1 ゴ 䱣ᴎџӊঞὖ⥛
1.1 ໜऐ๚ॲ
ϔǃ䱣ᴎ䆩偠 ҎӀᅲ䰙⌏ࡼЁӮ䘛ࠄϸ㉏⦄䈵ˈϔ㉏⿄ПЎ⹂ᅮᗻ⦄䈵˄ᖙ✊⦄䈵˅ˈ՟ བˈぎЁᡯᦋϔᄤˈᄤ㨑ഄ˗ৠᗻ⬉㥋Ⳍ᭹ˈᓖᗻ⬉㥋Ⳍㄝˊϔ㉏⿄ ПЎ䱣ᴎ⦄䈵˄ي✊⦄䈵˅ˈ՟བˈᡯϔᵮ䋼ഄഛࣔⱘ⹀Ꮥˈ݊㒧ᵰৃ㛑ᰃℷ䴶˄㾘 ᅮࠏ᳝ᖑⱘϔ䴶Ўℷ䴶˅ᳱϞˈгৃ㛑ᰃড䴶ᳱϞ˗ড়Ḑક⥛Ў 80%ⱘѻક Ёӏপϔӊѻકˈ᳝ৃ㛑পࠄⱘᰃড়Ḑકˈг᳝ৃ㛑পࠄⱘᰃϡড়Ḑકˊ ᅲ䰙䯂乬Ёˈ䳔㽕خ⾡ḋⱘ㾖ᆳϢ䆩偠ˈϔ㠀ⱘˈ⒵䎇ҹϟϝϾᴵӊ ⱘ䆩偠⿄Ў䱣ᴎ䆩偠ˊ ˄1˅䆩偠ৃҹⳌৠᴵӊϟ䞡䖯㸠˗ ˄2˅䆩偠ⱘ᠔᳝ৃ㛑㒧ᵰᰃџܜᯢ⹂ৃⶹⱘ˗ ˄3˅↣䆩偠Пࠡϡ㛑⹂ᅮાϔϾ㒧ᵰϔᅮӮߎ⦄ˊ 䱣ᴎ䆩偠ࣙᣀᇍ䱣ᴎ⦄䈵䖯㸠㾖ᆳǃ⌟䞣ǃ䆄ᔩ䖯㸠⾥ᄺᅲ偠ㄝˊ៥Ӏҹ ৢᦤࠄⱘ䆩偠䛑ᰃᣛ䱣ᴎ䆩偠ˈгㅔ⿄Ў䆩偠ˈ䗮ᐌ⫼ᄫ↡E㸼⼎ˊ՟བ˖ 1 E˖ᇚ䋼ഄഛࣔⱘϔᵮ⹀Ꮥᡩᦋϔˈ㾖ᆳℷ䴶ড䴶ᳱϞⱘᚙމˊ 2 E ˖ϔㆅЁ㺙᳝ᷛোҢ 1 ࠄ 30 ⱘ 30 া㑶ǃⱑϸ⾡买㡆ⱘЦЧ⧗ˈҢㆅЁӏ ᛣᢑপ 1 া⧗ˈ˄1˅㾖ᆳ݊ো᭄˗˄2˅㾖ᆳ݊买㡆ˊ 3 E ˖䆄ᔩᶤ㔥キ 1 ߚ䩳ݙⱘ⚍ߏ᭄ˊ 4 E ˖㾖ᆳᶤॖ⫳ѻⱘ♃⊵ⱘՓ⫼ᇓੑˊ ᇍѢ䱣ᴎ⦄䈵ˈҎӀ㒣䖛䭓ᳳⱘ㾖ᆳ䖯㸠䞣ⱘ䆩偠থ⦄˖䖭ѯথ⫳ⱘ㒧 ᵰᑊ䴲ᰃᴖх᮴ゴⱘˈ㗠ᰃ᳝㾘ᕟৃᇏⱘˊ՟བˈ䞣䞡ഄᡯᦋϔᵮ⹀Ꮥˈᕫ ࠄℷ䴶ᳱϞⱘ᭄Ϣℷ䴶ᳱϟⱘ᭄㟈䛑ᰃᡯᦋᘏ᭄ⱘϔञ˗ৠϔ䮼⚂থᇘ থ⚂ᔍᇘߏৠϔⳂᷛˈᔍⴔ⚍ᣝ✻ϔᅮⱘ㾘ᕟߚᏗˊ䞣ⱘ䞡䆩偠㾖ᆳ Ё᠔ਜ⦄ߎⱘ᳝㾘ᕟᗻˈህᰃ៥Ӏ᠔䇈ⱘ㒳䅵㾘ᕟᗻˊ㗠ὖ⥛䆎Ϣ᭄⧚㒳䅵ℷ ᰃⷨおᧁ⼎䱣ᴎ⦄䈵㒳䅵㾘ᕟᗻⱘϔ䮼᭄ᄺᄺ⾥ˊ Ѡǃḋᴀぎ䯈 ᇍѢ䱣ᴎ䆩偠ˈҎӀᛳ݈䍷ⱘᰃ䆩偠㒧ᵰˈे↣䱣ᴎ䆩偠ৢ᠔থ⫳ⱘ㒧2 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 ᵰˊ៥Ӏᇚ䱣ᴎ䆩偠Eⱘ↣ϔϾৃ㛑ⱘ㒧ᵰˈ⿄Ў䱣ᴎ䆩偠EⱘϔϾḋᴀ⚍ˈ䗮 ᐌ䆄ωˊ䱣ᴎ䆩偠Eⱘ᠔᳝ḋᴀ⚍㒘៤ⱘ䲚ড়িخ䆩偠Eⱘḋᴀぎ䯈ˈ 䗮ᐌ⫼ ᄫ↡Ω㸼⼎ˊ ՟ 1 E1˖ᇚ䋼ഄഛࣔⱘϔᵮ⹀Ꮥᡩᦋϔˈ㾖ᆳℷ䴶ড䴶ᳱϞⱘᚙމˊ Āℷ䴶ᳱϞāĀড䴶ᳱϞāᰃE1ⱘϸϾḋᴀ⚍ˈ᠔ҹḋᴀぎ䯈ৃㅔ䆄Ў Ω={ℷˈড}ˊ ՟ 2 E2˖ϔㆅЁ㺙᳝ᷛোҢ 1 ࠄ 20 ⱘ 20 Ͼ㑶ǃⱑϸ⾡买㡆ⱘЦЧ⧗ˈҢ ㆅЁӏᛣᢑপ 1 Ͼ⧗ˈ˄1˅㾖ᆳ݊ো᭄˗˄2˅㾖ᆳ݊买㡆ˊ ˄1˅㾖ᆳ݊ো᭄ Āপᕫiো⧗āˈi =1,2, ,20L ᰃE2ⱘḋᴀ⚍ˈ᠔ҹḋᴀぎ䯈ৃㅔ䆄Ў {1,2, ,20} Ω = L ˗ ˄2˅㾖ᆳ݊买㡆 Ҹω =1 Āপᕫ㑶⧗āˈω2=Āপᕫⱑ⧗āˈ߭ḋᴀぎ䯈 1 2 { , }ω ω Ω = ˊ ՟ 3 E3˖䆄ᔩᶤ㔥キ 1 ߚ䩳ݙⱘ⚍ߏ᭄ˊ Ā⚍ߏkāˈk=1,2, , ,L Ln ᰃE3ⱘḋᴀ⚍ˈ᠔ҹḋᴀぎ䯈ৃㅔ䆄Ў {1,2, , , }n Ω = L L ˊ ՟ 4 E4˖ϔᡍ♃⊵Ёӏᛣᢑপϔাˈ⌟䆩݊➗⚻ᇓੑˊ Ā⌟ᕫ♃⊵➗⚻ᇓੑЎtᇣᯊ˄ İ0 t < +∞˅āᰃE4ⱘḋᴀ⚍ˈ᠔ҹḋᴀぎ䯈ৃ 㸼⼎Ў Ω ={ 0t İt< +∞} ՟ 5 E5˖ᇚ䋼ഄഛࣔⱘϔᵮ⹀Ꮥᡩᦋϸˈ㾖ᆳℷ䴶ড䴶ᳱϞⱘᚙމˊ 䆩偠E5ⱘܼ䚼ḋᴀ⚍ᰃ˖(ℷˈℷ)ˈ(ℷˈড)ˈ(ডˈℷ)ˈ(ডˈড)ˈ݊Ё(ℷˈ ℷ)㸼⼎Āᦋϔℷ䴶ᳱϞˈᦋѠℷ䴶ᳱϞāˈձℸ㉏. ߭ḋᴀぎ䯈Ў Ω={(ℷˈℷ)ˈ(ℷˈড)ˈ(ডˈℷ)ˈ(ডˈড)}. ҢϞ䗄՟乬ৃҹⳟࠄ˖ḋᴀぎ䯈ৃҹᰃϔ㓈⚍䲚㓈⚍䲚ˈৃҹᰃ⾏ᬷ⚍ 䲚ˈгৃҹᰃᶤϾऎඳˈৃҹᰃ᳝䰤䲚᮴䰤䲚˄ᇍᑨⱘ⿄Ў᳝䰤ḋᴀぎ䯈᮴ 䰤ḋᴀぎ䯈˅ˊ ϝǃ䱣ᴎџӊ 䱣ᴎџӊ ϔϾ䱣ᴎ䆩偠Ёৃ㛑থ⫳гৃ㛑ϡথ⫳ⱘџӊ⿄Ў䆹䆩偠ⱘ䱣ᴎ џӊ˄ㅔ⿄џӊ˅ˈ䗮ᐌ⫼ᄫ↡ AǃBǃC ㄝ㸼⼎ˊᅲ䰙Ϟˈ䱣ᴎџӊᰃ⬅㢹ᑆϾḋ ᴀ⚍㒘៤ⱘ䲚ড়ˈᰃḋᴀぎ䯈ⱘᄤ䲚ˊ ᴀџӊ 䆩偠ⱘ↣ϔৃ㛑ⱘ㒧ᵰিخᴀџӊˊϔϾḋᴀ⚍ω 㒘៤ⱘऩ⚍ 䲚{ω}ህᰃ䱣ᴎ䆩偠ⱘᴀџӊˊ ᖙ✊џӊ ↣䆩偠Ёᖙ✊থ⫳ⱘџӊ⿄Ўᖙ✊џӊˊᰒ✊ˈḋᴀぎ䯈ᰃᖙ
1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ ✊џӊˈг䆄ЎΩˊ ϡৃ㛑џӊ↣䆩偠Ёϡৃ㛑থ⫳ⱘџӊ⿄Ўϡৃ㛑џӊˊᅗϡӏԩḋ ᴀ⚍ˈৃ⧚㾷Ўぎ䲚ˈ䆄Ў∅ˊ ⊼ᛣ˖ ď1Đྂ·ࣂԅٲюಾဎೋིԅඨߑۤڔГԅંԅൎࢅՇĕ ď2Đݮ·ಹߑಾಹߑԅྡྷᄵēྡྷ͎ԅಹߑಾဎయؤّݮ·ಹߑ٫ලᆦюԅē ࿙֗ಾྂ·ࣂԅᆐނēඹШရыୣน؏ۦಹߑĕ ď3Đ഻ݯಹߑԅখྡྷّՇ࿌ġྂ·ࣂΩԅ੶ّᆐނĕಹߑA֟ಓӲயࠧӲ ೋིᄯѻຣAᄯͧۃԅ੶ّݮ·ಹߑĕ ಯǃџӊП䯈ⱘ݇㋏䖤ㅫ ϔϾḋᴀぎ䯈ΩЁˈৃҹࣙ䆌ⱘ䱣ᴎџӊˊⷨお䱣ᴎџӊⱘ㾘ᕟˈᕔ ᕔᰃ䗮䖛ᇍㅔऩџӊ㾘ᕟⱘⷨおএথ⦄Ўᴖџӊⱘ㾘ᕟˊЎℸˈ៥Ӏᓩܹџ ӊП䯈ⱘϔѯ䞡㽕݇㋏䖤ㅫˊ⬅Ѣӏϔ䱣ᴎџӊᰃḋᴀぎ䯈ⱘᄤ䲚ˈ᠔ҹџӊ П䯈ⱘ݇㋏ঞ䖤ㅫϢ䲚ড়П䯈ⱘ݇㋏ঞ䖤ㅫᰃᅠܼ㉏Ԑⱘˊ 1ĕಹߑԅͧۃރອԉ བᵰĀџӊ A ⱘথ⫳ᖙ✊ᇐ㟈џӊ B ⱘথ⫳āˈ߭⿄џӊ B ࣙџӊ Aˈг⿄ A ᰃ B ⱘᄤџӊˈ䆄 A୴B B୵Aˊ ՟བˈᦋϔᵮ傄ᄤˈ㾖ᆳߎ⦄ⱘ⚍᭄ˈ݊ḋᴀぎ䯈Ω ={1,2,3,4,5,6}ˈ䆒 A={2,4}ˈB={2,4,6}ˈᰒ✊A ୴ BˈेџӊAᰃџӊBⱘᄤџӊˊ ⊼ᛣ˖ճఉྡྷಹߑA՛ပᆐಹߑڑຂ ∅ ୴ A ୴ Ωˊ ៥Ӏ㒭ߎџӊࣙ݇㋏ⱘϔϾⳈ㾖ⱘԩ㾷䞞˖ᑇ䴶ⶽᔶऎඳ㸼⼎ḋᴀぎ䯈 ΩˈᔶऎඳAϢᔶऎඳBߚ߿㸼⼎џӊAϢџӊBˊ⬅ѢAЁⱘ᠔⫼ḋᴀ⚍ ܼ䚼ሲѢBˈ᠔ҹџӊBࣙџӊAˈ㾕 1-1ˊ 1.1 A ⊂ B བᵰ᳝ A୴B Ϩ B୵Aˈ߭⿄џӊ A Ϣџӊ B Ⳍㄝˈ䆄 A=Bˊ Ω A B
4 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 ᯧⶹˈⳌㄝⱘϸϾџӊ AˈB ᘏᰃৠᯊথ⫳ৠᯊϡথ⫳ˈѺे A=B ㄝӋѢ ᅗӀᰃ⬅Ⳍৠⱘḋᴀ⚍㒘៤ⱘˊ 2ĕಹߑԅۤďωĐ Āџӊ A Ϣ B Ё㟇ᇥ᳝ϔϾџӊথ⫳ā䖭ḋⱘџӊ⿄Ўџӊ A Ϣ B ⱘџ ӊˈ䆄A BĤ ˊ ৃ㾕ˈA BĤ ᰃ⬅᠔᳝ሲѢ A ሲѢ B ⱘḋᴀ⚍㒘៤ˊџӊ A Ϣ B ⱘџӊ A BĤ ᇍᑨ䲚ড় A Ϣ B ⱘᑊ䲚ˊབ 1.2 䰈ᕅ䚼ߚ᠔⼎ˊ 1.2 A BĤ ՟བˈᦋϔᵮ傄ᄤⱘ䆩偠Ёˈ㢹䆒џӊA={2,3}ˈџӊB={1,2}ˈ߭џ ӊЎC A B= Ĥ ={1,2,3}ˈ㸼⼎{ᦋߎⱘ⚍᭄ᇣѢ 4}ˊ џӊৃҹᑓࠄ᳝䰤ϾџӊϢৃ߫ϾџӊПⱘᚙᔶ˖ ᇍѢĀџӊA A1, , ,2 L AnЁ㟇ᇥ᳝ϔϾথ⫳ā䖭ϔџӊˈ៥Ӏ⿄ЎA A1, , ,2 L An ⱘџӊˈ⫼A A1Ĥ Ĥ Ĥ2 L An㸼⼎ˈㅔ䆄Ў 1 n i iU= Aˊ ᇍѢĀৃ߫ϾџӊA A1, , , ,2 L An LЁ㟇ᇥ᳝ϔϾথ⫳ā䖭ϔџӊˈ៥Ӏ⿄Ў 1, , , ,2 n A A L A Lⱘџӊˈ⫼A A1Ĥ Ĥ Ĥ Ĥ2 L An L㸼⼎ˈㅔ䆄Ў 1 i i A ∞ = U ˊ 3ĕಹߑԅݲď߬Đ Āџӊ A Ϣ B ৠᯊথ⫳ā䖭ḋⱘџӊ⿄џӊ A Ϣ B ⱘ⿃˄Ѹ˅џӊˈ䆄 A Bģ ABˊ ABᰃ⬅᮶ሲѢ A জሲѢ B ⱘḋᴀ⚍㒘៤ˊབᵰᇚџӊ⫼䲚ড়㸼⼎ˈ߭џ ӊ A Ϣ B ⱘ⿃џӊ AB ᇍᑨ䲚ড় A Ϣ B ⱘѸ䲚ˊ݊ԩᛣНབ 1-3 Ёⱘ䰈ᕅ 䚼ߚ᠔⼎ˊ ՟བˈᦋ傄ᄤ䆩偠Ёˈ㢹䆒џӊ A={1,3,5}ˈџӊB={1,2}ˈ߭⿃џӊЎ {1} C A B= I = ˈ㸼⼎{ᦋߎⱘ⚍᭄ᰃ 1 ⚍}ˊ Ω A B
1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ 1.3 A Bģ ㉏Ԑഄˈгৃҹᇚ⿃џӊᑓࠄ᳝䰤ϾϢৃ߫ϾџӊП⿃ⱘᚙᔶ˖ ⫼A A1ģ ģ ģ2 L An 1 n i iI= A㸼⼎A A1, ,2 L,Anৠᯊথ⫳ⱘџӊ˗ ⫼ 1 i i A ∞ = I 㸼⼎A A1, , , ,2 L An Lৠᯊথ⫳ⱘџӊˊ 4ĕಹߑԅЕ Āџӊ A থ⫳㗠џӊ B ϡথ⫳ā䖭ḋⱘџӊ⿄Ўџӊ A Ϣ B ⱘᏂџӊˈ䆄 A B− ˊ A B− ᰃ⬅᠔᳝ሲѢ A 㗠ϡሲѢ B ⱘḋᴀ⚍㒘៤ˈ݊ԩᛣНབ 1.4 Ёⱘ䰈 ᕅ䚼ߚ᠔⼎ˊ 1.4 A B− ՟བˈᦋ傄ᄤ䆩偠Ёˈ㢹䆒џӊA ={2,3,4}ˈџӊB ={1, 2}ˈ߭Ꮒџӊ 3,4 { } A B− = ˊ 5ĕಹߑܚϢອఘ Āџӊ A Ϣџӊ B ϡ㛑ৠᯊথ⫳āˈгህᰃ䇈ˈAB ᰃϔϾϡৃ㛑џӊˈे AB = ∅ˈ ℸᯊ⿄џӊ A Ϣ B ᰃѦϡⳌᆍⱘ˄Ѧ᭹ⱘ˅ˊ 䗮ᐌᡞϸϾѦϡⳌᆍⱘџӊ A Ϣ B ⱘᑊ䆄 A B+ ˊ Ω A B B A Ω
6 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 A Ϣ B ѦϡⳌᆍㄝӋѢᅗӀ≵᳝Ⳍৠⱘḋᴀ⚍ˈे≵᳝݀݅ⱘḋᴀ⚍ˊ㢹⫼ 䲚ড়㸼⼎џӊˈ߭ A Ϣ B ѦϡⳌᆍेЎ A Ϣ B ᰃϡⳌѸⱘˈབ 1.5 ᠔⼎ˊ 1.5 AB = ∅ བᵰ n ϾџӊA A1, ,2 L,AnЁˈӏᛣϸϾџӊ䛑ϡৃ㛑ৠᯊথ⫳ˈे i j A A = ∅I ˈi j≠ ˈi j, =1,2, ,L n. ߭⿄䖭nϾџӊA A1, ,2 L,AnᰃϸϸѦϡⳌᆍⱘ˄ϸϸѦ᭹ⱘ˅ˊ 䗮ᐌᡞnϾѦϡⳌᆍⱘџӊA A1, ,2 L,Anⱘᑊ䆄 1 2 n A A+ + +L A ˄ㅔ䆄Ў 1 n i i A =
∑
˅. ᆍᯧⳟߎˈ䱣ᴎ䆩偠ЁˈӏԩϸϾϡৠⱘᴀџӊ䛑ᰃѦϡⳌᆍⱘ˖ӏϔ џӊ A ϢB A− ᰃѦϡⳌᆍⱘˈे ( ) AģB A− = ∅ˊ 6ĕճोಹߑďಹߑĐ 㢹 A ᰃϔϾџӊˈҸ A= Ω −Aˈ⿄AᰃAⱘᇍゟџӊˈ⿄Ўџӊ A ⱘ 䗚џӊˊ гህᰃ䇈ˈ A ᰃ⬅ḋᴀぎ䯈ΩЁ᠔᳝ϡሲѢ A Ёⱘḋᴀ⚍ᵘ៤ⱘˊབᵰᡞџ ӊ A ⳟ䲚ড়ˈ䙷МAህᰃ A ⱘ㸹䲚ˊ 1.6 Ёⱘ䰈ᕅ䚼ߚ㸼⼎ A ˊ 1.6 A ᰒ✊ˈϔ䆩偠Ёˈ㢹 A থ⫳ˈ߭Aᖙϡথ⫳ˈডПѺ✊˗A ϢAЁᖙ✊ ᳝ϔϾথ⫳ˈϨҙ᳝ϔϾথ⫳ˈेџӊ A ϢA⒵䎇݇㋏ A A Ω B A Ω 1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ A A = ∅ģ ˈ A A+ = Ωˊ ᇍӏᛣⱘџӊ AˈBˈ᳝ A B AB− = ˊ ᖙ✊џӊΩϢϡৃ㛑џӊ∅ᰃᇍゟџӊˈৠᯊজᰃѦϡⳌᆍџӊˊ ⊼ᛣ˖యಹߑ AēB ܚนճोಹߑē႕ಹߑ AēB ΡܚϢອఘĢӬಾēయಹߑ AēB ܚϢອఘē႕ಹߑ AēB รΡಾܚนճोಹߑĕ ՟བˈᦋ傄ᄤ䆩偠Ёˈ㢹A ={1,2}ˈB ={3,5}ˈ߭ A Ϣ B ѦϡⳌᆍˊԚᰃˈ џӊ B ϡᰃ A ⱘᇍゟџӊˈA ⱘᇍゟџӊA ={3,4,5,6}ˊ 7ĕಹߑᆦ ᇚџӊ A ϢAⱘ݇㋏ᑓࠄ n Ͼџӊⱘᚙᔶ˖ བᵰ n ϾџӊA A1, ,2 L,AnЁ㟇ᇥ᳝ϔϾথ⫳ˈϨӏᛣϸϾџӊϡৃ㛑ৠᯊথ ⫳ˈे 1 2 n A AĤ Ĥ ĤL A =Ωˈ A Aiģ j=∅ˈi j≠ ˈi j, =1,2, ,L nˈ ߭⿄䖭 n ϾџӊA A1, ,2 L,Anᵘ៤ϔϾᅠџӊ㒘ˈজ⿄Ўḋᴀぎ䯈ΩⱘϔϾߦ ߚˊ㾕 1.7ˊ 1.7 ḋᴀぎ䯈 Ω ⱘϔϾߦߚ ՟བˈᦋ傄ᄤ䆩偠ЁˈџӊA ={1,2}ˈB ={3,5}ˈC ={4,6}ᵘ៤ϔϾᅠ џӊ㒘ˊ Ѩǃџӊ䖤ㅫ⊩߭ ⬅џӊ݇㋏Ϣ䖤ㅫⱘᅮНৃҹⳟߎˈᅗӀϢ䲚ড়ⱘ݇㋏Ϣ䖤ㅫᰃϔ㟈ⱘˊ ℸˈ䲚ড়ⱘ䖤ㅫᗻ䋼ᇍџӊⱘ䖤ㅫг䛑䗖⫼ˊ џӊⱘ䖤ㅫ⊩᳝߭˖ 1ˊѸᤶᕟ A B B AĤ = Ĥ ˈAB BA= ˊ ˄1.1.1˅ 2. 㒧ড়ᕟ A B C =U U (A B C A B CU )U = U( U )ˈ ˄1.1.2˅
8 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 ( ) ( ) ABC= AB C A BC= . ˄1.1.3˅ 3. ߚ䜡ᕟ (AB C)Ĥ =(A C B CĤ )( Ĥ )ˈ ˄1.1.4˅ (A B CU ) =(AC) (U BC). ˄1.1.5˅ 4. ᖋᨽḍ݀ᓣ A B =Ĥ A BˈAB = A BĤ ˈ ˄1.1.6˅ ᇍѢ n ϾџӊA A1, , ,2 L Anˈ᳝ 1 1 1 1 n n n n i i i i iU= A =iI= A ˈǂiI= A =iU= A . ˄1.1.7˅ ՟ 6 䆒 AˈBˈC ᰃϝϾџӊˈ䆩⫼ AˈBˈC ⱘ䖤ㅫ݇㋏㸼⼎ϟ߫џӊ. ˄1˅BˈC 䛑থ⫳ˈ㗠Aϡথ⫳˗ ˄2˅AˈBˈC Ё㟇ᇥ᳝ϔϾথ⫳˗ ˄3˅AˈBˈC Ёᙄ᳝ϔϾথ⫳˗ ˄4˅AˈBˈC Ёᙄ᳝ϸϾথ⫳˗ ˄5˅AˈBˈC ЁϡѢϔϾথ⫳˗ ˄6˅AˈBˈC ЁϡѢϸϾথ⫳. 㾷 ˄1˅ĀBˈC 䛑থ⫳ˈ㗠Aϡথ⫳ā㸼⼎Ў˖ABC˗ ˄2˅ĀAˈBˈC Ё㟇ᇥ᳝ϔϾথ⫳ā㸼⼎Ў˖A B CĤ Ĥ ˗ ˄3˅ĀAˈBˈC Ёᙄ᳝ϔϾথ⫳ā㸼⼎Ў˖ABC ABC ABCĤ Ĥ ˗ ˄4˅ĀAˈBˈC Ёᙄ᳝ϸϾথ⫳ā㸼⼎Ў˖ABC ABC ABCĤ Ĥ ˗
˄5˅ĀAˈBˈC ЁϡѢϔϾথ⫳ā㸼⼎Ў˖ABC ABC ABC ABCĤ Ĥ Ĥ ˗ ˄6˅ĀAˈBˈCЁϡѢϸϾথ⫳ā㸼⼎Ў˖ABCˈ㗙A B CĤ Ĥ .
1.2 ໜऐ๚ॲڦ߁୲
៥Ӏ㾖ᆳϔ乍䱣ᴎ䆩偠᠔থ⫳ⱘϾ㒧ᵰˈህ݊ϔԧⱘ䆩偠㗠㿔ˈ↣ϔ џӊߎ⦄Ϣ৺䛑ᏺ᳝ᕜⱘي✊ᗻˈԐТ≵᳝㾘ᕟৃ㿔. Ԛᰃ䞣ⱘ䞡䆩偠 ৢˈህӮথ⦄˖ᶤѯџӊথ⫳ⱘৃ㛑ᗻѯˈϔѯџӊথ⫳ⱘৃ㛑ᗻᇣѯˈ 㗠᳝ѯџӊথ⫳ⱘৃ㛑ᗻ㟈Ⳍৠ. ՟བˈϔϾㆅᄤЁ㺙᳝ 100 ӊѻકˈ݊Ё 95 ӊᰃড়Ḑકˈ5 ӊᰃક. Ң݊Ёӏᛣপߎϔӊˈ߭পࠄড়Ḑકⱘৃ㛑ᗻህ↨পࠄ કⱘৃ㛑ᗻ. ؛བ䖭 100 ӊѻકЁⱘড়ḐકϢક䛑ᰃ 50 ӊˈ߭পࠄড়Ḑક Ϣপࠄકⱘৃ㛑ᗻህᑨ䆹Ⳍৠ. ᠔ҹˈϔϾџӊথ⫳ⱘৃ㛑ᗻᇣᰃᅗᴀ䑿᠔ ᳝ⱘϔ⾡ᅶ㾖ⱘᑺ䞣. ᕜ㞾✊ˈҎӀᏠᳯ⫼ϔϾ᭄ᴹᦣ䗄џӊথ⫳ⱘৃ㛑ᗻᇣˈ 㗠Ϩџӊথ⫳ⱘৃ㛑ᗻˈ䖭Ͼ᭄ህ˗џӊথ⫳ⱘৃ㛑ᗻᇣˈ䖭Ͼ᭄ህᇣ. Ўℸˈ៥Ӏ佪ܜᓩܹĀ乥⥛āⱘὖᗉˈᅗᦣ䗄њџӊⳌৠᴵӊϟ䞡 䆩偠᠔থ⫳ⱘ乥㐕ᑺˈ䖯㗠ᓩߎ㸼ᕕџӊϔ䆩偠Ёথ⫳ⱘৃ㛑ᗻᇣⱘ᭄ 䞣ᣛᷛ——ὖ⥛. ϔǃ乥⥛ ᅮН 1 Ⳍৠⱘᴵӊϟˈ䖯㸠њ n 䆩偠ˈ䖭 n 䆩偠ЁˈџӊAথ⫳ⱘ 1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ ᭄nA⿄ЎџӊAথ⫳ⱘ乥᭄˗↨ؐ A n n ⿄ЎџӊAথ⫳ⱘ乥⥛ˈᑊ䆄Ў f An( ). ՟ 1 ৠḋᴵӊϟˈᡯϔᵮ䋼ഄഛࣔⱘ⹀Ꮥˈ㗗ᆳĀℷ䴶ᳱϞāⱘ᭄. 䖭Ͼ䆩偠ग़Ϟ᳒㒣᳝Ҏخ䖛ˈᕫࠄབ㸼 1-1 ᠔⼎ⱘ᭄. 㸼 1-1 ᡩᦋ⹀Ꮥ䆩偠᭄ ᅲ偠㗙 ᡩᦋ᭄ n ߎ⦄ℷ䴶᭄nA˄乥᭄˅ 乥⥛nA n ᖋgᨽḍ 2048 1061 0.518 㪆 Є 4040 2048 0.5069 K·Ⲃᇨ䗞 12000 6019 0.5016 K·Ⲃᇨ䗞 24000 12012 0.5005 ՟ 1 Ёˈৃ㾕Āℷ䴶ᳱϞāⱘ乥⥛ 0.5 䰘䖥ᨚࡼˈᔧ n ᯊˈ䗤⏤〇ᅮ Ѣ 0.5. ᰒ✊ˈ乥⥛᳝ϟ߫ᗻ䋼˖ ᗻ䋼 1 䴲䋳ᗻ 0İ f An( )İ1. ᗻ䋼 2 㾘㣗ᗻ 䆒ΩЎᖙ✊џӊˈ߭ f Ω =n( ) 1. ᗻ䋼 3 ৃࡴᗻ 㢹џӊ AˈB ѦϡⳌᆍˈ߭ ( ) n f A BĤ = f An( )+ f Bn( ). 㒣偠㸼ᯢ˖㱑✊ n 䆩偠ЁˈџӊAߎ⦄ⱘ᭄nAϡ⹂ᅮˈ㗠џӊAⱘ 乥⥛nA n гϡ⹂ᅮˈԚᰃᔧ䆩偠䞡ᯊˈџӊAߎ⦄ⱘ乥⥛᳝ϔᅮⱘ〇ᅮᗻ. 䖭ህᰃ䇈ˈᔧ䆩偠᭄ܙߚᯊˈџӊAߎ⦄ⱘ乥⥛ϔϾᐌ᭄䰘䖥ᨚࡼ. 䖭⾡乥 ⥛ⱘ〇ᅮᗻˈ䇈ᯢ䱣ᴎџӊথ⫳ⱘৃ㛑ᗻᇣᰃџӊᴀ䑿᳝ⱘˈ⫼䖭Ͼᐌ᭄ᴹ 㸼⼎џӊAথ⫳ⱘৃ㛑ᗻᇣ↨䕗ᙄᔧ. 䖭ᰃ៥Ӏϟ䴶㒭ߎὖ⥛ⱘ㒳䅵ᅮНⱘᅶ 㾖⸔. Ѡǃὖ⥛ⱘ㒳䅵ᅮН ᅮН 2 䆩偠ᴵӊϡবⱘᚙމϟˈ䞡خ n 䆩偠ˈᔧ䆩偠᭄ n ܙߚᯊˈ џӊAথ⫳ⱘ乥⥛nA n 〇ᅮࠄᶤϔᐌ᭄pˈ߭⿄䖭Ͼᐌ᭄pЎџӊAϔ䆩偠Ё থ⫳ⱘὖ⥛ˈ䆄P A( ). े ( ) P A = p. ᭄P A( )ህᰃϔ䆩偠ЁᇍџӊAথ⫳ⱘৃ㛑ᗻᇣⱘϔ⾡᭄䞣ᦣ䗄. ៥Ӏ ⿄ᅮН 2 ᰃὖ⥛ⱘ㒳䅵ᅮН. ՟བˈ՟ 1 Ё⫼ 0.5 ᴹᦣ䗄ᦋϔᵮࣔ䋼⹀ᏕĀℷ䴶 ᳱϞāߎ⦄ⱘৃ㛑ᗻᇣ.
10 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 ϝǃὖ⥛ⱘᗻ䋼 ⬅ὖ⥛ⱘᅮНˈৃҹᕫὖ⥛ⱘϔѯ䞡㽕ᗻ䋼. ᗻ䋼 1 ᇍѢӏᛣџӊAˈ䛑᳝ 0İP A( ) 1İ . ˄1.2.1˅ ᗻ䋼 2 ( ) 1 P Ω = ˈP ∅ =( ) 0. ˄1.2.2˅ ᗻ䋼 3 ˄᳝䰤ৃࡴᗻ˅ᇍѢϸϸѦϡⳌᆍⱘn ϾџӊA A1, , ,2 L Anˈ᳝߭ 1 2 (A A An) P Ĥ Ĥ ĤL = P A( )1 +P A( )2 + +L P A( )n . ˄1.2.3˅ ⡍߿ഄˈᇍѢѦϡⳌᆍџӊAˈBˈ᳝ ( ) P A B =Ĥ P A P( )+ ( )B . ˄1.2.4˅ ᗻ䋼 4 ˄ὖ⥛ޣ⊩݀ᓣ˅䆒AˈBЎӏᛣϸϾџӊˈ᳝߭ ( ) P B A− = (P B AB− )= ( )P B −P AB( ). ˄1.2.5˅ ⡍߿ഄˈ㢹џӊA B⊂ ˈ᳝߭ ( ) P B A− = ( )P B −P A( ). ˄1.2.6˅ ᗻ䋼 5 ˄ᇍゟџӊⱘὖ⥛˅䆒 A ᰃ䱣ᴎџӊAⱘᇍゟџӊˈ᳝߭ ( ) P A = −1 P A( ). ˄1.2.7˅ 䆕 ЎA AU =ΩˈϨ AA =∅ˈ⬅㾘㣗ᗻ᳝䰤ৃࡴᗻᕫࠄ 1= Ω =P( ) P A A( + )= ( )P A P A+ ( ). ेᕫ ( )P A = −1 P A( ). ⡍߿ഄˈP A A( 1Ĥ Ĥ Ĥ2 L A =n) 1−P A A( 1Ĥ Ĥ Ĥ2 L An) 1 2 1 P A A( An). = − L ˄1.2.8˅ ᗻ䋼 6 ˄ὖ⥛ϔ㠀ࡴ⊩݀ᓣ˅ᇍѢӏᛣⱘџӊAˈBˈ᳝ ( ) P A B =Ĥ P A P B P AB( )+ ( )− ( ). ˄1.2.9˅ ᑓ ˄བ 1.8 ᠔⼎˅䆒A A A1, ,2 3ᰃϝϾџӊˈ᳝߭ 1 2 3 ( ) P A AĤ ĤA = P A( )1 +P A( )2 +P A( )3 −P A A( 1 2)−P A A( 1 3) 2 3 1 2 3 ( ) ( ) P A A P A A A − + . ˄1.2.10˅ 1.8 ϝϾџӊⱘὖ⥛݀ᓣ⼎㾷䞞 Ω 1 A A A1 2 A2 3 A 1 3 A A A A2 3 1 2 3 A A A
1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ 1 1 1 1 2 ( ) ( ) ( ) ( ) n i i j i j k i i j n i j k n n P A A A P A P A A P A A A = < < < =
∑
−∑
+∑
L İ İ İ İ Ĥ Ĥ Ĥ 1 2 1 ( 1)n ( ) n P A A A − − + +L L . ˄1211˅ ՟ 2 Ꮖⶹ ( ) 0.4P A = ˈ (P AB =) 0.3ˈP B =( ) 0.6ˈ∖ ˄1˅P AB( )˗ ˄2˅P A B( U )˗ ˄3˅P AB( ). 㾷 ˄1˅Ў AB AB B+ = ϨABϢAB ᰃϡⳌᆍⱘˈᬙ᳝ (P AB P AB)+ ( )=P B( )ˈ ѢᰃP AB( ) =P B P AB( )− ( )=0.6 0.3− =0.3˗ ˄2˅P A( )= −1 P A( ) = −1 0.4 =0.6ˈ P A B( U ) =P A P B P AB( )+ ( )− ( ) =0.6 0.6 0.3 0.9+ − = ˗ ˄3˅P AB( ) =P A B( Ĥ )= −1 P A B( Ĥ ) = −1 0.9 0.1= . ՟ 3 ᶤ䰶᷵Ѡᑈ㑻ᄺ⫳ϔᄺᳳᓔ䆒AˈBϸ䮼䗝ׂ䇒. Ꮖⶹᄺ⫳䗝ׂA䇒 ⱘ᳝ 45%ˈ䗝ׂB䇒ⱘ᳝ 35%ˈϸ䮼䇒䛑䗝ⱘ᳝ 10%. ⦄Ң䆹᷵Ѡᑈ㑻ӏ 䗝ϔᄺ⫳ˈ∖˖ ˄1˅䆹⫳㟇ᇥ䗝ׂϔ䮼䇒ⱘὖ⥛˗ ˄2˅䆹⫳া䗝ׂϔ䮼䇒ⱘὖ⥛. 㾷 䆒A ={䗝ׂA䮼䇒}ˈB ={䗝ׂB䮼䇒}ˈ߭ A B =Ĥ {㟇ᇥ䗝ׂϔ 䮼䇒} . ˄1˅P A B =( Ĥ ) P A P B( )+ ( )−P AB( )= 0.45 0.35 0.1 0.7+ − = ˄2˅ AB AB+ = {া䗝ׂϔ䮼䇒}ˈ (P AB AB+ )=P A B P AB( Ĥ )− ( ) =0.7 0.1 0.6− = .1.3 ࠟۆ߁୲
সҷˈҎӀ߽⫼ⷨおᇍ䈵ⱘ⠽⧚ԩᗻ䋼᠔᳝ⱘᇍ⿄ᗻ⹂ᅮњ䅵ㅫὖ ⥛ⱘϔ⾡ᮍ⊩. ՟བˈᡯᦋ⹀Ꮥ䆩偠ЁˈҸω1㸼⼎Āߎ⦄ℷ䴶āˈω2㸼⼎Āߎ⦄ড䴶āˈ߭ ḋᴀぎ䯈ΩЁϸϾᴀџӊ{ }ω1 { }ω2 থ⫳ⱘৃ㛑ᗻᰃⳌㄝⱘˈ㗠ৃҹ㾘ᅮ 1 ( ) P ω = ( 2) 1 2 P ω = . ेĀߎ⦄ℷ䴶āĀߎ⦄ড䴶āⱘὖ⥛ऴϔञ. ϟ䴶៥Ӏ㒭ߎㄝৃ㛑ὖൟⱘᅮНঞ݊ὖ⥛䅵ㅫ݀ᓣ. ϔǃসὖൟঞ݊ὖ⥛䅵ㅫ ᅮН 1 བᵰ䱣ᴎ䆩偠E⒵䎇ϟ䗄ᴵӊ˖ ˄1˅᳝䰤ᗻ˖䆩偠᠔ⱘᴀџӊϾ᭄ᰃ᳝䰤Ͼˈेḋᴀぎ䯈ⱘḋᴀ⚍া᳝ ᳝䰤Ͼ˗12 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 ˄2˅ㄝৃ㛑ᗻ˖↣Ͼᴀџӊথ⫳ⱘৃ㛑ᗻᰃⳌৠⱘ. ߭⿄䖭Ͼ䆩偠Ўসὖൟˈজ⿄Ўㄝৃ㛑ὖൟ. ᅮ⧚ সὖൟЁˈӏϔ䱣ᴎџӊA᠔ࣙⱘᴀџӊ᭄ m Ϣḋᴀぎ䯈Ω ᠔ࣙⱘᴀџӊᘏ᭄n ⱘ↨ؐˈ⿄Ў䱣ᴎџӊAⱘὖ⥛ˈे ( ) P A = A m n Ω = џӊ ࣙⱘᴀџӊ᭄ ࣙⱘᴀџӊᘏ᭄ . ˄1.3.1˅ ϞᓣህᰃসὖൟЁџӊAⱘὖ⥛䅵ㅫ݀ᓣ. ᇍѢসὖൟᑨ⊼ᛣ˖䅵ㅫসὖൟὖ⥛ᯊˈ佪ܜ㽕߸ᮁĀ᳝䰤ᗻāĀㄝ ৃ㛑ᗻāᰃ৺⒵䎇.Ā᳝䰤ᗻā䕗ᆍᯧⳟߎˈĀㄝৃ㛑ᗻā䳔㽕ḍᅲ䰙䯂乬ᴹ߸ᅮ. ݊㽕ᓘ⏙Ἦḋᴀぎ䯈ᰃᗢḋᵘ៤ⱘˈҢ㗠∖ߎᴀџӊⱘᘏ᭄n ˈৠᯊ∖ߎ᠔䅼 䆎џӊAࣙⱘᴀџӊ᭄ m ˈ✊ৢ߽⫼সὖ⥛䅵ㅫ݀ᓣ∖ᕫP A( ). ϟ䴶䗮䖛ԧ՟ᄤᴹᄺдসὖ⥛ⱘ䅵ㅫ. ՟ 1 Ңϔᡍ⬅ 45 ӊℷકǃ5 ӊક㒘៤ⱘѻકЁӏᛣᢑপ 3 ӊˈ∖পࠄ 2 ӊℷક 1 ӊકⱘὖ⥛. 㾷 䆒A={পࠄ 2 ӊℷક 1 ӊક}. Ң 50 ӊѻકЁӏপ 3 ӊ᳝݅ 3 50 C ⾡ϡৠপ⊩ˈे䆩偠᠔ⱘᴀџӊᘏ᭄ᰃ 3 50 C ϾˈџӊAࣙⱘᴀџӊ᭄ᰃ 2 1 45 5 C C Ͼ. 1 2 5 45 3 50 C C ( ) 0.252 C P A ∴ = ≈ ϔ㠀ഄˈ䆒᳝Nӊѻકˈ݊Ё᳝M ӊકˈ⦄ҢЁӏপn n˄ İ ˅ӊ˄ϡᬒN ಲ˅ˈ߭䖭 n ӊЁᙄ᳝ m n˄ İ ˅M ӊકⱘὖ⥛ᰃ C C C m n m M N M n N p= −− . ՟ 2 ⲦЁ᳝ 12 Ͼ⧗ˈ݊Ё 8 Ͼⱑ⧗ˈ4 Ͼ咥⧗. ⦄ҢЁӏপ 3 Ͼ. ∖˖ ˄1˅পࠄⱘ䛑ᰃⱑ⧗ⱘὖ⥛˗ ˄2˅পࠄϸϾⱑ⧗ˈϔϾ咥⧗ⱘὖ⥛˗ ˄3˅পࠄⱘϝϾ⧗Ё㟇ᇥ᳝ϔϾ咥⧗ⱘὖ⥛˗ ˄4˅পࠄⱘϝϾ⧗买㡆䛑Ⳍৠⱘὖ⥛. 㾷 䆒A =1 {পࠄⱘ䛑ᰃⱑ⧗}ˈA =2 {পࠄϸϾⱑ⧗ˈϔϾ咥⧗}ˈA =3 {পࠄ ⱘϝϾ⧗Ё㟇ᇥ᳝ϔϾ咥⧗}ˈA =4 {পࠄⱘ䛑ᰃ咥⧗}ˈB ={পࠄⱘϝϾ⧗买㡆 䛑Ⳍৠ}. ձ乬ᛣˈᕫࠄ ˄1˅ 83 1 3 12 ( ) C 0.255 C P A = = ˗ ˄2˅ 82 14 2 3 12 ) C C ( 0.509 C P A = = ˗ ˄3˅P A( 3)=1−P A( 3)=1−P A( 1)= −1 0.255 0.745= ˗
1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ ˄4˅ᰒ✊B A A= Ĥ1 4. ЎA1ϢA4ѦϡⳌᆍˈϨ 3 4 4 3 12 ) C ( 0.018 C P A = = ˈ᠔ҹ P B( )=P A( 1)+P A( 4)=0.255 0.018 0.273.+ = ՟ 3 ˄ᢑㅒⱘ݀ᑇᗻ˅ⲦЁ᳝a Ͼ咥⧗ˈ b Ͼⱑ⧗ˈᡞ⧗䱣ᴎഄϔাাপ ߎ˄ϡᬒಲ˅ˈ∖џӊAĀ k ˄ 0 k< İa b+ ˅পࠄ咥⧗āⱘὖ⥛. 㾷 ᮍ⊩ 1 ᡞ a + b Ͼ⧗㓪Ϟ 1 㟇 a + b োˈᇚ⧗ϔাϔাপߎৢᥦ៤ϔᥦˈ 㗗㰥ࠄপ⧗ⱘܜৢ乎ᑣˈℸ᳝݅(a b+ )!⾡প⊩ˈ⬅⧗ⱘഛࣔᗻⶹ↣⾡প⊩ᴎӮ 䛑ⳌৠˈᬙሲѢসὖൟˈAথ⫳ৃҹܜҢ a Ͼ咥⧗ЁӏপϔϾᬒ k Ͼԡ㕂Ϟˈ ✊ৢᇚ࠽ϟⱘa b+ −1Ͼ⧗䱣ᛣᥦa b+ −1Ͼԡ㕂Ϟˈ᳝݅C (1 1)! a a b+ − ⾡ᥦ ⊩ˈᬙ 1 C ( 1)! ( ) ( )! a a b a P A a b a b + − = = + + . ᮍ⊩ 2 া㗗㰥 k প⧗ˈ↣Ͼ⧗ҡ✊㓪᳝ϡৠⱘোⷕˈ⬅Ѣ↣Ͼ⧗䛑᳝Ⳍৠ ⱘᴎӮ㹿ᬒ k Ͼԡ㕂Ϟˈa + b Ͼϡৠⱘ⧗ˈᘏ᳝݅ a + b ⾡ᬒ⊩ˈᣝসὖൟˈA থ⫳ᖙ乏ᰃa Ͼ咥⧗ЁⱘϔϾᬒ k Ͼԡ㕂ϞˈेAথ⫳ⱘᬒ⊩᳝a ⾡ˈᬙ ( ) a P A a b = + . ℸ䯂乬᳝⾡ϡৠⱘ㗗㰥ᮍᓣˈᔧ✊㒧ᵰ䛑Ⳍৠˈk পࠄ咥⧗ⱘὖ⥛Ϣ k ᮴݇. ℸ՟䇈ᯢӴ㒳ⱘᢑㅒⱘ㒧ᵰϢܜৢ乎ᑣ᮴݇. ՟ 4 1 ~ 100݅ϔⱒϾ᭄ЁӏপϔϾ᭄ˈ∖˖˄1˅䖭Ͼ᭄㛑㹿2ᭈ䰸ⱘὖ⥛˗ ˄2˅䖭Ͼ᭄㛑㹿235ᭈ䰸ⱘὖ⥛. 㾷 䆒A={䖭Ͼ᭄㛑㹿2ᭈ䰸}˗B={䖭Ͼ᭄㛑㹿3ᭈ䰸}˗C={䖭Ͼ᭄㛑㹿5 ᭈ䰸}. ߭ ˄1˅䖭Ͼ᭄㛑㹿2ᭈ䰸ⱘὖ⥛Ў˖ 50 ( ) 0.5 100 P A = = ˗ ˄2˅䖭Ͼ᭄㛑㹿235ᭈ䰸ⱘὖ⥛Ў˖ ( B C) P AĤ Ĥ = P A P B( )+ ( )+P C( ) −P AB( )−P AC( ) −P BC( )+P ABC( ) 50 33 20 16 10 6 3 0.74 100 + + − − − + = = . Ѡǃԩὖ⥛ ὖ⥛䆎ⱘথሩ߱ᳳˈҎӀህ䅸䆚ࠄˈҙ؛ᅮḋᴀぎ䯈Ў᳝䰤ḋᴀぎ䯈ᰃϡ ⱘˈ᳝ᯊ䳔㽕໘⧚᳝᮴かϾḋᴀ⚍ⱘᚙᔶ. ៥Ӏܜⳟϟ䴶ϸϾ՟ᄤ. ՟ 5 ऎ䯈[1,6]Ϟ䱣ᴎഄӏᛣѻ⫳ϔϾ᭄xˈ∖xϡѢ3ⱘὖ⥛.
14 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 ՟ 6 䱣ᴎഄऩԡඳݙӏᦋϔ⚍Mˈ∖⚍M ࠄॳ⚍䎱⾏ϡѢ1 5ⱘὖ⥛. ҹϞϸϾ՟ᄤ䛑᳝Āㄝৃ㛑ᗻāⱘ⡍ᕕ. ࠡϔ՟Ёˈ៥Ӏ䅸ЎĀ䱣ᴎ᭄x ऎ䯈[1,6]Ϟӏԩϔ໘ߎ⦄ⱘᴎӮഛㄝāˈা㽕x㨑ܹऎ䯈[1,3]ݙᇍᑨⱘџӊህӮ থ⫳ˈὖ⥛ᑨ䆹Ўऎ䯈[1,3]䭓ᑺϢऎ䯈[1,6]䭓ᑺП↨ˈेὖ⥛ᑨ䆹ㄝѢ2 5˗ৢ ϔ՟Ёˈ៥Ӏৃ䅸ЎĀऩԡඳݙ↣ϔ⚍㹿ᦋࠄⱘᴎӮഛㄝāˈা㽕⚍M 㨑ܹҹॳ ⚍Ўᖗˈҹ1 5ЎञᕘⱘᇣݙᇍᑨⱘџӊህӮথ⫳ˈ݊ὖ⥛ᑨ䆹Ўᇣ䴶⿃Ϣ 䴶⿃П↨ 2 2 1 π 5 π 1 ⎛ ⎞ ⋅⎜ ⎟⎝ ⎠ ⋅ ˈेὖ⥛Ў 1 25. ᦣ䗄䖭ḋϔѯ䱣ᴎ䆩偠ⱘḋᴀぎ䯈Ωˈ䛑ᰃϔϾऎ䯈ऎඳˈ݊ḋᴀ⚍ऎ ඳΩݙ᳝Āㄝৃ㛑ߚᏗāⱘ⡍⚍. 䆒ऎඳA Ω୴ ˈབᵰḋᴀ⚍㨑ܹAЁˈ៥Ӏ ህ䇈џӊAথ⫳њ. 䖭ḋৃҹϟᅮН. ᅮН 2 䆒ḋᴀぎ䯈ΩЎϔϾ᳝䰤ऎඳˈҹSΩ㸼⼎Ωⱘᑺ䞣˄ϔ㓈Ў䭓ᑺˈ Ѡ㓈Ў䴶⿃ˈϝ㓈Ўԧ⿃ㄝ˅. A Ω୴ ᰃΩЁϔϾৃҹᑺ䞣ⱘᄤ䲚ˈSA㸼⼎Aⱘ ᑺ䞣ˈᅮН ( ) SA P A SΩ = ˄1.3.2˅ ЎџӊAথ⫳ⱘὖ⥛ˈ⿄݊Ўԩὖ⥛. ՟ 7 䆒ᶤ⬉ৄ↣ࠄᭈ⚍ഛᯊ. ϔҎᮽϞ䝦ᴹৢᠧᓔᬊ䷇ᴎ. ∖Ҫㄝᕙᯊ䯈 ϡ䍙䖛 10 ߚ䩳ህ㛑ࠄ䆹⬉ৄᯊⱘὖ⥛. 㾷 ᰒ✊ˈḋᴀぎ䯈Ω=[0,60]˄ऩԡ˖ߚ䩳˅. 䆒A㸼⼎Āㄝᕙᯊ䯈ϡ䍙䖛 10 ߚ䩳āˈ߭A =[50,60]. Ң㗠 ( ) SA P A SΩ = 10 1 60 6 = = .
1.4 ཉॲ߁୲
ϔǃᴵӊὖ⥛ϢЬ⊩݀ᓣ 㞾✊⬠ঞҎ㉏ⱘ⌏ࡼЁˈᄬⴔ䆌ѦⳌ㘨㋏ǃѦⳌᕅડⱘџӊ. 䰸њ㽕ߚ ᵤ䱣ᴎџӊAথ⫳ⱘὖ⥛P A( )ˈ᳝ᯊ៥Ӏ䖬㽕ᦤߎ䰘ࡴⱘ䰤ࠊᴵӊˈгህᰃ㽕 ߚᵤĀџӊBᏆ㒣থ⫳ⱘࠡᦤϟџӊAথ⫳ⱘὖ⥛āˈ៥Ӏ䆄ЎP A B( | ). 䖭ህ ᰃᴵӊὖ⥛䯂乬. 1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ ᓩ՟ ᶤ⧁᳝ 40 ৡᄺ⫳ˈ݊Ё 25 ৡ⬋⫳ˈ15 ৡཇ⫳˗䖭 40 ৡᄺ⫳Ё䑿催 1.72 ㉇ҹϞⱘ᳝ 22 ৡˈ݊Ё 20 ৡ⬋⫳ˈ2 ৡཇ⫳. ˄1˅ӏ䗝ϔৡᄺ⫳ˈ䯂䆹ᄺ⫳ⱘ䑿催 1.72 ㉇ҹϞⱘὖ⥛ᰃᇥ? ˄2˅ӏ䗝ϔৡᄺ⫳ˈ䗝ߎᴹৢথ⦄ᰃϔԡ⬋⫳ˈ䯂䆹ৠᄺⱘ䑿催 1.72 ㉇ҹ Ϟⱘὖ⥛ᰃᇥ? ߚᵤ 䆒 A ={ӏ䗝ϔৡᄺ⫳ˈ䆹ᄺ⫳䑿催 1.72 ㉇ҹϞ}ˈ B ={ӏ䗝ϔৡᄺ⫳ˈ䆹ᄺ⫳ᰃ⬋⫳} . ৃҹⳟࠄˈϔϾ䯂乬∖ⱘᰃP A( )ˈ㗠ѠϾ䯂乬ˈᰃĀᏆⶹџӊBথ⫳ā ⱘ䰘ࡴᴵӊϟˈ∖Aথ⫳ⱘὖ⥛ˈे∖ⱘᰃP A B( | ).Ѣᰃ᳝ 22 ( ) 0.55 40 P A = = ˈ ( | ) 20 0.8 25 P A B = = . ᑊϨᆍᯧⳟߎ˖ ( ) 25 40 P B = ˈ ( ) 20 40 P AB = ˈ Ң㗠 ( | ) 20 20 / 40 ( ) 25 25/ 40 ( ) P AB P A B P B = = = . ⬅ℸ៥Ӏৃҹ㒭ߎᴵӊὖ⥛ⱘϔ㠀ᅮН. ᅮН 1 䆒AˈBᰃϸϾ䱣ᴎџӊˈϨP B >( ) 0ˈ⿄ ( ) ( | ) ( ) P AB P A B P B = . ˄1.4.1˅ ЎᏆⶹџӊBথ⫳ⱘᴵӊϟˈџӊAথ⫳ⱘᴵӊὖ⥛A݇ѢBⱘᴵӊὖ⥛. ৠ⧚ˈᔧP A >( ) 0ᯊˈгৃ㉏ԐഄᅮНB݇ѢAⱘᴵӊὖ⥛ˈे ( ) ( | ) ( ) P AB P B A P A = ˈ ⬅䖭ϾᅮНৃⶹˈᇍӏᛣϸϾџӊAˈBˈ᳝ ( ) ( ) ( | ), ( ) 0 P AB =P B P A B P B > ˈ ˄1.4.2˅ ㉏Ԑഄ᳝ P AB( )=P A P B A( ) ( | ), P A( ) 0> ˈ ⿄ҹϞϸᓣЎὖ⥛ⱘЬ⊩݀ᓣ. ὖ⥛ⱘЬ⊩݀ᓣৃҹᑓࠄ᳝䰤Ͼџӊ⿃ⱘᚙᔶ˖ 㢹 1 1 (n i) 0 i P − A = >
I
ˈ߭ 1 2 3 1 2 1 3 1 2 ( ) ( ) ( | ) ( | ). P A A A =P A P A A P A A A ˄143˅ 1 2 3 1 2 1 3 1 2 1 2 1 ( n) ( ) ( | ) ( | ) ( |n n ). P A A ALA =P A P A A P A A A LP A A ALA− ˄144˅ ՟ 1 ᡯᦋϸᵮ傄ᄤˈϔᵮ傄ᄤߎ⦄⚍᭄㛑㹿 3 ᭈ䰸ⱘᴵӊϟˈ∖ϸᵮ 傄ᄤߎ⦄ⱘ⚍᭄ПѢ 8 ⱘὖ⥛. 㾷 䆒A ={ϔᵮ傄ᄤߎ⦄⚍᭄㛑㹿 3 ᭈ䰸}ˈ߭ ( ) 2 1 6 3 P A = = . ˈ ˈ16 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 B ={ϸᵮ傄ᄤߎ⦄ⱘ⚍᭄ПѢ 8}. Ўᡯᦋϸᵮ傄ᄤ᳝݅62 =36⾡ৃ㛑ⱘ㒧ᵰˈ㗠џӊAB ={(3,6),(6,3), (6,4),(6,5),(6,6)}. ℸ ( ) 5 36 P AB = ˈ 5 ( ) 36 5 ( ) 1 ( ) 12 3 P AB P B A P A = = = . ՟ 2 ᏆⶹϔᆊᒁЁϸϾᄽᄤ⬆Эϔᑈݙ⫳⮙ⱘ᭄ߚᏗᰃഛㄝⱘˈϨ⫳ ⮙᭄ⱘ↨՟⬆ऴ 20%ˈЭऴ 18%ˈ⬆⫳⮙ⱘᴵӊϟˈЭг⫳⮙ⱘὖ⥛Ў 0.6. ϸᄽᄤৠᯊ⫳⮙ऴ 12%. ∖ ˄1˅ϸᄽᄤৠᯊ⫳⮙ⱘὖ⥛˗ ˄2˅㟇ᇥ᳝ϔϾᄽᄤ⫳⮙ⱘὖ⥛˗ ˄3˅Э⫳⮙ⱘᴵӊϟˈ⬆г⫳⮙ⱘὖ⥛˗ 㾷 䆒A ={⬆⫳⮙}ˈB={Э⫳⮙}. ߭ ˄1˅P AB( )=P A P B A( ) ( | ) 0.2 0.6 0.12= × = ˗ ˄2˅P A B =( Ĥ ) P A P B P AB( )+ ( )− ( ) =0.2 0.18 0.12 0.26+ − = ˗ ˄3˅ ( | ) ( ) 0.12 0.67 ( ) 0.18 P AB P A B P B = = = . Ѡǃܼὖ⥛݀ᓣ䋱ᮃ݀ᓣ Ўњ䅵ㅫ↨䕗ᴖⱘџӊⱘὖ⥛ˈҎӀ㒣ᐌᡞᴖџӊߚ㾷Ў㢹ᑆϾѦϡⳌ ᆍⱘㅔऩџӊПˈ䗮䖛ߚ߿䅵ㅫ䖭ѯㅔऩџӊⱘὖ⥛ˈݡᑨ⫼ὖ⥛ⱘࡴ⊩݀ᓣ ϢЬ⊩݀ᓣ∖ᕫ᠔䳔㒧ᵰ. ᅮ⧚ 1 ܼὖ⥛݀ᓣ 䆒ΩЎϔḋᴀぎ䯈ˈA A1, , ,2 L AnЎḋᴀぎ䯈ΩⱘϔϾ ߦߚˈे i j A A = ∅ ˄i j≠ ˅˗A A1+ 2+ +L An = Ω . ϨP A >( ) 0i ˄i=1,2, ,L n˅ˈ߭ᇍӏϔџӊBˈ᳝ 1 ( ) n ( ) (i i) i P B P A P B A = =
∑
. ˄1.4.5˅ 䆕ᯢ B B= ģΩ =Bģ(A A1+ 2+ +L An) =BA BA1+ 2+ +L BAn . ⬅ѢBA BA1, 2, ,L BAnϸϸѦ᭹ˈ᠔ҹ᳝ 1 2 1 1 ( ) ( n) n ( i) n ( ) ( | )i i i i P B P BA BA BA P BA P A P B A = = = + + +L =∑
=∑
՟ 3 ؛ᅮᶤᯊᳳݙᕅડ㙵⼼ӋḐব࣪ⱘ㋴া᳝䫊㸠ᄬℒ߽⥛ⱘব࣪.㒣 ߚᵤˈ䆹ᯊᳳݙ߽⥛ϡӮϞ⍼ˈ߽⥛ϟ䇗ⱘὖ⥛Ў 70%ˈ߽⥛ϡবⱘὖ⥛Ў 30%. 1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ ḍ㒣偠ˈ߽⥛ϟ䇗ᯊᶤᬃ㙵⼼Ϟ⍼ⱘὖ⥛Ў 60%ˈ߽⥛ϡবᯊˈ䖭ᬃ㙵⼼ Ϟ⍼ⱘὖ⥛Ў 20%. ∖䖭ᬃ㙵⼼Ϟ⍼ⱘὖ⥛. 㾷 䆒A ={䆹ᬃ㙵⼼Ϟ⍼}ˈB={߽⥛ϟ䇗}ˈ߭ B = {߽⥛ϡব}. ߭P B =( ) 0.7ˈP B =( ) 0.3ˈP A B =( ) 0.6ˈP A B =( ) 0.2. A AB AB= + ( )P A =P B P A B( ) ( | )+P B P A B( ) ( | ) =0.7 0.6 0.3 0.2 0.48× + × = . ՟ 4 ⬆ǃЭǃϭϝাⲦᄤˈ⬆ⲦЁ㺙᳝ 2 Ͼ㑶⧗ˈ4 Ͼⱑ⧗˗ЭⲦЁ㺙᳝ 4 Ͼ㑶⧗ˈ2 Ͼⱑ⧗˗ϭⲦЁ㺙᳝ 3 Ͼ㑶⧗ˈ3 Ͼⱑ⧗. ᑊ؛䆒ࠄϝাⲦᄤЁপ⧗ⱘ ᴎӮഛㄝ. ⦄ҢϝাⲦᄤЁӏপϔⲦˈݡҢℸⲦЁӏপϔ⧗ˈ∖পࠄ㑶⧗ⱘὖ⥛. 㾷 䆒B ={পࠄ㑶⧗}. Ai˄i =1,2,3˅ߚ߿㸼⼎পࠄ⬆ǃЭǃϭϝϾⲦᄤˈ߭ P A( )1 =P A( )2 =P A( )3 =13ˈ P B A =( 1) 26ˈP B A =( 2) 64ˈP B A =( 3) 36. P B( )=P A P B A( ) ( | )1 1 +P A P B A( ) ( | )2 2 +P A P B A( ) ( | )3 3 1 2 1 4 1 3 0.5 3 6 3 6 3 6 = × + × + × = . ՟ 5 ⦏⩗ᵃ៤ㆅߎଂˈ↣ㆅ 20 া. ؛䆒ㆅ 0ˈ1ˈ2 া⅟કⱘὖ⥛ߚ ߿Ў 0.8ˈ0.1 0.1. ϔ乒ᅶ℆䌁ϔㆅ⦏⩗ᵃˈ䌁фᯊˈଂ䋻ਬӏপϔㆅˈ㗠乒 ᅶ䱣ᴎⱘᆳⳟ 4 াˈ㢹᮴⅟કˈ߭фϟ䆹ㆅ⦏⩗ᵃˈ৺߭䗔䖬. 䆩∖乒ᅶфϟ䆹 ㆅⱘὖ⥛. 㾷 䆒B ={фϟ䆹ㆅ}ˈA =i {↣ㆅ᳝iাક}˄i =0,1,2˅ˈ߭ P A =( ) 0.80 ˈP A =( ) 0.11 ˈP A =( ) 0.12 . 194 184 0 1 4 2 4 20 20 C C ( ) 1 ( ) ( ) C C P B A = ˈ P B A = ˈ P B A = . P B( )=P A P B A( ) ( | )0 0 +P A P B A( ) ( | )1 1 +P A P B A( ) ( | )2 2 194 184 4 4 20 20 C C 0.8 1 0.1 0.1 0.94 C C = × + × + × ≈ . ܼὖ⥛݀ᓣ㒭ߎњ៥ӀϔϾ䅵ㅫফࠄϾᕅડ݇㋏ⱘџӊὖ⥛ⱘ݀ᓣ˖ ؛䆒 1, , ,2 n A A L A ᰃΩⱘϔϾߦߚˈᑊϨᏆⶹџӊAiⱘὖ⥛P A( )i ˄ᅗӀᰃ䆩偠ࠡⱘᏆ ⶹὖ⥛ˈ⿄Ўܜ偠ὖ⥛˅ঞџӊBAiᏆথ⫳ⱘᴵӊϟⱘᴵӊὖ⥛P B A( i) ˄ i=1,2, ,L n˅ˈ߭⬅ܼὖ⥛݀ᓣህৃㅫߎP B( ). ⦄ⱘ䯂乬ᰃ˖៥Ӏ䖯㸠њϔ 䆩偠ˈབᵰџӊB⹂ᅲথ⫳њˈ߭ᇍѢџӊA i˄i =1,2, ,L n˅ⱘὖ⥛ᑨ㒭ќ䞡ᮄԄ 䅵ˈгህᰃ㽕䅵ㅫџӊAiџӊBᏆথ⫳ⱘᴵӊϟⱘᴵӊὖ⥛P A B( i )˄ᅗӀᰃ䆩
18 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 偠ৢⱘџӊὖ⥛ˈᐌ⿄Ўৢ偠ὖ⥛˅. ϟ䴶ⱘ䋱ᮃ݀ᓣህ㒭ߎњ䅵ㅫৢ偠ὖ⥛ ( i ) P A B ⱘ݀ᓣ. ᅮ⧚ 2 䋱ᮃ˄Bayes˅݀ᓣ 䆒џӊ A A1, , ,2 L AnЎḋᴀぎ䯈ΩⱘϔϾߦ ߚˈϨP B >( ) 0ˈP A >( ) 0i ˄ i=1,2, ,L n˅ˈ߭ ( ) ( ) ( )i i P A B P A B P B = ( ) ( ) ( ) i i P A P B A P B = 1 ) ) ) ) ( ( | ( ( | n i i i i i P A P B A P A P B A = =
∑
˄ i=1,2, ,L n˅ ˄1.4.6˅ г⿄ϞᓣЎ䗚ὖ⥛݀ᓣ. ՟ 6 Ϟ䗄՟ 5 Ёˈབᵰ乒ᅶфϟϔㆅ⦏⩗ᵃˈ߭фϟⱘ䖭ㆅЁ⹂ᅲ≵᳝⅟ કⱘὖ⥛ᰃᇥ˛ 㾷 ᠔∖ὖ⥛Ў˖ 0 0 ( ) ( ) ( ) P A B P A B P B = ( ) (0 0) 0.8 1 0.85 ( ) 0.94 P A P B A P B × = = ≈ . ՟ 7 䆒Ꮉॖ⬆ᎹॖЭⱘѻકક⥛ߚ߿Ў 1% 2%ˈ⦄Ң⬅⬆ॖЭ ॖⱘѻકߚ߿ऴ 60% 40%ⱘϔᡍѻકЁ䱣ᴎᢑপϔӊˈথ⦄ᰃકˈ߭䆹ક ᰃ⬆ॖ⫳ѻⱘὖ⥛ᰃᇥ? 㾷 䆒B ={পࠄϔӊક}ˈA ={পࠄ⬆ॖѻક}ˈ A = {পࠄЭॖѻક}. ߭ P A =( ) 0.6ˈ ( ) 0.4P A = ˈP B A =( ) 0.01ˈP B A =( ) 0.02. ( )P B =P A P B A P A P B A( ) ( | )+ ( ) ( | ) 0.6 0.01 0.4 0.02 0.014= × + × = . ( ) ( ) ( ) P AB P A B P B = ( ) ( ) 0.6 0.01 3 0.43 ( ) 0.014 7 P A P B A P B × = = = ≈ .1.5 ๚ॲڦ܀૬Ⴀ
՟ 1 ϔ㹟Ё㺙᳝ 10 Ͼ⧗ˈ݊Ё 3 Ͼ㑶⧗ˈ7 Ͼ咥⧗. ҢЁӏপϸˈ↣প ϔ⧗. 䆒A ={ϔপࠄ㑶⧗}ˈB ={Ѡপࠄ㑶⧗}ˈ∖P B A( )ঞP B( ). 㾷 ˄1˅㢹ᰃϡᬒಲᢑḋˈ⬅乬ᛣᯧⶹ 2 ( ) 9 P B A = . ( ) ( ) ( ) ( ) ( ) P B =P A P B A P A P B A+ 3 2 7 3 10 9 10 9 = × + × 3 10 = . ৃ㾕 P B A( ) ≠P B( ). 䖭䇈ᯢџӊAⱘথ⫳Ϣ৺ᇍџӊBথ⫳ⱘὖ⥛ᰃ᳝ᕅડⱘ. 1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ ˄2˅㢹ᰃ᳝ᬒಲᢑḋˈ⬅乬ᛣᕫࠄ ( ) 3 10 P B A = ˈ ( ) 3 10 P B = . ৃ㾕 P B A( ) =P B( ). 䖭䇈ᯢџӊAⱘথ⫳ϡᕅડџӊBথ⫳ⱘὖ⥛ˈ䖭ᯊ⿄џӊAϢBᰃⳌѦ⣀ ゟⱘ ⬅ὖ⥛Ь⊩݀ᓣⶹˈབᵰP B A( ) =P B( )ˈ߭ ( ) ( ) ( ) P AB =P A P B A =P A P B( ) ( ) ᅮН 1 ᇍӏᛣⱘϸϾџӊAǃBˈ㢹 ( ) ( ) ( ) P AB =P A P B ߭⿄џӊAǃBᰃⳌѦ⣀ゟⱘ. ᅮ⧚ 1 㢹џӊAϢBⳌѦ⣀ゟˈϨP A( ) 0> ˈP B( ) 0> ˈ߭ P A B( )=P A( )ˈP B A( | )=P B( ) . ᅮ⧚ 2 㢹џӊAϢB⣀ゟˈ߭ϟ߫ᇍџӊ˖AϢB ˈ A ϢBˈA Ϣ B 䛑 ᰃⳌѦ⣀ゟⱘ. 䆕 ⬅ѢAϢBⳌѦ⣀ゟˈᬙP AB( )=P A P B( ) ( ). ℸ᳝ ( ) ( ) ( ) ( ) ( ) ( ) ( ) P AB =P A AB− =P A P AB− =P A P A P B− ( )[1 ( )] ( ) ( ) P A P B P A P B = − = . ℸˈAϢ B ⳌѦ⣀ゟ. ݇Ѣ A ϢB A Ϣ B ⱘ⣀ゟᗻৠ⧚ৃ䆕. ᅮ⧚ 2 䖬ৃভ䗄Ў˖㢹ಯᇍџӊAϢBˈAϢB ˈ A ϢBˈA Ϣ B Ё᳝ϔᇍ ⳌѦ⣀ゟˈ߭ϝᇍгⳌѦ⣀ゟˈे䖭ಯᇍџӊ㗙䛑ⳌѦ⣀ゟˈ㗙䛑ϡⳌ Ѧ⣀ゟ. ᅲ䰙ᑨ⫼ЁˈᇍѢџӊⱘ⣀ゟᗻˈ៥ӀᐌᐌϡᰃḍᅮНᴹ߸ᮁˈ㗠ᰃḍ ϔџӊⱘথ⫳ᰃ৺ᕅડϔџӊⱘথ⫳ᴹ߸ᮁ. ՟ 2 ⬆ǃЭϸᇘⳌৠᴵӊϟ䖯㸠ᇘߏˈҪӀߏЁⳂᷛⱘὖ⥛ߚ߿ᰃ 0.9 0.8. བᵰϸϾᇘৠᯊ⣀ゟᇘߏϔˈ䯂Ⳃᷛ㹿ߏЁⱘὖ⥛ᰃᇥ˛ 㾷 䆒A ={⬆ߏЁⳂᷛ}ˈB ={ЭߏЁⳂᷛ}ˈC={Ⳃᷛ㹿ߏЁ}. Ѣᰃ P A =( ) 0.9ˈP B =( ) 0.8. জЎC A B= U ˈϨAϢBⳌѦ⣀ゟˈᬙ P C( )=P A B( U )=P A P B P AB( )+ ( )− ( ) =P A P B P A P B( )+ ( )− ( ) ( ) =0.9 0.8 0.9 0.8 0.98+ − × = . џӊⱘ⣀ゟᗻὖᗉˈৃҹᑓࠄϝϾϝϾҹϞⱘџӊⱘᚙᔶ. ᅮН 2 ᇍӏᛣϝϾџӊAˈBˈCˈབᵰ᳝ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P AB P A P B P BC P B P C P CA P C P A P ABC P A P B P C = = = =
20 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 ಯϾㄝᓣৠᯊ៤ゟˈ߭⿄џӊAˈBˈCⳌѦ⣀ゟ. ᅮН 3 㢹 n ϾџӊA A1, , ,2 L An⒵䎇ᇍѢӏᛣⱘk˄1<kİ ˅n ӏᛣⱘϔ㒘 1 2 1İi i< <L<ikİn䛑᳝ㄝᓣ P A A( i1 i2LAik)=P A P A( ) ( )i1 i2 LP A( )ik ៤ゟˈ߭⿄џӊA A1, , ,2 L AnⳌѦ⣀ゟ. ⊼ᛣ˖ ď1Đఢڴಹߑ A A1, , ,2L Anອܚ՟ोē႕ୣᄯఉ࿉ k ّಹߑྙອܚ՟ो 1 k< n ˄ İ ˅. ď2ĐఢڴಹߑA A1, , ,2L Anອܚ՟ोē႕ 1 1 ( n i) n ( )i i i P A P A = = =
∏
I
ˈ 1 1 ( n i) 1 n ( )i i i P A P A = = = −∏
U
. ˄1.5.1˅ ՟ 3 ϝҎ⣀ゟഄএخৠϔϾ䆩偠ˈᏆⶹ↣ϾҎ䆩偠៤ࡳⱘὖ⥛ߚ߿Ў 0.8ˈ 0.7ˈ0.5ˈ䯂ϝҎЁ㟇ᇥ᳝ϔҎ䆩偠៤ࡳⱘὖ⥛ᰃᇥ˛ 㾷ᇚϝҎ㓪োЎ 1ˈ2ˈ3ˈ䆄A =i {iϾҎ䆩偠៤ࡳ}˄i=1,2,3˅. P A A A( 1Ĥ Ĥ2 3)= −1 P A A A( 1Ĥ Ĥ2 3) 1= −P A A A( 1 2 3)= −1 P A P A P A( ) ( ) ( )1 2 3 = −1 0.2 0.3 0.5 0.97× × = .
1.6 ܀૬ᄓႾଚ
؛䆒䆩偠Eা᳝ϸ⾡ৃ㛑ⱘ㒧ᵰ˖Aঞ A ˈⳌৠⱘᴵӊϟᇚ䆩偠E䞡䖯 㸠 n ˈ㢹䆩偠ⱘ㒧ᵰѦϡᕅડˈ߭⿄䖭 n 䆩偠ᰃ n 䞡⣀ゟ䆩偠ᑣ߫˄г⿄ Ў䋱ࡾ䞠ὖൟ˅. ᇍѢ n 䞡⣀ゟ䆩偠ᑣ߫ˈ៥ӀЏ㽕ⷨお n 䆩偠ЁˈџӊAথ⫳ m ⱘὖ⥛˖ ( ) n P m . ᅮ⧚ 1 བᵰ⣀ゟ䆩偠ᑣ߫Ёˈ↣䆩偠া᳝ϸ⾡ৃ㛑ⱘ㒧ᵰ˖Aঞ A ˈ ᑊϨ ( ) P A =pˈP A( ) 1= − =p q˄݊Ё0< <p 1˅ˈ ߭ n 䆩偠ЁџӊAথ⫳ m ⱘὖ⥛ ( ) n P m C ! 0 !( )! m m n m m n m np q − m n mn p q − m n = = − ˄ İ İ ˅. ˄1.6.1˅ ՟ 1 ᶤᇘ↣ᇘߏߏЁⳂᷛⱘὖ⥛Ў 0.8ˈ⦄䖯㸠 20 ⣀ゟᇘߏ.∖ ˄1˅ᙄ᳝ 15 ߏЁⳂᷛⱘὖ⥛˗ ˄2˅ߏЁⳂᷛⱘ᭄ϡ䍙䖛 18 ⱘὖ⥛. 㾷 ˄1˅20 ⣀ゟᇘߏЁᙄ᳝ 15 ߏЁⳂᷛⱘὖ⥛ᰃ 15 15 5 20(15) C20 0.8 0.2 0.175 P = × × ≈ ˗ 1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ ˄2˅ߏЁⳂᷛⱘ᭄ϡ䍙䖛 18 ⱘὖ⥛ 20 20 ( 18) 1 (19) (20) P mİ = −P −P 19 19 1 20 20 1 C 0.8 0.2 0.8 0.93 = − × × − ≈ . ՟ 2 ᶤᄺⱘᄺ⫳ᥦ⧗䯳ϢᬭᎹᥦ⧗䯳䖯㸠↨䌯. Ꮖⶹ↣ϔሔᄺ⫳ᥦ⧗䯳㦋 㚰ⱘὖ⥛Ў 0.6ˈᬭᎹᥦ⧗䯳㦋㚰ⱘὖ⥛Ў 0.4. ∖ ˄1˅䞛⫼ϝሔϸ㚰ࠊᯊˈᄺ⫳ᥦ⧗䯳㦋㚰ⱘὖ⥛˗ ˄2˅䞛⫼Ѩሔϝ㚰ࠊᯊˈᄺ⫳ᥦ⧗䯳㦋㚰ⱘὖ⥛. 㾷˄1˅䞛⫼ϝሔϸ㚰ࠊᯊˈᄺ⫳ᥦ⧗䯳㦋㚰᳝ϟ߫ϸ⾡ᚙᔶ˖ A1˖2:0 ˄ᄺ⫳䯳䖲㚰ϸሔ˅˗ A2˖2:1 ˄ࠡϸሔ㚰ϔሔˈϝሔᄺ⫳䯳㚰˅. ⬅乬ᛣ 2 1 ( ) 0.6 0.36 P A = = ˈ 1 2 2 ( ) C 0.6 0.4 0.6 0.288 P A = × × × = . ℸᄺ⫳ᥦ⧗䯳㦋㚰ⱘὖ⥛Ў P A A( 1+ 2)=P A( )1 +P A( ) 0.36 0.288 0.6482 = + = . ˄2˅䞛⫼Ѩሔϝ㚰ࠊᯊˈᄺ⫳ᥦ⧗䯳㦋㚰᳝ϟ߫ϝ⾡ᚙᔶ˖ B1˖3:0 ˄ᄺ⫳䯳䖲㚰ϝሔ˅˗ B2˖3:1 ˄ࠡϝሔᄺ⫳䯳㚰ϸሔˈ䋳ϔሔˈಯሔᄺ⫳䯳㚰˅˗ B3˖3:2 ˄ࠡಯሔ㚰ϸሔˈѨሔᄺ⫳䯳㚰˅. ⬅乬ᛣ 3 1 ( ) 0.6 0.216 P B = = ˈ 2 2 2 3 ( ) C 0.6 0.4 0.6 0.259 P B = × × × ≈ ˈ 2 2 2 3 4 ( ) C 0.6 0.4 0.6 0.207 P B = × × × ≈ . ℸᄺ⫳ᥦ⧗䯳㦋㚰ⱘὖ⥛Ў ( 1 2 3) ( )1 ( )2 ( )3 0.216 0.259 0.207 0.682 . P B B+ +B =P B +P B +P B = + + ≈
သ༶ᅃ
1ˊ⬆ǃЭǃϭϝҎᇘϔ䵊ˈ䆄A ={⬆Ё䵊}ˈB ={ЭЁ䵊}ˈC ={ϭЁ 䵊} ˈ䆩⫼Ϟ䗄ϝϾџӊⱘ䖤ㅫˈ㸼⼎ϟ߫џӊ˖ ˄1˅⬆Ё䵊˗ ˄2˅⬆Ё䵊㗠ЭЁ䵊˗ ˄3˅ϝҎЁা᳝ϭЁ䵊˗ ˄4˅ϝҎЁᙄད᳝ϔҎЁ䵊˗ ˄5˅ϝҎЁ㟇ᇥ᳝ϔҎЁ䵊˗ ˄6˅ϝҎЁ㟇ᇥ᳝ϔҎЁ䵊˗ ˄7˅ϝҎЁᙄ᳝ϸҎЁ䵊˗ ˄8˅ϝҎЁ㟇ᇥϸҎЁ䵊˗ ˄9˅ϝҎഛЁ䵊˗ ˄10˅ϝҎЁ㟇ϔҎЁ䵊˗ ˄11˅ϝҎЁ㟇ϸҎЁ䵊. 2ˊᶤජᏖ݅থ㸠AǃBǃCϝ⾡㒌. 䇗ᶹ㸼ᯢˈሙ⇥ᆊᒁЁ䅶䌁Cⱘऴ22 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 30%ˈৠᯊ䅶䌁AǃBϸⱘऴ 10%ˈৠᯊ䅶䌁AC㗙BCⱘ ऴ 8%ǃ5%ˈϝ⾡㒌䛑䅶ⱘऴ 3%. Ҟ䆹ජᏖЁӏᡒϔሙ⇥ᆊᒁˈ∖˖ ˄1˅䆹᠋া䅶ABϸ⾡㒌ⱘὖ⥛˗˄2˅䆹᠋া䅶Cⱘὖ⥛. 3ˊᇚϔᵮࣔ⿄ⱘ傄ᄤᡯᦋϸˈ∖ϸߎ⦄ⱘ⚍᭄ПㄝѢ 8 ⱘὖ⥛. 4ˊҢ㺙᳝ 3 া㑶⧗ˈ2 াⱑ⧗ⱘⲦᄤЁӏᛣপߎϸা⧗ˈ∖݊Ё᳝ᑊϨা᳝ ϔা㑶⧗ⱘὖ⥛. 5ˊᡞ 10 ᴀкӏᛣᬒкᶊϞˈ∖݊Ёᣛᅮⱘ 3 ᴀкᬒϔ䍋ⱘὖ⥛. 6ˊЎњޣᇥ↨䌯എˈᡞ 20 Ͼ⧗䯳ӏᛣߚ៤ϸ㒘ˈ↣㒘 10 䯳䖯㸠↨䌯ˈ∖ ᳔ᔎⱘϸϾ䯳㹿ߚϡৠ㒘ݙⱘὖ⥛. 7ˊᶤѻક᳝ǃЁǃᇣϝ⾡ൟো. ᶤ݀ৌথߎ 17 ӊℸѻકˈ݊Ё 10 ӊোˈ 4 ӊЁোˈ3 ӊᇣো. Ѹ䋻Ҏ㉫ᖗ䱣ᛣᇚ䖭ѯѻકথ㒭乒ᅶ. 䯂ϔϾ䅶䋻Ў 4 ӊ োǃ3 ӊЁো 2 ӊᇣোⱘ乒ᅶˈ㛑ᣝ᠔ᅮൟোབ᭄ᕫࠄ䅶䋻ⱘὖ⥛ᰃᇥ˛ 8ˊ䆒ⲦЁ᳝ 3 Ͼⱑ⧗ˈ2 Ͼ㑶⧗ˈ⦄ҢⲦЁӏᢑ 2 Ͼ⧗ˈ∖পࠄϔϾ㑶⧗ ϔϾⱑ⧗ⱘὖ⥛. 9ˊᇚ 3 Ͼ⧗䱣ᴎⱘᬒܹ 3 ϾⲦᄤЁএˈ䯂˖ ˄1˅↣Ⲧᙄ᳝ϔ⧗ⱘὖ⥛ᰃᇥ˛ ˄2˅ぎϔⲦⱘὖ⥛ᰃᇥ˛ 10ˊϔᡍѻક݅ 20 ӊˈ݊Ёϔㄝક 9 ӊˈѠㄝક 7 ӊˈϝㄝક 4 ӊ. Ң䖭ᡍ ѻકЁӏপ 3 ӊˈ∖˖ ˄1˅পߎⱘ 3 ӊѻકЁᙄ᳝ 2 ӊㄝ㑻Ⳍৠⱘὖ⥛˗ ˄2˅পߎⱘ 3 ӊѻકЁ㟇ᇥ᳝ 2 ӊㄝ㑻Ⳍৠⱘὖ⥛. 11ˊᏆⶹ 10 ӊѻકЁ᳝ 7 ӊℷકˈ3 ӊક. ˄1˅ϡᬒಲഄ↣ҢЁӏপϔӊˈ݅প 3 ˈ∖পࠄ 3 ӊકⱘὖ⥛˗ ˄2˅↣ҢЁӏপϔӊˈ᳝ᬒಲഄপ 3 ˈ∖পࠄ 3 ӊકⱘὖ⥛˗ ˄3˅ҢЁӏপϝӊˈ∖㟇ᇥপࠄ 1 ӊકⱘὖ⥛. 12ˊ⣢Ҏ䎱⾏ 100 ㉇໘ᇘߏϔࡼ⠽ˈߏЁⱘὖ⥛Ў 0.6˗བᵰϔߏЁˈ ߭䖯㸠ѠᇘߏˈԚ⬅Ѣࡼ⠽䗗䎥㗠Փ䎱⾏ব៤Ў 150 ㉇˗བᵰѠজߏ Ёˈ߭䖯㸠ϝᇘߏˈ䖭ᯊ䎱⾏বЎ 200 ㉇ˊ؛ᅮ᳔䖯㸠ϝᇘߏˈ䆒ߏЁ ⱘὖ⥛Ϣ䎱⾏៤ড↨ˈ∖⣢ҎߏЁࡼ⠽ⱘὖ⥛. 13ˊ100 Ͼ䳊ӊЁˈ᳝ 10 Ͼકˈ↣᮴ᬒಲⱘӏপϔϾˈ㾘ᅮབᵰপᕫϔ Ͼড়Ḑકˈህϡݡ㒻㓁প䳊ӊ. ∖ϝݙপᕫড়Ḑકⱘὖ⥛. 14ˊৠᯊᡯᦋϸᵮ⹀ᏕˈᏆⶹ݊Ё᳝ϔᵮ⹀Ꮥᰃℷ䴶Ϟˈ䯂䖭ᯊϔᵮ⹀ Ꮥгᰃℷ䴶Ϟⱘὖ⥛Ў˛ 15ˊ⬆ǃЭǃϭϝҎ䗮䖛ᢑㅒއᅮϸᓴৠϔഎⱘখ㾖⼼ⱘᔦሲˈ⬆ܜǃЭ ǃϭ᳔ৢ. ∖ ˄1˅Эᢑࠄখ㾖⼼ⱘὖ⥛α ˗ ˄2˅བᵰᏆⶹЭᢑࠄњখ㾖⼼ˈ߭⬆гᢑࠄњখ㾖⼼ⱘὖ⥛β.
1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ 16ˊ䆒 10 ӊѻકЁ᳝ 4 ӊϡড়ḐકˈҢЁӏপϸӊ. ᏆⶹϸӊЁ᳝ϔӊᰃϡ ড়Ḑકˈ߭ϔӊгᰃϡড়Ḑકⱘὖ⥛. 17ˊ䆒ᶤ⾡ࡼ⠽⬅ߎ⫳ㅫ䍋⌏ࠄ 20 ቕҹϞⱘὖ⥛Ў 0.8ˈ⌏ࠄ 25 ቕҹϞⱘὖ ⥛Ў 0.4. བᵰϔাࡼ⠽⦄Ꮖ㒣⌏ࠄ 20 ቕˈ䯂ᅗ㛑⌏ࠄ 25 ቕҹϞⱘὖ⥛ᰃᇥ˛ 18ˊЎњ䰆ℶᛣˈⷓݙৠᯊ䆒᳝⬆ǃЭϸ⾡䄺㋏㒳ˈ↣⾡㋏㒳ऩ⣀Փ ⫼ᯊˈ᳝݊ᬜⱘὖ⥛˖㋏㒳⬆Ў 0.92ˈ㋏㒳ЭЎ 0.93. ⬆Эϸ㋏㒳ৠᯊՓ⫼ᯊ䛑᳝ ᬜⱘὖ⥛Ў 0.862. ∖ ˄1˅থ⫳ᛣᯊˈ䖭ϸϾ䄺㋏㒳㟇ᇥ᳝ϔϾ᳝ᬜⱘὖ⥛˗ ˄2˅㋏㒳Э᳝ᬜⱘᴵӊϟˈ㋏㒳⬆г᳝ᬜⱘὖ⥛˗ ˄3˅㋏㒳⬆᳝ᬜⱘᴵӊϟˈ㋏㒳Эг᳝ᬜⱘὖ⥛. 19ˊҞ᳝ 3 ㆅ䋻⠽ˈ݊Ё⬆ॖ⫳ѻⱘ᳝ 2 ㆅˈЭॖ⫳ѻⱘ᳝ 1 ㆅ. Ꮖⶹ⬆ॖ⫳ ѻⱘ↣ㆅЁ᳝ 98 Ͼড়Ḑકˈ2 Ͼક˗㗠Эॖ⫳ѻⱘ 1 ㆅЁ᳝ 90 Ͼড়Ḑકˈ10 Ͼક. ⦄Ң 3 ㆅЁӏপ 1 ㆅˈҢপߎⱘ䖭ϔㆅЁӏপ 1 ӊѻક. ∖˖ ˄1˅䖭ӊѻકᰃ⬆ॖ⫳ѻⱘড়Ḑકⱘὖ⥛˗ ˄2˅䖭ӊѻકᰃড়Ḑકⱘὖ⥛. 20ˊ䆒᳝ᴹ㞾ϝϾഄऎⱘ 10 ৡǃ15 ৡ 25 ৡ㗗⫳ⱘৡ㸼. ݊Ёཇ⫳ⱘ ৡ㸼ߚ߿Ў 3 ӑǃ7 ӑ 5 ӑ. 䱣ᴎⱘপߎϔϾഄऎⱘৡ㸼ˈҢЁӏᢑϔӑ. ∖ পߎⱘᰃཇ⫳㸼ⱘὖ⥛. 21ˊϔҎҢഄࠄ⌢फᴹখࡴӮ䆂ˈҪЬ☿䔺ⱘὖ⥛Ў0.5ˈЬ亲ᴎⱘὖ⥛Ў 0.3ˈЬ≑䔺ⱘὖ⥛Ў0.2. བᵰЬ☿䔺ᴹˈ䖳ࠄⱘὖ⥛Ў0.25ˈЬ亲ᴎᴹ䖳ࠄⱘ ὖ⥛Ў0.12ˈЬ≑䔺ᴹ䖳ࠄⱘὖ⥛Ў0.08. ∖ℸҎ䖳ࠄⱘὖ⥛. 22ˊᶤ⾡Ҿ఼⬅ 3 Ͼ䚼ӊ㒘㺙㗠៤. ؛䆒䚼ӊ䋼䞣ѦϡᕅડϨᅗӀⱘӬ䋼ક ⥛ߚ߿Ў 0.8ǃ0.7 Ϣ 0.9. ᏆⶹབᵰϝϾ䚼ӊ䛑ᰃӬ䋼કˈ߭㒘㺙ৢⱘҾ఼ϔᅮড় Ḑ˗བᵰ᳝ϔϾ䚼ӊϡᰃӬ䋼કˈ߭㒘㺙ৢⱘҾ఼ড়Ḑ⥛Ў 0.8˗བᵰ᳝ϸϾ䚼ӊ ϡᰃӬ䋼કˈ߭㒘㺙ৢⱘҾ఼ড়Ḑ⥛Ў 0.4˗བᵰϝϾ䚼ӊ䛑ϡᰃӬ䋼કˈ߭㒘㺙 ৢⱘҾ఼ড়Ḑ⥛Ў 0.1. 䆩∖Ҿ఼ⱘড়Ḑ⥛. 23ˊⲦЁᬒ᳝ 12 ϾЦЧ⧗ˈ݊Ё 9 Ͼᰃᮄⱘ. ϔ↨䌯ᯊҢЁӏপ 3 Ͼᴹ ⫼ˈ䌯ৢᬒಲˈѠ↨䌯ᯊݡҢЁӏপ 3 Ͼ. ∖Ѡ↨䌯ᯊপߎⱘ⧗䛑ᰃᮄ⧗ ⱘὖ⥛. 24ˊ䆩ोЁ᳝ϔ䘧䗝ᢽ乬ˈ᳝݅ 4 ϾㄨḜৃկ䗝ᢽˈ݊Ёা᳝ϔϾㄨḜᰃℷ ⹂ⱘ. 㗗⫳བᵰӮ㾷䖭䘧乬ˈ߭ϔᅮ㛑䗝ߎℷ⹂ㄨḜ˗བᵰҪϡӮ㾷䖭䘧乬ˈ߭ӏ 䗝ϔϾㄨḜ. 䆒㗗⫳Ӯ㾷䖭䘧乬ⱘὖ⥛ᰃ 0.8ˈ∖ ˄1˅㗗⫳䗝ߎℷ⹂ㄨḜⱘὖ⥛˗ ˄2˅Ꮖⶹ䆹㗗⫳᠔䗝ㄨḜᰃℷ⹂ⱘˈ߭Ҫ⹂ᅲӮ㾷䖭䘧乬ⱘὖ⥛. 25ˊᶤॖ⬆ǃЭǃϭϝϾ䔺䯈⫳ѻৠϔ⾡ѻકˈ݊ѻ䞣ߚ߿ऴܼॖᘏѻ䞣ⱘ 40%ǃ38%ǃ22%ˈ㒣Ẕ偠ⶹ䔺䯈ⱘક⥛ߚ߿Ў 0.04ǃ0.03ǃ0.05. ⦄Ң䆹⾡ѻ કЁӏᛣপϔӊ䖯㸠Ẕᶹ.
24 㒣⌢᭄ᄺ 嗨 ὖ⥛䆎Ϣ᭄ ⧚㒳䅵 ˄1˅∖䖭ӊѻકᰃકⱘὖ⥛˗ ˄2˅Ꮖⶹᢑᕫⱘϔӊᰃકˈ䯂ℸકᴹ㞾⬆ǃЭǃϭ䔺䯈ⱘὖ⥛ߚ߿ᰃ ᇥ˛ 26ˊϝҎ⣀ゟഄএ⸈䆥ϔӑᆚⷕˈᏆⶹҎ㛑䆥ߎⱘὖ⥛ߚ߿Ў 1/5ǃ1/3ǃ1/4ˈ 䯂ϝҎЁ㟇ᇥ᳝ϔҎ㛑ᇚᆚⷕ䆥ߎⱘὖ⥛ᰃᇥ˛ 27ˊᶤҎ᳝ϔІ m ᡞᔶⳌৠⱘ䩹࣭ˈ݊Ёা᳝ϔᡞ㛑ᠧᓔᆊ䮼. ᳝ϔ䆹Ҏ 䜦䝝ৢಲᆊˈϟᛣ䆚ഄ↣Ңm ᡞ䩹࣭Ё䱣֓ᣓϔাএᓔ䮼ˈ䯂䆹Ҏ k ᠡ ᡞ䮼ᠧᓔⱘὖ⥛˛ 28ˊϔᎹҎⳟㅵϝৄᴎᑞˈϔᇣᯊݙ⬆ǃЭǃϭϝৄᴎᑞ䳔䆹ᎹҎ✻ⳟⱘ ὖ⥛ߚ߿Ў 0.9ǃ0.8 0.85. ∖˖ϔᇣᯊЁˈ ˄1˅≵᳝ᴎᑞ䳔㽕✻ⳟⱘὖ⥛˗ ˄2˅㟇ᇥ᳝ϔৄᴎᑞϡ䳔㽕✻ⳟⱘὖ⥛˗ ˄3˅㟇᳝ϔৄᴎᑞ䳔㽕✻ⳟⱘὖ⥛. 29ˊࡴᎹᶤϔ䳊ӊ݅䳔ಯ䘧Ꮉᑣˈ䆒ϔǃѠǃϝǃಯ䘧Ꮉᑣߎકⱘὖ⥛ ߚ߿Ў 0.02ǃ0.03ǃ0.05ǃ0.04ˈϨ䘧ᎹᑣѦϡᕅડˈ∖ࡴᎹߎⱘ䳊ӊⱘક⥛. 30ˊ؛㢹↣ϾҎⱘ㸔⏙Ё᳝ᶤ⮙↦ⱘὖ⥛Ў 0.004ˈ⏋ড় 100 ϾҎⱘ㸔⏙ˈ ∖ℸ㸔⏙Ё᳝ℸ⮙↦ⱘὖ⥛. 31ˊ䆒ᶤ⾡㥃⠽ᇍᶤ⾡⮒⮙ⱘ⊏ᛜ⥛Ў80%ˈ⦄᳝ 10 ৡᙷ᳝䖭⾡⮒⮙ⱘ⮙Ҏ ৠᯊ᳡⫼䖭⾡㥃ˈ∖݊Ё㟇ᇥ᳝ 6 Ҏ㹿⊏ᛜⱘὖ⥛. 32ˊᶤᇘ↣ᇘߏߏЁⳂᷛⱘὖ⥛Ў 0.8ˈ⦄䖯㸠 20 ⣀ゟᇘߏ.∖ ˄1˅ᙄ᳝ 15 ߏЁⳂᷛⱘὖ⥛˗ ˄2˅ߏЁⳂᷛⱘ᭄ϡ䍙䖛 18 ⱘὖ⥛. 33ˊᶤ⾡⮒⮙ⱘ㞾✊⮞ᛜ⥛Ў 0.1ˈЎњẔ偠ϔ⾡⊏⭫䆹⮙ⱘᮄ㥃ᰃ৺᳝ᬜˈ ᇚᅗ㒭ᙷ䆹⮙ⱘ 10 ԡᖫᜓ㗙᳡⫼ˈᣝ㑺ᅮ˖བᵰ 10 ৡফ䆩㗙Ё㟇ᇥ᳝ 3 Ҏ⮞ᛜ ህ䅸Ў䆹㥃᳝ᬜˈ৺߭䅸Ўᅠܼ᮴ᬜ. ᣝℸ㑺ᅮˈ∖ᮄ㥃ᅲ䰙Ϟᅠܼ᮴ᬜԚ㹿⹂ᅮ Ў᳝ᬜⱘὖ⥛. 34ˊ䆒ᶤൟোⱘ催ᇘ⚂ˈ↣䮼⚂থᇘϔথ⚂ᔍߏЁ亲ᴎⱘὖ⥛Ў 0.6. ⦄䜡㕂 㢹ᑆ䮼⚂⣀ゟⱘথᇘϔথ⚂ᔍˈ䯂℆ҹ 99%ⱘᡞᦵߏЁᴹ⢃ⱘϔᶊᬠᴎˈ㟇ᇥ 䳔䜡㕂䮼催ᇘ⚂˛ 35ˊᇘߏ䖤ࡼЁˈϔᇘߏ᳔㛑ᕫ 10 ⦃ˊ䆒ᶤ䖤ࡼਬϔᇘߏЁᕫ 10 ⦃ⱘὖ⥛Ў 0.4ˈᕫ 9 ⦃ⱘὖ⥛Ў 0.3ˈᕫ 8 ⦃ⱘὖ⥛Ў 0.2ˈ∖䆹䖤ࡼਬѨ⣀ ゟⱘᇘߏЁᕫࠄϡᇥѢ 48 ⦃ⱘὖ⥛. 36ˊ䞥Ꮉ䔺䯈᳝ 10 ৄৠ㉏ൟⱘᴎᑞˈ↣ৄᴎᑞ䜡ⱘ⬉ࡼᴎࡳ⥛Ў 10 ग⪺ˈ Ꮖⶹ↣ৄᴎᑞᎹᯊˈᑇഛ↣ᇣᯊᅲ䰙ᓔࡼ 12 ߚ䩳ˈϨᓔࡼϢ৺ᰃⳌѦ⣀ゟⱘˊ⦄ ᔧഄ⬉կᑨ㋻ᓴˈկ⬉䚼䮼াᦤկ 50 ग⪺ⱘ⬉㒭䖭 10 ৄᴎᑞˈ䯂䖭 10 ৄ ᴎᑞ㛑ℷᐌᎹⱘὖ⥛Ў˛ 37ˊ⬆ǃЭǃϭϝҎৠᯊᇍ亲ᴎ䖯㸠ᇘߏˈϝҎߏЁⱘὖ⥛ߚ߿Ў 0.4ǃ0.5ǃ
1 ゴ 䱣 ᴎ џ ӊ ঞ ὖ ⥛ 0.7. 亲ᴎ㹿ϔҎߏЁ㗠ߏ㨑ⱘὖ⥛Ў 0.2ˈ㹿ϸҎߏЁ㗠ߏ㨑ⱘὖ⥛Ў 0.6ˈ㢹ϝ Ҏ䛑ߏЁˈ亲ᴎᖙᅮ㹿ߏ㨑ˈ∖亲ᴎ㹿ߏ㨑ⱘὖ⥛. 38ˊ؛ᅮϔॖᆊ⫳ѻⱘ↣ৄҾ఼ҹὖ⥛ 0.70 ৃҹⳈߎॖˈҹὖ⥛ 0.30 䳔䖯 ϔℹ䇗䆩ˈ㒣䇗䆩ৢҹὖ⥛ 0.80 ৃҹߎॖ˗ҹὖ⥛ 0.20 ᅮЎϡড়Ḑકϡ㛑ߎॖ. ⦄䆹ॖ⫳ѻњ˄ ı ˅n n 2 ৄҾ఼˄؛ᅮৄҾ఼ⱘ⫳ѻ䖛ⳌѦ⣀ゟ˅.∖˖ ˄1˅ n ৄܼ䚼㛑ߎॖⱘὖ⥛α ˗ ˄2˅݊Ёᙄད᳝ϸৄϡ㛑ߎॖⱘὖ⥛β˗ ˄3˅݊Ё㟇ᇥ᳝ϸৄϡ㛑ߎॖⱘὖ⥛θ.
