Quiz — CoursProb1.pdf

Question 1 sur 10 20:00

[{"id":151998,"question":"Quelle est la formule de l'espérance d'une variable aléatoire X suivant une loi binomiale B(n, p) ?","option_a":"E(X) = n × p","option_b":"E(X) = p × (1-p)","option_c":"E(X) = n × (1-p)","option_d":"E(X) = p \/ n","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi binomiale B(n, p) est E(X) = n × p, où n est le nombre d'essais et p la probabilité de succès.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"E(X) = n × p\", \"b\": \"E(X) = p × (1-p)\", \"c\": \"E(X) = n × (1-p)","_debug_options_count":4},{"id":151999,"question":"La variance d'une variable aléatoire X est toujours positive.","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 et est toujours positive ou nulle.","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":152000,"question":"Dans une loi uniforme continue sur [a, b], quelle est la probabilité P(X ≤ c) pour c ∈ [a, b] ?","option_a":"P(X ≤ c) = (c - a) \/ (b - a)","option_b":"P(X ≤ c) = c \/ b","option_c":"P(X ≤ c) = (b - c) \/ (b - a)","option_d":"P(X ≤ c) = 1 \/ (b - a)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi uniforme continue sur [a, b], P(X ≤ c) = (c - a) \/ (b - a).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"P(X ≤ c) = (c - a) \/ (b - a)\", \"b\": \"P(X ≤ c) = c \/ b\", \"c\": ","_debug_options_count":4},{"id":152001,"question":"Quelle est la probabilité d'obtenir exactement 3 succès en 5 essais avec une probabilité de succès de 0,4 ?","option_a":"0,2304","option_b":"0,3456","option_c":"0,1234","option_d":"0,4567","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité est donnée par la formule de la loi binomiale : C(5,3) × (0,4)^3 × (0,6)^2 = 0,3456.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0,2304\", \"b\": \"0,3456\", \"c\": \"0,1234\", \"d\": \"0,4567\"}}","_debug_options_count":4},{"id":152002,"question":"La loi normale centrée réduite a une espérance égale à 0 et un écart-type égal à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La loi normale centrée réduite N(0,1) a une espérance de 0 et un écart-type de 1.","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":152003,"question":"Quelle est la variance d'une variable aléatoire X suivant une loi uniforme discrète sur {1, 2, 3, 4, 5} ?","option_a":"2","option_b":"3","option_c":"4","option_d":"5","option_e":"","option_f":"","bonne_reponse":"b","explication":"La variance est calculée par Var(X) = (n² - 1)\/12, où n est le nombre de valeurs. Ici, Var(X) = (25 - 1)\/12 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"3\", \"c\": \"4\", \"d\": \"5\"}}","_debug_options_count":4},{"id":152004,"question":"Dans une expérience aléatoire, si P(A) = 0,3 et P(B) = 0,5, quelle est la probabilité P(A ∪ B) si A et B sont incompatibles ?","option_a":"0,8","option_b":"0,2","option_c":"0,15","option_d":"0,65","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si A et B sont incompatibles, P(A ∪ B) = P(A) + P(B) = 0,3 + 0,5 = 0,8.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0,8\", \"b\": \"0,2\", \"c\": \"0,15\", \"d\": \"0,65\"}}","_debug_options_count":4},{"id":152005,"question":"La fonction de répartition d'une variable aléatoire X est toujours croissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction de répartition est toujours croissante car elle représente la probabilité cumulée.","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":152006,"question":"Quelle est la probabilité d'obtenir au moins un succès en 4 essais avec une probabilité de succès de 0,2 ?","option_a":"0,5904","option_b":"0,4096","option_c":"0,8","option_d":"0,2","option_e":"","option_f":"","bonne_reponse":"a","explication":"P(au moins un succès) = 1 - P(aucun succès) = 1 - (0,8)^4 = 0,5904.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0,5904\", \"b\": \"0,4096\", \"c\": \"0,8\", \"d\": \"0,2\"}}","_debug_options_count":4},{"id":152007,"question":"L'écart-type d'une variable aléatoire est toujours inférieur ou égal à sa variance.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'écart-type est la racine carrée de la variance. Il peut être supérieur ou inférieur selon la valeur de la variance.","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}]