Quiz — probabilité2015.pdf

Question 1 sur 10 20:00

[{"id":36220,"question":"Soit X une variable aléatoire suivant une loi binomiale B(n=5, p=0.3). Quelle est P(X=2) ?","option_a":"C(5,2) × 0.3² × 0.7³","option_b":"C(5,2) × 0.3³ × 0.7²","option_c":"5 × 0.3 × 0.7","option_d":"0.3²","option_e":"","option_f":"","bonne_reponse":"a","explication":"La probabilité P(X=k) pour une loi binomiale est donnée par C(n,k) × p^k × (1-p)^(n-k). Ici, k=2, n=5, p=0.3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"C(5,2) × 0.3² × 0.7³\", \"b\": \"C(5,2) × 0.3³ × 0.7²\", \"c\": ","_debug_options_count":4},{"id":36221,"question":"L'espérance d'une variable aléatoire suivant une loi de Bernoulli de paramètre p est toujours égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'espérance d'une loi de Bernoulli est p, pas nécessairement 1. Par exemple, si p=0.5, l'espérance vaut 0.5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":36222,"question":"Quelle formule donne la variance d'une variable aléatoire X suivant une loi binomiale B(n,p) ?","option_a":"n × p","option_b":"n × p × (1-p)","option_c":"n² × p × (1-p)","option_d":"p × (1-p)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La variance d'une loi binomiale B(n,p) est n × p × (1-p).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"n × p\", \"b\": \"n × p × (1-p)\", \"c\": \"n² × p × (1-p)\", \"d\": \"","_debug_options_count":4},{"id":36223,"question":"Si X suit une loi binomiale B(10, 0.4), quelle est l'espérance de X ?","option_a":"4","option_b":"6","option_c":"0.4","option_d":"40","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi binomiale B(n,p) est n × p = 10 × 0.4 = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"4\", \"b\": \"6\", \"c\": \"0.4\", \"d\": \"40\"}}","_debug_options_count":4},{"id":36224,"question":"La loi binomiale peut modéliser le nombre de succès dans une série d'épreuves indépendantes à deux issues.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la définition même de la loi binomiale : elle modélise le nombre de succès dans n épreuves indépendantes avec deux issues possibles.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":36225,"question":"Quelle est la probabilité d'obtenir au moins un succès en 3 lancers d'une pièce équilibrée ?","option_a":"0.125","option_b":"0.5","option_c":"0.875","option_d":"0.375","option_e":"","option_f":"","bonne_reponse":"c","explication":"P(au moins un succès) = 1 - P(aucun succès) = 1 - (0.5)^3 = 1 - 0.125 = 0.875.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0.125\", \"b\": \"0.5\", \"c\": \"0.875\", \"d\": \"0.375\"}}","_debug_options_count":4},{"id":36226,"question":"Si X suit une loi binomiale B(n,p), alors Var(X) = E(X²) - [E(X)]².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété générale de la variance : Var(X) = E(X²) - [E(X)]², valable pour toute variable aléatoire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":36227,"question":"Une variable aléatoire X prend les valeurs 0, 1 et 2 avec les probabilités respectives 0.2, 0.5 et 0.3. Quelle est E(X) ?","option_a":"0.8","option_b":"1.1","option_c":"1.5","option_d":"0.5","option_e":"","option_f":"","bonne_reponse":"b","explication":"E(X) = 0×0.2 + 1×0.5 + 2×0.3 = 0 + 0.5 + 0.6 = 1.1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0.8\", \"b\": \"1.1\", \"c\": \"1.5\", \"d\": \"0.5\"}}","_debug_options_count":4},{"id":36228,"question":"La variance d'une variable aléatoire est toujours positive ou nulle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La variance mesure la dispersion des valeurs autour de l'espérance. Elle est toujours positive ou nulle (cas où X est constante).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":36229,"question":"Dans une loi binomiale B(7, 0.2), quelle est la probabilité d'obtenir exactement 3 succès ?","option_a":"C(7,3) × 0.2³ × 0.8⁴","option_b":"C(7,3) × 0.2⁴ × 0.8³","option_c":"7 × 0.2 × 0.8","option_d":"0.2³","option_e":"","option_f":"","bonne_reponse":"a","explication":"P(X=3) = C(7,3) × (0.2)^3 × (0.8)^(7-3) = C(7,3) × 0.2³ × 0.8⁴.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"C(7,3) × 0.2³ × 0.8⁴\", \"b\": \"C(7,3) × 0.2⁴ × 0.8³\", \"c\"","_debug_options_count":4}]