Après réalisation d'une enquête, on estime que le temps en minutes, consacré
quotidiennement par un élève à faire ses devoirs scolaires, est une variable
aléatoire \(X\) suivant une loi normale d'espérance 75 minutes et d'écart-type
28 minutes.
L'allure de la courbe de densité de cette loi normale est représentée ci-dessous.
L'égalité \(P\left(X \le 46 \right) = 0,15 \) est illustrée graphiquement.
{"graphInit": {"range": [[-1, 159], [0, 0.006252515621467317]], "scale": [3.75, 63974.25040037397], "axisArrows": "->", "axisOpacity": 0.5, "gridOpacity": 0.0, "gridStep": [5, 5], "tickStep": [1, 1], "labelStep": [2, 2], "unityLabels": true, "xLabel": "Temps en minutes", "yLabel": "Probabilit\u00e9"}, "plot": [["function(x){ return 0.00568410511042483*Math.exp(-Math.pow(x - 75, 2)/1568);}", [-1, 159], {"stroke": "#6495ED"}]], "label": [[[22.666666666666668, 0.0028136320296602924], "0,15", "above", {"color": "#A55446"}]], "path": [[[[-2, 0], [-2, 0.00012956452004848898]], false, {"stroke": "#ED7864"}], [[[-1, 0], [-1, 0.00014284434086867328]], false, {"stroke": "#ED7864"}], [[[0, 0], [0, 0.00015728454168305328]], false, {"stroke": "#ED7864"}], [[[1, 0], [1, 0.0001729637513205357]], false, {"stroke": "#ED7864"}], [[[2, 0], [2, 0.00018996351788724245]], false, {"stroke": "#ED7864"}], [[[3, 0], [3, 0.0002083681636053639]], false, {"stroke": "#ED7864"}], [[[4, 0], [4, 0.0002282646067593372]], false, {"stroke": "#ED7864"}], [[[5, 0], [5, 0.0002497421489452068]], false, {"stroke": "#ED7864"}], [[[6, 0], [6, 0.0002728922259672514]], false, {"stroke": "#ED7864"}], [[[7, 0], [7, 0.0002978081209062048]], false, {"stroke": "#ED7864"}], [[[8, 0], [8, 0.0003245846380965753]], false, {"stroke": "#ED7864"}], [[[9, 0], [9, 0.00035331773699723946]], false, {"stroke": "#ED7864"}], [[[10, 0], [10, 0.0003841041252198227]], false, {"stroke": "#ED7864"}], [[[11, 0], [11, 0.00041704081029311356]], false, {"stroke": "#ED7864"}], [[[12, 0], [12, 0.00045222461008816704]], false, {"stroke": "#ED7864"}], [[[13, 0], [13, 0.0004897516222065956]], false, {"stroke": "#ED7864"}], [[[14, 0], [14, 0.0005297166530420724]], false, {"stroke": "#ED7864"}], [[[15, 0], [15, 0.0005722126076599737]], false, {"stroke": "#ED7864"}], [[[16, 0], [16, 0.0006173298420995011]], false, {"stroke": "#ED7864"}], [[[17, 0], [17, 0.0006651554801831183]], false, {"stroke": "#ED7864"}], [[[18, 0], [18, 0.0007157726974157137]], false, {"stroke": "#ED7864"}], [[[19, 0], [19, 0.0007692599750660225]], false, {"stroke": "#ED7864"}], [[[20, 0], [20, 0.0008256903280404152]], false, {"stroke": "#ED7864"}], [[[21, 0], [21, 0.0008851305106786245]], false, {"stroke": "#ED7864"}], [[[22, 0], [22, 0.0009476402051163039]], false, {"stroke": "#ED7864"}], [[[23, 0], [23, 0.001013271197364039]], false, {"stroke": "#ED7864"}], [[[24, 0], [24, 0.001082066546739819]], false, {"stroke": "#ED7864"}], [[[25, 0], [25, 0.001154059754754937]], false, {"stroke": "#ED7864"}], [[[26, 0], [26, 0.0012292739399846415]], false, {"stroke": "#ED7864"}], [[[27, 0], [27, 0.001307721025847238]], false, {"stroke": "#ED7864"}], [[[28, 0], [28, 0.0013894009485614616]], false, {"stroke": "#ED7864"}], [[[29, 0], [29, 0.001474300892844582]], false, {"stroke": "#ED7864"}], [[[30, 0], [30, 0.0015623945631458825]], false, {"stroke": "#ED7864"}], [[[31, 0], [31, 0.001653641498375259]], false, {"stroke": "#ED7864"}], [[[32, 0], [32, 0.0017479864381784837]], false, {"stroke": "#ED7864"}], [[[33, 0], [33, 0.001845358748823627]], false, {"stroke": "#ED7864"}], [[[34, 0], [34, 0.001945671916692249]], false, {"stroke": "#ED7864"}], [[[35, 0], [35, 0.0020488231172102046]], false, {"stroke": "#ED7864"}], [[[36, 0], [36, 0.0021546928668030245]], false, {"stroke": "#ED7864"}], [[[37, 0], [37, 0.0022631447651176467]], false, {"stroke": "#ED7864"}], [[[38, 0], [38, 0.0023740253343147166]], false, {"stroke": "#ED7864"}], [[[39, 0], [39, 0.0024871639617038022]], false, {"stroke": "#ED7864"}], [[[40, 0], [40, 0.0026023729513690143]], false, {"stroke": "#ED7864"}], [[[41, 0], [41, 0.0027194476897172983]], false, {"stroke": "#ED7864"}], [[[42, 0], [42, 0.002838166929079953]], false, {"stroke": "#ED7864"}], [[[43, 0], [43, 0.002958293192614996]], false, {"stroke": "#ED7864"}], [[[44, 0], [44, 0.00307957330280034]], false, {"stroke": "#ED7864"}], [[[45, 0], [45, 0.0032017390347832296]], false, {"stroke": "#ED7864"}], [[[46, 0], [46, 0.0033245078947689677]], false, {"stroke": "#ED7864"}], [[[28.333333333333332, 0.0007086720821033746], [22.666666666666668, 0.0028136320296602924]], false, {"stroke": "#A55446", "arrow-start": "classic-wide-long"}]]}
Dans tout l'exercice, on donnera des réponses à \(10^{-3}\) près.
Déterminer la probabilité qu'un élève consacre quotidiennement moins de 46 minutes à faire ses devoirs scolaires.