{"id":100,"date":"2026-10-03T06:05:25","date_gmt":"2026-10-03T03:05:25","guid":{"rendered":"https:\/\/wpcalc.com\/fr\/?p=100"},"modified":"2026-10-03T06:05:25","modified_gmt":"2026-10-03T03:05:25","slug":"calculateur-loi-binomiale","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/fr\/statistiques\/calculateur-loi-binomiale\/","title":{"rendered":"Calculateur de loi binomiale"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Calculer des probabilit\u00e9s binomiales<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"100\" data-favorite-title=\"Loi binomiale\" data-favorite-url=\"https:\/\/wpcalc.com\/fr\/statistiques\/calculateur-loi-binomiale\/\" data-favorite-excerpt=\"Calculez les probabilit\u00e9s binomiales\u00a0: P(X = k), la probabilit\u00e9 cumul\u00e9e P(X \u2264 k), P(X \u2265 k) ou la probabilit\u00e9 d\u2019un intervalle de nombres de succ\u00e8s. Saisissez le nombre d\u2019essais, la probabilit\u00e9 de succ\u00e8s et k pour obtenir \u00e9galement l\u2019esp\u00e9rance, la variance et l\u2019\u00e9cart-type.\" aria-label=\"Ajouter aux favoris\" data-favorite-icon=\"icon icon-wpcalc-distribution-histogram\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/fr\/statistiques\/calculateur-loi-binomiale\/\" method=\"POST\" class=\"calculator\" id=\"calculator-100\" data-post=\"100\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Champs de saisie<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Nombre d\u2019essais (n)<\/label><div class=\"term\">n<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"nombre_dessais_n\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 10\" min=\"1\" max=\"10000\" value=\"10\"\/><small id=\"calc-info-1\">Nombre d\u2019essais de Bernoulli ind\u00e9pendants. Nombre entier de 1 \u00e0 10 000.<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Probabilit\u00e9 de succ\u00e8s (p)<\/label><div class=\"term\">p<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"probabilit_de_succs_p\" id=\"field-2\" aria-describedby=\"calc-unit-2 calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 50\" min=\"0\" max=\"100\" value=\"50\"\/><div class=\"unit\" id=\"calc-unit-2\">%<\/div><small id=\"calc-info-2\">Probabilit\u00e9 de succ\u00e8s lors d\u2019un essai, de 0 % \u00e0 100 %.<\/small><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">Nombre de succ\u00e8s \u00e9tudi\u00e9s<\/label><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"nombre_de_succs_tudis\"><option value=\"exact\" selected>Exactement k succ\u00e8s<\/option><option value=\"range\">Entre k\u2081 et k\u2082 succ\u00e8s<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">Nombre de succ\u00e8s (k)<\/label><div class=\"term\">k<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"nombre_de_succs_k\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 5\" min=\"0\" value=\"5\"\/><small id=\"calc-info-4\">Nombre entier de 0 \u00e0 n. En mode intervalle, il s\u2019agit de la borne inf\u00e9rieure k\u2081.<\/small><\/div><div class=\"field has-term\" id=\"input-5\" hidden><label for=\"field-5\">Borne sup\u00e9rieure (k\u2082)<\/label><div class=\"term\">k\u2082<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"borne_suprieure_k\" id=\"field-5\" aria-describedby=\"calc-info-5\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 7\" min=\"0\" value=\"7\"\/><small id=\"calc-info-5\">Borne sup\u00e9rieure de l\u2019intervalle. Elle doit \u00eatre au moins \u00e9gale \u00e0 k\u2081.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalculer\" id=\"calculate-button\" data-post=\"100\">Calculer<\/button><button type=\"reset\">R\u00e9initialiser<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculer automatiquement<\/label><small>Si activ\u00e9, le r\u00e9sultat se met \u00e0 jour automatiquement d\u00e8s que vous modifiez une valeur.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Vos r\u00e9sultats<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>P(X = k)<\/span><span class=\"result-term\">pmf<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>P(X \u2264 k)<\/span><span class=\"result-term\">cdf<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>P(X \u2265 k)<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>P(k\u2081 \u2264 X \u2264 k\u2082)<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>Esp\u00e9rance<\/span><span class=\"result-term\">\u03bc<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-6\"><span class=\"field-title\"><span>Variance<\/span><span class=\"result-term\">\u03c3\u00b2<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-7\"><span class=\"field-title\"><span>\u00c9cart-type<\/span><span class=\"result-term\">\u03c3<\/span><\/span><span class=\"field-value\" id=\"result-7\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-8\"><span class=\"field-title\"><span>Interpr\u00e9tation et d\u00e9tails<\/span><\/span><span class=\"field-value\" id=\"result-8\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copier le lien vers ce calcul<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Laisser un commentaire<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"100\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Que pensez-vous de ce r\u00e9sultat\u00a0?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">J\u2019aime <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Utile <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confus <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">D\u00e9\u00e7u <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inexact <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Formule de la loi binomiale<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formule<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ P(X = k) = \\binom{n}{k} p^k (1-p)^{n-k}, \\qquad \\binom{n}{k} = \\frac{n!}{k!\\,(n-k)!} \\\\ P(X \\le k) = \\sum_{i=0}^{k} \\binom{n}{i} p^i (1-p)^{n-i}, \\qquad P(X \\ge k) = 1 - P(X \\le k-1) \\\\ \\mu = np, \\qquad \\sigma^2 = np(1-p), \\qquad \\sigma = \\sqrt{np(1-p)} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Avec&nbsp;:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 nombre d\u2019essais ind\u00e9pendants<\/li>\n<li>\\(p\\) \u2014 probabilit\u00e9 de succ\u00e8s lors d\u2019un essai (entre 0 et 1)<\/li>\n<li>\\(k\\) \u2014 nombre de succ\u00e8s consid\u00e9r\u00e9 (\\(0 \\le k \\le n\\))<\/li>\n<li>\\(\\binom{n}{k}\\) \u2014 coefficient binomial, soit le nombre de fa\u00e7ons de choisir \\(k\\) succ\u00e8s parmi \\(n\\) essais<\/li>\n<li>\\(\\mu\\), \\(\\sigma^2\\), \\(\\sigma\\) \u2014 esp\u00e9rance, variance et \u00e9cart-type de la loi<\/li>\n<\/ul>\n<p>La premi\u00e8re formule donne la fonction de masse : la probabilit\u00e9 d\u2019obtenir exactement \\(k\\) succ\u00e8s en \\(n\\) essais. La deuxi\u00e8me donne la fonction de r\u00e9partition : la probabilit\u00e9 d\u2019obtenir au plus \\(k\\) succ\u00e8s, calcul\u00e9e en additionnant les probabilit\u00e9s de 0 \u00e0 \\(k\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Loi binomiale : exemple de calcul<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemple&nbsp;:<\/strong> on lance 10 fois une pi\u00e8ce \u00e9quilibr\u00e9e. Quelle est la probabilit\u00e9 d\u2019obtenir exactement 5 fois pile&nbsp;? Ici, \\(n = 10\\), \\(p = 0.5\\), \\(k = 5\\).<\/p>\n<p>\\[ P(X = 5) = \\binom{10}{5} (0.5)^5 (0.5)^{10-5} = 252 \\cdot \\frac{1}{1024} = 0.246094 \\approx 24.61\\% \\]<\/p>\n<p>\\[ P(X \\le 5) = \\sum_{i=0}^{5} \\binom{10}{i}(0.5)^{10} = \\frac{638}{1024} = 0.623047 \\approx 62.30\\% \\]<\/p>\n<p>\\[ \\mu = 10 \\cdot 0.5 = 5, \\qquad \\sigma^2 = 10 \\cdot 0.5 \\cdot 0.5 = 2.5, \\qquad \\sigma = \\sqrt{2.5} \\approx 1.5811 \\]<\/p>\n<p><strong>R\u00e9sultat&nbsp;:<\/strong> P(X = 5) = 0,246094 (24,61&nbsp;%), P(X \u2264 5) = 0,623047 (62,30&nbsp;%), P(X \u2265 5) = 0,623047 (62,30&nbsp;%), esp\u00e9rance = 5, variance = 2,5, \u00e9cart-type \u2248 1,5811.<\/p>\n<p>\n\r\n<\/section>\n<h2>Comment utiliser le calculateur<\/h2>\n<ol>\n<li>Saisissez le <strong>Nombre d\u2019essais (n)<\/strong> : un nombre entier de 1 \u00e0 10&nbsp;000.<\/li>\n<li>Saisissez la <strong>Probabilit\u00e9 de succ\u00e8s (p)<\/strong> sous forme de pourcentage entre 0 % et 100 % (par exemple, saisissez 50 pour une pi\u00e8ce \u00e9quilibr\u00e9e).<\/li>\n<li>Choisissez une option sous <strong>Nombre de succ\u00e8s consid\u00e9r\u00e9<\/strong> : \u00ab&nbsp;Exactement k succ\u00e8s&nbsp;\u00bb pour une seule valeur ou \u00ab&nbsp;Entre k\u2081 et k\u2082 succ\u00e8s&nbsp;\u00bb pour un intervalle.<\/li>\n<li>Saisissez le <strong>Nombre de succ\u00e8s (k)<\/strong>. En mode intervalle, il s\u2019agit de la borne inf\u00e9rieure k\u2081.<\/li>\n<li>Si vous avez choisi le mode intervalle, saisissez \u00e9galement la <strong>Borne sup\u00e9rieure (k\u2082)<\/strong>, qui doit \u00eatre au moins \u00e9gale \u00e0 k\u2081.<\/li>\n<li>Consultez les r\u00e9sultats&nbsp;: P(X = k), P(X \u2264 k), P(X \u2265 k) et, en mode intervalle, P(k\u2081 \u2264 X \u2264 k\u2082), ainsi que l\u2019esp\u00e9rance, la variance et l\u2019\u00e9cart-type. Le bloc d\u2019interpr\u00e9tation indique aussi des probabilit\u00e9s compl\u00e9mentaires, comme P(X &lt; k) et P(X &gt; k).<\/li>\n<\/ol>\n<h2>Qu\u2019est-ce que la loi binomiale ?<\/h2>\n<p>La loi binomiale mod\u00e9lise le nombre de succ\u00e8s dans un nombre fixe d\u2019essais ind\u00e9pendants, chacun ayant la m\u00eame probabilit\u00e9 de succ\u00e8s \\(p\\). Chaque essai a exactement deux issues \u2014 succ\u00e8s ou \u00e9chec \u2014&nbsp;: on parle donc d\u2019\u00e9preuve de Bernoulli. Parmi les exemples classiques figurent le nombre de fois o\u00f9 l\u2019on obtient pile lors d\u2019une s\u00e9rie de lancers de pi\u00e8ce, le nombre d\u2019articles d\u00e9fectueux dans un \u00e9chantillon ou le nombre de patients qui r\u00e9pondent \u00e0 un traitement.<\/p>\n<p>L\u2019esp\u00e9rance de la loi est \\(\\mu = np\\) : en moyenne, on s\u2019attend \u00e0 \\(n \\cdot p\\) succ\u00e8s. La dispersion est mesur\u00e9e par la variance \\(\\sigma^2 = np(1-p)\\) et l\u2019\u00e9cart-type \\(\\sigma\\). Lorsque \\(p = 0.5\\), la loi est sym\u00e9trique autour de \\(n\/2\\) ; lorsque \\(p\\) est proche de 0 ou de 1, elle devient asym\u00e9trique, avec des valeurs concentr\u00e9es pr\u00e8s de 0 ou de \\(n\\).<\/p>\n<p>Le calculateur distingue les probabilit\u00e9s exactes des probabilit\u00e9s cumul\u00e9es. La fonction de masse r\u00e9pond \u00e0 la question \u00ab&nbsp;quelle est la probabilit\u00e9 d\u2019obtenir exactement k succ\u00e8s&nbsp;?&nbsp;\u00bb, tandis que la fonction de r\u00e9partition r\u00e9pond \u00e0 \u00ab&nbsp;quelle est la probabilit\u00e9 d\u2019obtenir au plus k succ\u00e8s&nbsp;?&nbsp;\u00bb. Les probabilit\u00e9s cumul\u00e9es sont souvent plus utiles en pratique : par exemple, si k est un seuil de qualit\u00e9, P(X \u2264 k) donne la probabilit\u00e9 de rester dans la limite, et 1 \u2212 P(X \u2264 k) donne la probabilit\u00e9 de la d\u00e9passer. Notez que P(X \u2265 k) inclut k&nbsp;; si vous cherchez strictement plus de k succ\u00e8s, utilisez P(X &gt; k) = 1 \u2212 P(X \u2264 k), indiqu\u00e9 dans le bloc d\u2019interpr\u00e9tation.<\/p>\n<h2>Conditions d\u2019application, limites et conseils<\/h2>\n<ul>\n<li><strong>Nombre fixe d\u2019essais.<\/strong> \\(n\\) doit \u00eatre d\u00e9fini \u00e0 l\u2019avance. Si le nombre d\u2019essais est lui-m\u00eame al\u00e9atoire, un autre mod\u00e8le (par exemple la loi de Poisson) est plus adapt\u00e9.<\/li>\n<li><strong>Ind\u00e9pendance.<\/strong> Le r\u00e9sultat d\u2019un essai ne doit pas influencer celui d\u2019un autre. Un \u00e9chantillonnage sans remise dans une petite population ne respecte pas cette condition ; utilisez plut\u00f4t la loi hyperg\u00e9om\u00e9trique.<\/li>\n<li><strong>Probabilit\u00e9 de succ\u00e8s constante.<\/strong> \\(p\\) doit \u00eatre identique \u00e0 chaque essai. Si elle varie d\u2019un essai \u00e0 l\u2019autre, le mod\u00e8le binomial ne s\u2019applique plus.<\/li>\n<li><strong>Grand n.<\/strong> Le calculateur utilise une m\u00e9thode logarithmique stable fond\u00e9e sur la fonction gamma : il traite donc les grandes valeurs de \\(n\\) (jusqu\u2019\u00e0 10&nbsp;000) sans d\u00e9passement de capacit\u00e9. Les tr\u00e8s petites probabilit\u00e9s sont affich\u00e9es en notation scientifique.<\/li>\n<li><strong>Interpr\u00e9tation des tr\u00e8s petites probabilit\u00e9s.<\/strong> Un r\u00e9sultat comme \u00ab&nbsp;\u2248&nbsp;0&nbsp;%&nbsp;\u00bb ne signifie pas que l\u2019issue est impossible : la probabilit\u00e9 peut simplement \u00eatre inf\u00e9rieure \u00e0 environ \\(10^{-8}\\). V\u00e9rifiez la valeur d\u00e9cimale affich\u00e9e \u00e0 c\u00f4t\u00e9 du pourcentage.<\/li>\n<li><strong>R\u00e8gle pratique pour l\u2019approximation.<\/strong> Pour un grand \\(n\\), lorsque \\(np\\) et \\(n(1-p)\\) sont tous deux sup\u00e9rieurs \u00e0 environ 5, la loi binomiale est bien approch\u00e9e par une loi normale d\u2019esp\u00e9rance \\(np\\) et d\u2019\u00e9cart-type \\(\\sqrt{np(1-p)}\\), ce qui est pratique pour faire des estimations rapides de t\u00eate.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calculez les probabilit\u00e9s binomiales\u00a0: P(X = k), la probabilit\u00e9 cumul\u00e9e P(X \u2264 k), P(X \u2265 k) ou la probabilit\u00e9 d\u2019un intervalle de nombres de succ\u00e8s. Saisissez le nombre d\u2019essais, la probabilit\u00e9 de succ\u00e8s et k pour obtenir \u00e9galement l\u2019esp\u00e9rance, la variance et l\u2019\u00e9cart-type.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[28],"tags":[56],"class_list":["post-100","post","type-post","status-publish","format-standard","hentry","category-statistiques","tag-distributions"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/100","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/comments?post=100"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/100\/revisions"}],"predecessor-version":[{"id":5948,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/100\/revisions\/5948"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/media?parent=100"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/categories?post=100"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/tags?post=100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}