Quiz — Serie 51- Probabilités.pdf

Question 1 sur 10 20:00

[{"id":30910,"question":"Quelle est la formule de la probabilité d'une variable aléatoire suivant une loi binomiale B(n, p) ?","option_a":"P(X=k) = C(n,k) * p^k * (1-p)^(n-k)","option_b":"P(X=k) = p^k * (1-p)^(n-k)","option_c":"P(X=k) = n! \/ (k!(n-k)!) * p * (1-p)","option_d":"P(X=k) = (p + (1-p))^n","option_e":"","option_f":"","bonne_reponse":"a","explication":"La loi binomiale B(n, p) modélise le nombre de succès dans n épreuves indépendantes de probabilité p. La formule correcte est P(X=k) = C(n,k) * p^k * (1-p)^(n-k).","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=k) = C(n,k) * p^k * (1-p)^(n-k)\", \"b\": \"P(X=k) = p^k * (1-p)^","_debug_options_count":4},{"id":30911,"question":"L'espérance d'une variable aléatoire X est toujours égale à sa variance.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'espérance et la variance sont deux mesures différentes : l'espérance est la moyenne pondérée des valeurs, tandis que la variance mesure la dispersion autour de cette moyenne.","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":30912,"question":"Si deux événements A et B sont indépendants, alors P(A ∩ B) est égale à :","option_a":"P(A) + P(B)","option_b":"P(A) * P(B)","option_c":"P(A) \/ P(B)","option_d":"P(A) - P(B)","option_e":"","option_f":"","bonne_reponse":"b","explication":"Par définition, deux événements A et B sont indépendants si et seulement si P(A ∩ B) = P(A) * P(B).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"P(A) + P(B)\", \"b\": \"P(A) * P(B)\", \"c\": \"P(A) \/ P(B)\", \"d\": \"P(A) ","_debug_options_count":4},{"id":30913,"question":"Quelle est la probabilité d'obtenir un 6 en lançant un dé équilibré une fois ?","option_a":"1\/2","option_b":"1\/3","option_c":"1\/6","option_d":"1\/12","option_e":"","option_f":"","bonne_reponse":"c","explication":"Un dé équilibré a 6 faces numérotées de 1 à 6. La probabilité d'obtenir un 6 est donc 1\/6.","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\/2\", \"b\": \"1\/3\", \"c\": \"1\/6\", \"d\": \"1\/12\"}}","_debug_options_count":4},{"id":30914,"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 est une mesure de dispersion et est toujours positive ou nulle, car elle est calculée à partir de carrés.","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":30915,"question":"Si X suit une loi uniforme sur [a, b], alors son espérance est égale à :","option_a":"(a + b)\/2","option_b":"a + b","option_c":"b - a","option_d":"a * b","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une loi uniforme sur [a, b], l'espérance est la moyenne des bornes, soit (a + b)\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"(a + b)\/2\", \"b\": \"a + b\", \"c\": \"b - a\", \"d\": \"a * b\"}}","_debug_options_count":4},{"id":30916,"question":"Deux événements A et B sont incompatibles si et seulement si P(A ∩ B) = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Deux événements sont incompatibles (ou disjoints) si leur intersection est vide, donc P(A ∩ B) = 0.","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":30917,"question":"Quelle est la probabilité d'obtenir deux fois pile en lançant une pièce équilibrée deux fois ?","option_a":"1\/2","option_b":"1\/4","option_c":"1\/3","option_d":"1\/8","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité d'obtenir pile en un lancer est 1\/2. Pour deux lancers indépendants, la probabilité est (1\/2) * (1\/2) = 1\/4.","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\/2\", \"b\": \"1\/4\", \"c\": \"1\/3\", \"d\": \"1\/8\"}}","_debug_options_count":4},{"id":30918,"question":"Si X suit une loi exponentielle de paramètre λ, alors son espérance est égale à :","option_a":"λ","option_b":"1\/λ","option_c":"λ^2","option_d":"2λ","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi exponentielle de paramètre λ, l'espérance est 1\/λ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"λ\", \"b\": \"1\/λ\", \"c\": \"λ^2\", \"d\": \"2λ\"}}","_debug_options_count":4},{"id":30919,"question":"La probabilité conditionnelle P(A|B) est toujours inférieure ou égale à P(A).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité conditionnelle P(A|B) = P(A ∩ B)\/P(B) peut être supérieure, inférieure ou égale à P(A), selon la relation entre A et B.","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}]