Quiz — serie 3( énoncé)lois uniformes.pdf

Question 1 sur 10 20:00

[{"id":4556,"question":"Quelle est la probabilité d'obtenir un nombre pair en lançant un dé équilibré à 6 faces ?","option_a":"1\/2","option_b":"1\/3","option_c":"2\/3","option_d":"1\/6","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un dé équilibré a 6 faces numérotées de 1 à 6. Les nombres pairs sont 2, 4 et 6, soit 3 issues. La probabilité est donc 3\/6 = 1\/3.","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\/3\", \"c\": \"2\/3\", \"d\": \"1\/6\"}}","_debug_options_count":4},{"id":4557,"question":"Pour une variable aléatoire X suivant une loi uniforme continue sur [0, 1], quelle est la probabilité que X soit inférieur ou égal à 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":"Dans une loi uniforme continue sur [0, 1], la probabilité est proportionnelle à la longueur de l'intervalle. P(X ≤ 0,5) = 0,5 - 0 = 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":4558,"question":"L'espérance d'une loi uniforme discrète sur {1, 2, 3, 4, 5} est égale à 3.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance est (1+2+3+4+5)\/5 = 15\/5 = 3. L'affirmation est donc vraie.","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":4559,"question":"Quelle est la variance d'une loi uniforme discrète sur {1, 2, 3, 4} ?","option_a":"1","option_b":"1,25","option_c":"2,5","option_d":"5","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'espérance est (1+2+3+4)\/4 = 2,5. La variance est ((1-2,5)² + (2-2,5)² + (3-2,5)² + (4-2,5)²)\/4 = 1,25.","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\", \"b\": \"1,25\", \"c\": \"2,5\", \"d\": \"5\"}}","_debug_options_count":4},{"id":4560,"question":"Pour une loi uniforme continue sur [a, b], la densité de probabilité est constante et égale à 1\/(b-a).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La densité de probabilité d'une loi uniforme continue sur [a, b] est bien 1\/(b-a), car l'aire sous la courbe doit être égale à 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":4561,"question":"Si X suit une loi uniforme discrète sur {2, 4, 6, 8}, quelle est la probabilité que X soit supérieur ou égal à 6 ?","option_a":"1\/2","option_b":"1\/4","option_c":"3\/4","option_d":"1\/8","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les issues ≥ 6 sont 6 et 8, soit 2 issues sur 4. La probabilité est donc 2\/4 = 1\/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\": \"1\/2\", \"b\": \"1\/4\", \"c\": \"3\/4\", \"d\": \"1\/8\"}}","_debug_options_count":4},{"id":4562,"question":"Quelle est l'espérance d'une loi uniforme continue sur [0, 10] ?","option_a":"5","option_b":"10","option_c":"0","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi uniforme continue sur [a, b] est (a+b)\/2. Ici, (0+10)\/2 = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"5\", \"b\": \"10\", \"c\": \"0\", \"d\": \"1\"}}","_debug_options_count":4},{"id":4563,"question":"La variance d'une loi uniforme continue sur [a, b] est égale à (b-a)²\/12.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule de la variance pour une loi uniforme continue sur [a, b] est bien (b-a)²\/12.","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":4564,"question":"Si X suit une loi uniforme discrète sur {1, 2, 3, 4, 5, 6}, quelle est la probabilité que X soit un nombre premier ?","option_a":"1\/3","option_b":"1\/2","option_c":"2\/3","option_d":"1\/6","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les nombres premiers dans {1, 2, 3, 4, 5, 6} sont 2, 3 et 5, soit 3 issues. 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\": \"a\", \"options\": {\"a\": \"1\/3\", \"b\": \"1\/2\", \"c\": \"2\/3\", \"d\": \"1\/6\"}}","_debug_options_count":4},{"id":4565,"question":"Pour une loi uniforme continue sur [0, 1], la probabilité que X soit dans l'intervalle [0,25 ; 0,75] est égale à 0,5.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La longueur de l'intervalle [0,25 ; 0,75] est 0,5. La probabilité est donc 0,5.","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}]