Maîtrisez les Probabilités Totales — Terminale Mathématiques

Question 1 sur 10 10:00

[{"id":27138,"question":"Soit deux événements A et B tels que P(A) = 0.4 et P(B|A) = 0.7. Quelle est la probabilité P(A ∩ B) ?","option_a":"0.28","option_b":"0.22","option_c":"0.18","option_d":"0.47","option_e":"","option_f":"","bonne_reponse":"A","explication":"P(A ∩ B) = P(B|A) × P(A) = 0.7 × 0.4 = 0.28.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27139,"question":"Le théorème des probabilités totales s'applique uniquement si les événements forment une partition de l'espace des possibles.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une condition nécessaire pour que le théorème soit valide.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27140,"question":"Soit une partition {A1, A2} de l'espace des possibles avec P(A1) = 0.3 et P(A2) = 0.7. Si P(B|A1) = 0.5 et P(B|A2) = 0.2, quelle est P(B) ?","option_a":"0.29","option_b":"0.31","option_c":"0.35","option_d":"0.41","option_e":"","option_f":"","bonne_reponse":"A","explication":"P(B) = P(B|A1)×P(A1) + P(B|A2)×P(A2) = 0.5×0.3 + 0.2×0.7 = 0.15 + 0.14 = 0.29.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27141,"question":"Si P(A) = 0.6 et P(B) = 0.5, peut-on avoir P(A ∩ B) = 0.4 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"P(A ∩ B) ne peut pas dépasser la plus petite des probabilités P(A) ou P(B). Ici, 0.4 \u003E 0.5 n'est pas possible.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27142,"question":"Dans un jeu de dés, on lance un dé équilibré. Quelle est la probabilité d'obtenir un nombre pair ou un 3 ?","option_a":"1\/2","option_b":"2\/3","option_c":"5\/6","option_d":"1\/3","option_e":"","option_f":"","bonne_reponse":"C","explication":"Événements : pair = {2,4,6}, 3 = {3}. Union = {2,3,4,6} → 4 cas favorables sur 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27143,"question":"Le théorème des probabilités totales permet de calculer P(A) à partir de P(A|B) et P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le théorème permet de calculer P(B) à partir de P(B|A) et P(A), pas l'inverse directement.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27144,"question":"Soit A et B deux événements incompatibles avec P(A) = 0.3 et P(B) = 0.4. Quelle est P(A ∪ B) ?","option_a":"0.7","option_b":"0.12","option_c":"0.5","option_d":"0.1","option_e":"","option_f":"","bonne_reponse":"A","explication":"P(A ∪ B) = P(A) + P(B) = 0.3 + 0.4 = 0.7 car A et B sont incompatibles.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27145,"question":"Si P(A|B) = 0.8 et P(B) = 0.25, quelle est P(A ∩ B) ?","option_a":"0.2","option_b":"0.32","option_c":"0.05","option_d":"0.16","option_e":"","option_f":"","bonne_reponse":"A","explication":"P(A ∩ B) = P(A|B) × P(B) = 0.8 × 0.25 = 0.2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27146,"question":"Le théorème des probabilités totales est une généralisation de la formule des probabilités conditionnelles.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Il étend la formule en décomposant l'événement en une partition.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27147,"question":"Dans une urne contenant 3 boules rouges et 2 boules bleues, on tire deux boules sans remise. Quelle est la probabilité d'obtenir une boule rouge puis une bleue ?","option_a":"3\/10","option_b":"6\/25","option_c":"3\/5","option_d":"1\/5","option_e":"","option_f":"","bonne_reponse":"A","explication":"P(rouge puis bleue) = (3\/5) × (2\/4) = 6\/20 = 3\/10.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]