Quiz — Magazine n°1 variables aléatoires.pdf

Question 1 sur 10 20:00

[{"id":45363,"question":"Quelle est la probabilité d'obtenir un nombre pair en lançant un dé équilibré à 6 faces ?","option_a":"1\/6","option_b":"1\/3","option_c":"1\/2","option_d":"2\/3","option_e":"","option_f":"","bonne_reponse":"c","explication":"Il y a 3 nombres pairs (2, 4, 6) sur 6 faces possibles. La probabilité est donc 3\/6 = 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1\/6\", \"b\": \"1\/3\", \"c\": \"1\/2\", \"d\": \"2\/3\"}}","_debug_options_count":4},{"id":45364,"question":"Si X est une variable aléatoire discrète prenant les valeurs 1, 2 et 3 avec les probabilités respectives 0.2, 0.5 et 0.3, alors P(X ≤ 2) = ?","option_a":"0.2","option_b":"0.5","option_c":"0.7","option_d":"0.8","option_e":"","option_f":"","bonne_reponse":"d","explication":"P(X ≤ 2) = P(X=1) + P(X=2) = 0.2 + 0.5 = 0.7.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"0.2\", \"b\": \"0.5\", \"c\": \"0.7\", \"d\": \"0.8\"}}","_debug_options_count":4},{"id":45365,"question":"L'espérance d'une variable aléatoire est toujours un entier.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'espérance peut être un nombre décimal, comme 2.5, selon les valeurs et probabilités de la variable.","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":45366,"question":"Quelle loi de probabilité modélise le nombre de succès dans une série d'essais indépendants à deux issues (succès\/échec) ?","option_a":"Loi normale","option_b":"Loi binomiale","option_c":"Loi uniforme","option_d":"Loi exponentielle","option_e":"","option_f":"","bonne_reponse":"b","explication":"La loi binomiale modélise le nombre de succès dans n essais indépendants avec une probabilité p de succès à chaque essai.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Loi normale\", \"b\": \"Loi binomiale\", \"c\": \"Loi uniforme\", \"d\": \"Lo","_debug_options_count":4},{"id":45367,"question":"Si X suit une loi binomiale B(5, 0.4), alors P(X=3) = ?","option_a":"0.2304","option_b":"0.3456","option_c":"0.4000","option_d":"0.6000","option_e":"","option_f":"","bonne_reponse":"b","explication":"P(X=3) = C(5,3) * (0.4)^3 * (0.6)^2 = 10 * 0.064 * 0.36 = 0.2304.","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\": \"0.4000\", \"d\": \"0.6000\"}}","_debug_options_count":4},{"id":45368,"question":"La variance d'une variable aléatoire est toujours positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La variance est toujours positive ou nulle, car elle est définie comme l'espérance du carré des écarts à la moyenne.","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":45369,"question":"Quelle est l'espérance d'une variable aléatoire X prenant les valeurs 0, 1 et 2 avec les probabilités 0.1, 0.3 et 0.6 ?","option_a":"1.5","option_b":"1.6","option_c":"1.7","option_d":"1.8","option_e":"","option_f":"","bonne_reponse":"b","explication":"E(X) = 0*0.1 + 1*0.3 + 2*0.6 = 0 + 0.3 + 1.2 = 1.5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"1.5\", \"b\": \"1.6\", \"c\": \"1.7\", \"d\": \"1.8\"}}","_debug_options_count":4},{"id":45370,"question":"Si X suit une loi uniforme sur [0, 1], alors P(X ≤ 0.5) = ?","option_a":"0.25","option_b":"0.5","option_c":"0.75","option_d":"1","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi uniforme sur [0, 1], la probabilité est proportionnelle à la longueur de l'intervalle. P(X ≤ 0.5) = 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\": \"0.25\", \"b\": \"0.5\", \"c\": \"0.75\", \"d\": \"1\"}}","_debug_options_count":4},{"id":45371,"question":"La loi normale est utilisée pour modéliser des variables aléatoires discrètes.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La loi normale modélise des variables aléatoires continues, tandis que les variables discrètes sont modélisées par des lois comme la loi binomiale ou de Poisson.","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":45372,"question":"Quelle est la probabilité qu'une variable aléatoire X suivant une loi normale centrée réduite prenne une valeur inférieure à -1.96 ?","option_a":"0.025","option_b":"0.05","option_c":"0.95","option_d":"0.975","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi normale centrée réduite, P(X \u003C -1.96) ≈ 0.025 (valeur issue de la table de la loi normale).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0.025\", \"b\": \"0.05\", \"c\": \"0.95\", \"d\": \"0.975\"}}","_debug_options_count":4}]