(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 33014, 1042] NotebookOptionsPosition[ 31212, 972] NotebookOutlinePosition[ 31797, 995] CellTagsIndexPosition[ 31754, 992] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Exercices r\[EAcute]solus - 2 - Probabilit\[EAcute]s", "Section", CellChangeTimes->{{3.539516964165578*^9, 3.539516965722534*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "EXPROBA21. Sur 10000 personnes d\[CloseCurlyQuote]un \[AHat]ge \ donn\[EAcute], il y a 400 cas de cancer du poumon. \nSur ces 10000 personnes, \ il y avait 6000 fumeurs dont 300 avaient le cancer du poumon. Quelle est la \ probabilit\[EAcute] qu\[CloseCurlyQuote]une personne tir\[EAcute]e au hasard \ \na) n\[CloseCurlyQuote]ait pas le cancer du poumon;\t\t\t\t", StyleBox["0.96", FontColor->RGBColor[0.5, 0, 0.5]], "\nb) soit un fumeur canc\[EAcute]reux;\t\t\t\t\t", StyleBox["0.03", FontColor->RGBColor[0.5, 0, 0.5]], "\nc) soit un fumeur si on sait qu\[CloseCurlyQuote]elle est \ canc\[EAcute]reuse?\t\t", StyleBox["0.75", FontColor->RGBColor[0.5, 0, 0.5]] }], "Titre2", CellChangeTimes->{{3.539516980747326*^9, 3.539517007024907*^9}, { 3.539517873102725*^9, 3.539517873446746*^9}, {3.6323906470482397`*^9, 3.6323906953739653`*^9}, {3.6323907283260202`*^9, 3.632390733873312*^9}}], Cell[CellGroupData[{ Cell["Solution d\[EAcute]taill\[EAcute]e", "Titre3", CellChangeTimes->{{3.632390526017165*^9, 3.632390529287654*^9}}], Cell[TextData[{ "En dressant le tableau suivant, on peut facilement compl\[EAcute]ter les \ cases manquantes \n", Cell[BoxData[GridBox[{ { StyleBox[" ", FontFamily->"Apple Garamond BT"], StyleBox["C", FontFamily->"Apple Garamond BT"], SuperscriptBox[ StyleBox["C", FontFamily->"Apple Garamond BT"], StyleBox["c", FontFamily->"Apple Garamond BT", FontSize->14.720000267028809`]], StyleBox[ RowBox[{"\[Sum]", "="}], FontFamily->"Apple Garamond BT"]}, { StyleBox["F", FontFamily->"Apple Garamond BT"], StyleBox[ RowBox[{"300", " "}], FontFamily->"Apple Garamond BT"], StyleBox[ RowBox[{ StyleBox["5700", FontColor->RGBColor[1, 0, 0]], " "}], FontFamily->"Apple Garamond BT"], StyleBox["6000", FontFamily->"Apple Garamond BT"]}, { SuperscriptBox[ StyleBox["F", FontFamily->"Apple Garamond BT"], StyleBox["c", FontFamily->"Apple Garamond BT", FontSize->14.720000267028809`]], StyleBox[ RowBox[{ StyleBox["100", FontColor->RGBColor[1, 0, 0]], " "}], FontFamily->"Apple Garamond BT"], StyleBox[ RowBox[{ StyleBox["3900", FontColor->RGBColor[1, 0, 0]], " "}], FontFamily->"Apple Garamond BT"], StyleBox[ RowBox[{ StyleBox["4000", FontColor->RGBColor[1, 0, 0]], " "}], FontFamily->"Apple Garamond BT"]}, { StyleBox[ RowBox[{"\[Sum]", "="}], FontFamily->"Apple Garamond BT"], StyleBox["400", FontFamily->"Apple Garamond BT"], StyleBox["9600", FontFamily->"Apple Garamond BT"], StyleBox["10000", FontFamily->"Apple Garamond BT"]} }, GridBoxDividers->{ "Columns" -> {{True}}, "ColumnsIndexed" -> {}, "Rows" -> {{True}}, "RowsIndexed" -> {}}]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "C", ")"}], "=", RowBox[{ FractionBox["400", "10000"], "=", "0.04"}]}], TraditionalForm]]], "\t\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "F", ")"}], "=", RowBox[{ FractionBox["6000", "10000"], "=", "0.6"}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"C", "|", "F"}], ")"}], "=", RowBox[{ FractionBox["300", "6000"], "=", "0.05"}]}], TraditionalForm]]], "\t\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"C", "\[Intersection]", "F"}], ")"}], "=", RowBox[{ FractionBox["300", "10000"], "=", "0.03"}]}], TraditionalForm]]], "\na) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", SuperscriptBox["C", "c"], ")"}], "=", "0.96"}], TraditionalForm]]], "\nb) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"F", "\[Intersection]", "C"}], ")"}], "=", RowBox[{ FractionBox["300", "10000"], "=", "0.03"}]}], TraditionalForm]]], "\nc) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"F", "|", "C"}], ")"}], "=", RowBox[{ FractionBox[ RowBox[{"P", "(", RowBox[{"F", "\[Intersection]", "C"}], ")"}], RowBox[{"P", "(", "C", ")"}]], "=", RowBox[{ FractionBox["0.03", "0.04"], "=", RowBox[{ FractionBox["3", "4"], "=", "0.75"}]}]}]}], TraditionalForm]]] }], "cadre", CellChangeTimes->{{3.539517035557723*^9, 3.53951710075218*^9}, { 3.539517137269816*^9, 3.539517299057014*^9}, {3.632390432048703*^9, 3.632390484709013*^9}, {3.6323905460917053`*^9, 3.632390670709915*^9}}] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "EXPROBA22. Un homme a dans sa poche un trousseau de 6 clefs. Avant de \ rentrer chez lui, il constate qu\[CloseCurlyQuote]il a perdu une clef. Il \ doit ouvrir sa porte qui comporte deux serrures. Trouver\na) la probabilit\ \[EAcute] qu\[CloseCurlyQuote]il sache ouvrir la porte;\t", StyleBox["2/3", FontColor->RGBColor[0.5, 0, 0.5]], "\nb) la probabilit\[EAcute] qu\[CloseCurlyQuote]il puisse ouvrir la porte \ avec les deux premi\[EGrave]res clefs qu\[CloseCurlyQuote]il essaie.\t", StyleBox["1/15", FontColor->RGBColor[0.5, 0, 0.5]] }], "Titre2", CellChangeTimes->{{3.5395173344109488`*^9, 3.53951735762424*^9}, { 3.5395178675282497`*^9, 3.5395178963513527`*^9}, {3.6323907368530188`*^9, 3.632390738153057*^9}, {3.632390837719533*^9, 3.632390854501008*^9}}], Cell[CellGroupData[{ Cell["Solution d\[EAcute]taill\[EAcute]e", "Titre3", CellChangeTimes->{{3.632390526017165*^9, 3.632390529287654*^9}}], Cell[TextData[{ "a) Il doit perdre une cl\[EAcute] mais cela ne peut-\[EHat]tre une des deux \ cl\[EAcute]s dont il a besoin donc ", Cell[BoxData[ FormBox[ RowBox[{"p", "=", RowBox[{ FractionBox["4", "6"], "=", FractionBox["2", "3"]}]}], TraditionalForm]]], "\nb) Il doit toujours pouvoir ouvrir sa porte ", Cell[BoxData[ FormBox[ RowBox[{"(", FractionBox["4", "6"], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " et prendre les 2 bonnes cl\[EAcute]s parmi les 5 restantes du trousseau: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox["4", "6"], FractionBox["2", "5"], FractionBox["1", "4"]}], "=", FractionBox["1", "15"]}], TraditionalForm]]] }], "cadre", CellChangeTimes->{{3.539517366565217*^9, 3.539517398783836*^9}, { 3.539517433741209*^9, 3.5395174483359747`*^9}, {3.63239075214952*^9, 3.6323908255182753`*^9}}] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ EXPROBA23. Dans une \[EAcute]cole, 40 \[EAcute]tudiants ont Abel comme \ professeur, 60 ont Baudouin comme professeur. 10 \[EAcute]tudiants d\[CloseCurlyQuote]Abel ont rat\[EAcute], et 12 \ \[EAcute]l\[EGrave]ves de Baudouin ont rat\[EAcute]. a) Un \[EAcute]l\[EGrave]ve ayant rat\[EAcute] est tir\[EAcute] au hasard. \ Quelle est la probabilit\[EAcute] qu\[CloseCurlyQuote]il ait Abel comme \ professeur\[NonBreakingSpace]? b) Quelle est la probabilit\[EAcute] qu\[CloseCurlyQuote]un \[EAcute]l\ \[EGrave]ve quelconque ait rat\[EAcute] et soit un \[EAcute]l\[EGrave]ve de \ Baudouin? c) Quelle est la probabilit\[EAcute] qu\[CloseCurlyQuote]un \[EAcute]l\ \[EGrave]ve choisi au hasard soit en \[EAcute]chec\[NonBreakingSpace]?\ \>", "Titre2", CellChangeTimes->{{3.539517493848287*^9, 3.53951753657019*^9}, { 3.539517869318508*^9, 3.539517869646741*^9}, {3.6323908696729107`*^9, 3.6323908730450974`*^9}}], Cell[CellGroupData[{ Cell["Solution d\[EAcute]taill\[EAcute]e", "Titre3", 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StyleBox["40", FontFamily->"Apple Garamond BT"], StyleBox["60", FontFamily->"Apple Garamond BT"], StyleBox["100", FontFamily->"Apple Garamond BT"]} }, GridBoxDividers->{ "Columns" -> {{True}}, "ColumnsIndexed" -> {}, "Rows" -> {{True}}, "RowsIndexed" -> {}}]]], " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "Abel", ")"}], "=", FractionBox["40", "100"]}], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "Baudouin", ")"}], "=", FractionBox["60", "100"]}], TraditionalForm]], FormatType->"TraditionalForm"], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"\[EAcute]chec", "|", "Abel"}], ")"}], "=", FractionBox["10", "40"]}], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"\[EAcute]chec", "|", "Baudouin"}], ")"}], "=", FractionBox["12", "60"]}], TraditionalForm]]], "\na) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"Abel", "|", "\[EAcute]chec"}], ")"}], " ", "=", " ", RowBox[{ FractionBox[ RowBox[{"P", "(", RowBox[{"Abel", " ", "\[Intersection]", " ", "\[EAcute]chec"}], ")"}], RowBox[{"P", "(", "\[EAcute]chec", ")"}]], "=", RowBox[{ FractionBox[ RowBox[{ RowBox[{"P", "(", "Abel", ")"}], RowBox[{"P", "(", " ", RowBox[{"\[EAcute]chec", "|", "Abel"}], ")"}]}], RowBox[{"P", "(", "\[EAcute]chec", ")"}]], "=", RowBox[{ FractionBox[ RowBox[{ FractionBox["40", "100"], " ", FractionBox[".10", "40"]}], FractionBox["22", "100"]], "=", FractionBox["10", "22"]}]}]}]}], TraditionalForm]]], "\nplus simplement,\nnombre d\[CloseCurlyQuote]\[EAcute]l\[EGrave]ves en \ \[EAcute]chec = 10+12=22\nnombre d\[CloseCurlyQuote]\[EAcute]l\[EGrave]ves en \ \[EAcute]chec chez Abel=10\nb) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"\[EAcute]chec", "\[Intersection]", "Baudouin"}], ")"}], "=", RowBox[{ RowBox[{ FractionBox["60", "100"], FractionBox["12", "60"]}], "=", FractionBox["12", "100"]}]}], TraditionalForm]]], "\nc) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "\[EAcute]chec", ")"}], "=", RowBox[{ RowBox[{ RowBox[{"P", "(", RowBox[{"\[EAcute]chec", " ", "\[Intersection]", "Abel"}], ")"}], " ", "+", " ", RowBox[{"P", "(", RowBox[{"\[EAcute]chec", " ", "\[Intersection]", " ", "Baudouin"}], ")"}]}], "=", RowBox[{ RowBox[{ RowBox[{ FractionBox["40", "100"], " ", FractionBox["10", "40"]}], "+", RowBox[{ FractionBox["60", "100"], FractionBox["12", "60"]}]}], "=", FractionBox["22", "100"]}]}]}], TraditionalForm]]], "\nplus simplement,\nnombre d\[CloseCurlyQuote]\[EAcute]l\[EGrave]ves en \ \[EAcute]chec =22\nnombre d\[CloseCurlyQuote]\[EAcute]l\[EGrave]ves =100" }], "cadre", CellChangeTimes->{{3.539517549298387*^9, 3.539517630424777*^9}, { 3.539517692007691*^9, 3.5395177745916767`*^9}, {3.632390894248735*^9, 3.6323911171933804`*^9}, {3.6323911539636927`*^9, 3.6323911806669188`*^9}}] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "EXPROBA24. *Un test rapide pour d\[EAcute]tecter le cancer a \[EAcute]t\ \[EAcute] mis au point. Si un individu est canc\[EAcute]reux, il y a 95 \ chances sur 100 que le test r\[EAcute]agisse positivement. Par contre, s'il \ est sain, il y a quand m\[EHat]me 10 chances sur 100 que le test \ r\[EAcute]agisse positivement. Dans une population donn\[EAcute]e, on compte \ 18,5 % de r\[EAcute]actions positives.\na) Trouver la proportion \ th\[EAcute]orique de canc\[EAcute]reux dans cette population.\t", StyleBox["0.1", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], "\nb) Si un individu est pris au hasard dans cette population, trouver la \ probabilit\[EAcute] qu'il soit canc\[EAcute]reux et r\[EAcute]agisse au test. \ ", StyleBox["0.095", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], "\nc) Si un individu choisi au hasard dans cette population a r\[EAcute]agi \ positivement au test, trouver la probabilit\[EAcute] que cet individu soit \ effectivement atteint du cancer.\t", StyleBox["0.513514", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]] }], "Titre2", CellChangeTimes->{{3.4715150432781878`*^9, 3.471515063443638*^9}, { 3.47151573086954*^9, 3.471515731122117*^9}, {3.4715164070828047`*^9, 3.471516414374674*^9}, {3.5071779815616903`*^9, 3.50717798245658*^9}, 3.5373368685332947`*^9, {3.53733696405436*^9, 3.537337001116192*^9}, 3.632391205420401*^9, {3.632392065126038*^9, 3.6323920695661297`*^9}}], Cell[CellGroupData[{ Cell["Solution d\[EAcute]taill\[EAcute]e", "Titre3", CellChangeTimes->{{3.632390526017165*^9, 3.632390529287654*^9}}], Cell[TextData[{ "Tout d\[CloseCurlyQuote]abord \[EAcute]crivons les probabilit\[EAcute]s \ donn\[EAcute]es dans l\[CloseCurlyQuote]\[EAcute]nonc\[EAcute]:\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", RowBox[{"(", RowBox[{"positif", " ", "|", " ", "cancer"}], ")"}]}], "=", "0.95"}], TraditionalForm]]], "\tet donc ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"n\[EAcute]gatif", "|", " ", "cancer"}], ")"}], "=", "0.05"}], TraditionalForm]], FormatType->"TraditionalForm"], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"positif", " ", "|", " ", SuperscriptBox["cancer", "c"]}], ")"}], " ", "=", " ", "0.1"}], TraditionalForm]]], "\tet donc ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"n\[EAcute]gatif", "|", " ", SuperscriptBox["cancer", "c"]}], ")"}], "=", "0.9"}], TraditionalForm]], FormatType->"TraditionalForm"], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", RowBox[{"(", "positif", ")"}]}], "=", "0.185"}], TraditionalForm]], FormatType->"TraditionalForm"], "\na) On cherche ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", "cancer", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " mais impossible de le calculer sous la forme ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"P", RowBox[{"(", RowBox[{"cancer", " ", "\[Intersection]", " ", "positif"}], ")"}]}], " ", "+", " ", RowBox[{"P", "(", RowBox[{"cancer", " ", "\[Intersection]", " ", "n\[EAcute]gatif"}], ")"}]}], "=", RowBox[{ RowBox[{ RowBox[{"P", "(", " ", "positif", ")"}], RowBox[{"P", "(", RowBox[{"cancer", " ", "|", " ", "positif"}], ")"}]}], "+", RowBox[{ RowBox[{"P", "(", "n\[EAcute]gatif", ")"}], RowBox[{"P", "(", RowBox[{"cancer", " ", "|", "n\[EAcute]gatif"}], ")"}]}]}]}], TraditionalForm]]], " car on ne connait pas ces probabilit\[EAcute]s conditionnelles.\nOn d\ \[EAcute]compose alors plut\[OHat]t ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", "positif", ")"}], TraditionalForm]], FormatType->"TraditionalForm"], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", RowBox[{"(", "positif", ")"}]}], "=", RowBox[{"0.185", "=", RowBox[{ RowBox[{ RowBox[{"P", RowBox[{"(", RowBox[{"cancer", " ", "\[Intersection]", " ", "positif"}], ")"}]}], " ", "+", " ", RowBox[{"P", "(", RowBox[{ SuperscriptBox["cancer", "c"], " ", "\[Intersection]", " ", "positif"}], ")"}]}], "=", RowBox[{ RowBox[{"0.95", " ", RowBox[{"P", "(", "cancer", ")"}]}], "+", RowBox[{"0.1", " ", RowBox[{"(", RowBox[{"1", "-", RowBox[{"P", "(", "cancer", ")"}]}], ")"}]}]}]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], "\n", Cell[BoxData[ FormBox[ RowBox[{"0.185", "=", RowBox[{"0.1", "+", RowBox[{"0.85", " ", RowBox[{"P", "(", "cancer", ")"}]}]}]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "cancer", ")"}], "=", RowBox[{ FractionBox["0.085", "0.85"], "=", "0.1"}]}], TraditionalForm]]], "\nb) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"cancer", " ", "\[Intersection]", " ", "positif"}], ")"}], "=", " ", RowBox[{ RowBox[{ RowBox[{"P", "(", "cancer", ")"}], " ", RowBox[{"P", "(", RowBox[{"positif", " ", "|", " ", "cancer"}], ")"}]}], "=", RowBox[{ RowBox[{"0.1", " ", "0.95"}], "=", "0.095"}]}]}], TraditionalForm]]], "\nc) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", RowBox[{"C", "|", "P"}], ")"}], "=", RowBox[{Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"P", "(", RowBox[{"C", " ", "\[Intersection]", " ", "P"}], ")"}], RowBox[{"P", "(", "P", ")"}]], "=", FractionBox[ RowBox[{"0.95", " ", "0.1"}], "0.185"]}], TraditionalForm]]]]], "=", RowBox[{"0.5135135135135135`", "=", FractionBox["95", "185"]}]}]}], TraditionalForm]]] }], "cadre", CellChangeTimes->{{3.471515047259499*^9, 3.471515060530604*^9}, { 3.47151573815571*^9, 3.471515917519438*^9}, {3.471515953069335*^9, 3.471515969336195*^9}, {3.4715161319829073`*^9, 3.4715161419216413`*^9}, { 3.471516174158284*^9, 3.471516367964025*^9}, {3.471516419266533*^9, 3.471516424394946*^9}, {3.632391218765897*^9, 3.6323914160349903`*^9}, { 3.632391449495296*^9, 3.632391618964415*^9}, {3.63239169888153*^9, 3.6323918650103283`*^9}, {3.632391954362273*^9, 3.632391976482315*^9}, { 3.632392020454208*^9, 3.632392022763598*^9}}] }, Closed]] }, Open ]], Cell[TextData[{ "EXPROBA25. Un ranch poss\[EGrave]de 20 chevaux (6 blancs, 5 noirs et 9 \ gris) et une cal\[EGrave]che pr\[EAcute]vue pour \[EHat]tre tir\[EAcute]e par \ 2 chevaux. Le cocher de la cal\[EGrave]che choisit au hasard les 2 chevaux de \ l\[CloseCurlyQuote]attelage, parmi les 20 chevaux du ranch.\na) Calculer la \ probabilit\[EAcute] des \[EAcute]v\[EAcute]nements \ suivants\[NonBreakingSpace]:\n\tA\[NonBreakingSpace]: \[LeftGuillemet]\ \[NonBreakingSpace]les 2 chevaux sont blancs\[NonBreakingSpace]\ \[RightGuillemet],\t\t\t", StyleBox["0.0789474", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], "\n\tB\[NonBreakingSpace]: \[LeftGuillemet]\[NonBreakingSpace]l\ \[CloseCurlyQuote]un des chevaux, au moins, est blanc\[NonBreakingSpace]\ \[RightGuillemet],\t\t", StyleBox["0.521053", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], "\n\tC\[NonBreakingSpace]: \[LeftGuillemet]\[NonBreakingSpace]les 2 chevaux \ sont de la m\[EHat]me couleur\[NonBreakingSpace]\[RightGuillemet].\t\t", StyleBox["0.321053", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], "\nb) Sachant que les deux chevaux de l\[CloseCurlyQuote]attelage sont de \ couleurs diff\[EAcute]rentes, quelle est la probabilit\[EAcute] pour que l\ \[CloseCurlyQuote]un soit blanc\[NonBreakingSpace]?\t\t", StyleBox["0.651163", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], "\nc) Des enfants effectuent un stage de 3 jours dans le ranch. le cocher \ organise alors, chaque jour, une promenade en cal\[EGrave]che. Chaque jour, \ il choisit au hasard l\[CloseCurlyQuote]attelage.\nCalculer la probabilit\ \[EAcute] des \[EAcute]v\[EAcute]nements suivants\[NonBreakingSpace]:\n\t", Cell[BoxData[ FormBox[ SubscriptBox["E", "0"], TraditionalForm]]], "\[NonBreakingSpace]: \ \[LeftGuillemet]\[NonBreakingSpace]l\[CloseCurlyQuote]attelage n\ \[CloseCurlyQuote]est jamais unicolore\[NonBreakingSpace]\[RightGuillemet],\t\ \t", StyleBox["0.312974", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]], "\n\t", Cell[BoxData[ FormBox[ SubscriptBox["E", "1"], TraditionalForm]]], "\[NonBreakingSpace]: \ \[LeftGuillemet]\[NonBreakingSpace]l\[CloseCurlyQuote]attelage est unicolore \ exactement une fois\[NonBreakingSpace]\[RightGuillemet].\t", StyleBox["0.443986", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]] }], "Titre2", CellChangeTimes->{ 3.471515073701281*^9, {3.471516374992196*^9, 3.471516382165702*^9}, { 3.507177987736833*^9, 3.507177988664795*^9}, {3.537337101417485*^9, 3.5373371571380167`*^9}, 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Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", RowBox[{"(", RowBox[{ RowBox[{"1", " ", "blanc"}], " ", "|", " ", RowBox[{"couleur", " ", Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{"\[NotEqual]", " "}], TraditionalForm]]]]]}]}], ")"}]}], "=", RowBox[{ FractionBox[ RowBox[{"P", "(", RowBox[{ RowBox[{ RowBox[{"1", " ", "blanc"}], "\[Intersection]", "couleur"}], FormBox["\[NotEqual]", TraditionalForm]}], ")"}], RowBox[{"P", "(", RowBox[{"couleur", FormBox["\[NotEqual]", TraditionalForm]}], ")"}]], "=", RowBox[{ FractionBox[ RowBox[{ RowBox[{ FractionBox["6", "20"], FractionBox["14", "19"]}], "+", RowBox[{ FractionBox["14", "20"], FractionBox["6", "19"]}]}], FractionBox["129", "190"]], "=", RowBox[{ FractionBox[ FractionBox[ RowBox[{"6", " ", "14"}], "190"], FractionBox["129", "190"]], "=", RowBox[{ FractionBox[ FractionBox["42", "95"], FractionBox["129", "190"]], "=", FractionBox["28", "43"]}]}]}]}]}], TraditionalForm]]], "\nc) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", 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Dans une classe, il y a 8 filles et 12 gar\[CCedilla]ons. Dans \ une autre classe, 11 filles et 7 gar\[CCedilla]ons. On tire une classe au \ sort (pile ou face) puis, dans la classe d\[EAcute]sign\[EAcute]e, on choisit \ au hasard un \[EAcute]l\[EGrave]ve. Une fille a-t-elle plus de chance d\ \[CloseCurlyQuote]\[EHat]tre d\[EAcute]sign\[EAcute]e qu\[CloseCurlyQuote]un \ gar\[CCedilla]on\[NonBreakingSpace]?\t", StyleBox["0.505556", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]] }], "Titre2", CellChangeTimes->{ 3.4715152259004908`*^9, 3.506432128660983*^9, 3.5071774660913773`*^9, { 3.5071788917562237`*^9, 3.507178892620008*^9}, {3.6323920831440983`*^9, 3.6323920858691187`*^9}, {3.6323922385043783`*^9, 3.6323922395439863`*^9}, 3.632392291611947*^9}], Cell[CellGroupData[{ Cell["Solution d\[EAcute]taill\[EAcute]e", "Titre3", CellChangeTimes->{{3.632390526017165*^9, 3.632390529287654*^9}}], Cell[TextData[{ Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{"\<\"P(fille) = P(fille \[Intersection] classe A) + P(fille \ \[Intersection] classe B) = \"\>", "\[InvisibleSpace]", FractionBox["91", "180"]}], SequenceForm[ "P(fille) = P(fille \[Intersection] classe A) + P(fille \[Intersection] \ classe B) = ", Rational[91, 180]], Editable->False], TraditionalForm]], "Print"], "\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "fille", ")"}], "=", RowBox[{ RowBox[{"P", "(", RowBox[{"fille", " ", "\[Intersection]", " ", RowBox[{"classe", " ", "A"}]}], ")"}], " ", "+", " ", RowBox[{"P", RowBox[{"(", RowBox[{"fille", " ", "\[Intersection]", " ", RowBox[{"classe", " ", "B"}]}], ")"}]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], "\n", Cell[BoxData[ FormBox[ RowBox[{"=", RowBox[{ RowBox[{ RowBox[{"P", RowBox[{"(", RowBox[{"classe", " ", "A"}], ")"}], "P", RowBox[{"(", RowBox[{"fille", " ", "|", RowBox[{"classe", " ", "A"}]}], ")"}]}], "+", RowBox[{ RowBox[{"P", "(", RowBox[{"classe", " ", "B"}], ")"}], RowBox[{"P", "(", RowBox[{"fille", " ", "|", RowBox[{"classe", " ", "B"}]}], ")"}]}]}], "=", RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "2"], FractionBox["8", "20"]}], "+", RowBox[{ FractionBox["1", "2"], FractionBox["11", "18"]}]}], "=", RowBox[{ FractionBox["91", "180"], "=", StyleBox["0.505556", FontSlant->"Italic", FontColor->RGBColor[0.5, 0, 0.5]]}]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], "\net donc ", Cell[BoxData[ FormBox[ InterpretationBox[ RowBox[{"\<\"P(gar\[CCedilla]on) = \"\>", "\[InvisibleSpace]", FractionBox["89", "180"]}], SequenceForm["P(gar\[CCedilla]on) = ", Rational[89, 180]], Editable->False], TraditionalForm]], "Print"] }], "cadre", CellChangeTimes->{ 3.471527090658173*^9, 3.50643212866129*^9, 3.507177466091627*^9, { 3.6323921602070637`*^9, 3.632392307512854*^9}}] }, Closed]] }, Open ]] }, WindowToolbars->"EditBar", CellGrouping->Manual, WindowSize->{990, 842}, WindowMargins->{{203, Automatic}, {Automatic, 53}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, PageHeaders->{{Inherited, Inherited, Inherited}, { "macformath.net", Inherited, Inherited}}, CellLabelAutoDelete->True, FrontEndVersion->"10.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (December 4, \ 2014)", StyleDefinitions->"stylemath.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[579, 22, 137, 1, 66, "Section"], Cell[CellGroupData[{ Cell[741, 27, 920, 18, 212, "Titre2"], Cell[CellGroupData[{ Cell[1686, 49, 118, 1, 47, "Titre3"], Cell[1807, 52, 3648, 129, 431, "cadre"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[5504, 187, 791, 14, 151, "Titre2"], Cell[CellGroupData[{ Cell[6320, 205, 118, 1, 47, "Titre3"], Cell[6441, 208, 915, 27, 171, "cadre"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[7405, 241, 921, 16, 212, "Titre2"], Cell[CellGroupData[{ Cell[8351, 261, 118, 1, 47, "Titre3"], Cell[8472, 264, 4547, 149, 575, "cadre"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[13068, 419, 1488, 28, 303, "Titre2"], Cell[CellGroupData[{ Cell[14581, 451, 118, 1, 47, "Titre3"], Cell[14702, 454, 4867, 155, 551, "cadre"] }, Closed]] }, Open ]], Cell[19596, 613, 2675, 58, 463, "Titre2"] }, Open ]], Cell[CellGroupData[{ Cell[22308, 676, 118, 1, 47, "Titre3"], Cell[22429, 679, 4251, 150, 464, "cadre"] }, Closed]], Cell[CellGroupData[{ Cell[26717, 834, 379, 7, 45, "Titre2"], Cell[CellGroupData[{ Cell[27121, 845, 118, 1, 47, "Titre3"], Cell[27242, 848, 732, 22, 108, "cadre"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell[28023, 876, 812, 15, 120, "Titre2"], Cell[CellGroupData[{ Cell[28860, 895, 118, 1, 47, "Titre3"], Cell[28981, 898, 2203, 70, 195, "cadre"] }, Closed]] }, Open ]] } ] *) (* End of internal cache information *)