n\n<\/form>\n\n<div id=\"probaResult\"><\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"cta\">\n\n<div style=\"font-size:43px\">P(A|B)<\/div>\n\n<h2>En probabilit\u00e9s, l&#8217;information change le calcul.<\/h2>\n\n<p>\nConditionnement, ind\u00e9pendance, Bayes, esp\u00e9rance et variance constituent\nla bo\u00eete \u00e0 outils fondamentale pour mod\u00e9liser l&#8217;incertitude et r\u00e9soudre\nles probl\u00e8mes probabilistes de niveau CPGE.\n<\/p>\n\n<div class=\"actions\" style=\"justify-content:center\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda REVOIR LE COURS<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 REFAIRE LE QCM<\/a>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<script>\nfunction mzaProbaHint(id){\n const el=document.getElementById(id);\n if(!el)return;\n el.style.display = el.style.display===\"block\" ? \"none\" : \"block\";\n}\n\nfunction mzaProbaScore(){\n const form=document.getElementById(\"probaQuiz\");\n const result=document.getElementById(\"probaResult\");\n\n let score=0;\n let complete=true;\n\n [\"p1\",\"p2\",\"p3\",\"p4\",\"p5\",\"p6\"].forEach(function(name){\n   const answer=form.querySelector('input[name=\"'+name+'\"]:checked');\n\n   if(!answer){\n     complete=false;\n   }else{\n     score+=Number(answer.value);\n   }\n });\n\n result.style.display=\"block\";\n\n if(!complete){\n   result.innerHTML=\"<strong>R\u00e9pondez aux 6 questions avant de valider.<\/strong>\";\n   return;\n }\n\n const pct=Math.round(score\/6*100);\n\n let level=\"Probabilit\u00e9s \u00e0 consolider\";\n if(pct>=50) level=\"Bases acquises\";\n if(pct>=67) level=\"Bon niveau\";\n if(pct>=84) level=\"Tr\u00e8s bonne ma\u00eetrise\";\n if(pct===100) level=\"Excellent niveau CPGE\";\n\n result.innerHTML=\n \"<strong style='font-size:30px;color:#efd68c'>\"+pct+\"%<\/strong>\"+\n \"<p><b>\"+level+\"<\/b><\/p>\"+\n \"<p>Score : \"+score+\" \/ 6<\/p>\"+\n \"<p>Les r\u00e9ponses correctes et la correction d\u00e9taill\u00e9e restent masqu\u00e9es.<\/p>\";\n}\n<\/script>\n\n<\/div>\n&#8220;`\n","protected":false},"excerpt":{"rendered":"<p>&#8220;`html id=&#8221;mza-probabilites-17&#8243; COURS 17 \u2022 PROBABILIT\u00c9S Mod\u00e9liser.Conditionner. Pr\u00e9voir. Les probabilit\u00e9s permettent de mesurer l&#8217;incertitude, raisonner avec des \u00e9v\u00e9nements, mettre \u00e0 jour une information gr\u00e2ce au conditionnement et mod\u00e9liser des exp\u00e9riences al\u00e9atoires \u00e0 l&#8217;aide de variables al\u00e9atoires. \ud83d\udcda COMMENCER \ud83e\udde0 M\u00c9THODE \ud83e\udde9 EXERCICES \u2753 QCM 1. UNIVERS &#038; \u00c9V\u00c9NEMENTS D\u00e9crire une exp\u00e9rience al\u00e9atoire. Univers \u03a9 \u03a9 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1563","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1563","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/comments?post=1563"}],"version-history":[{"count":1,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1563\/revisions"}],"predecessor-version":[{"id":1565,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1563\/revisions\/1565"}],"wp:attachment":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/media?parent=1563"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}