Bài giảng Xác suất thống kê: Chương 1 - Tôn Thất Tú

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Bài giảng "Xác suất thống kê" Chương 1 - Xác suất, cung cấp cho sinh viên những kiến thức như: kiến thức về tổ hợp; không gian mẫu và biến cố; định nghĩa và các tính chất của xác suất; xác suất có điều kiện; công thức nhân xác suất; biến cố độc lập;...Mời các bạn cùng tham khảo!

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Nội dung Text: Bài giảng Xác suất thống kê: Chương 1 - Tôn Thất Tú

❳⑩❈ ❙❯❻❚ ❚❍➮◆● ❑➊ ❚!♥ ❚❤➜% ❚& ✣➔ ◆➤♥❣✱ ✷✵✶✾ ❚!♥ ❚❤➜% ❚& ✶✴✹✽ ❍!❝ ♣❤➛♥✿ ❳→❝ *✉➜- -❤.♥❣ ❦➯ ◆+✐ ❞✉♥❣ ✯ ❍0❝ ♣❤➛♥ ❜❛♦ ❣%♠ ✹✺ )✐➳)✱ ✈.✐ ✻ ❝❤23♥❣ ♥❤2 5❛✉✿ ✲ ❈❤23♥❣ ✶✳ ❳→❝ 5✉➜) ✲ ❈❤23♥❣ ✷✳ ❇✐➳♥ ♥❣➝✉ ♥❤✐➯♥ ✲ ❈❤23♥❣ ✸✿ ❱❡❝)3 ♥❣➝✉ ♥❤✐➯♥ ✲ ❈❤23♥❣ ✹✿ ❚❤G♥❣ ❦➯ ♠I )↔ ✲ ❈❤23♥❣ ✺✿ K.❝ ❧2M♥❣ )❤❛♠ 5G ✲ ❈❤23♥❣ ✻✿ ❑✐➸♠ ✤Q♥❤ ❣✐↔ )❤✉②➳) )❤G♥❣ ❦➯ ✯ ❚➔✐ ❧✐➺✉ 8❤❛♠ ❦❤↔♦✿ ✶✳ ❏❛② ▲✳ ❉❡✈♦V❡ ✭✷✵✶✷✮✱ ZV♦❜❛❜✐❧✐)② ❛♥❞ ❙)❛)✐5)✐❝5 ❢♦V ❊♥❣✐♥❡❡V✐♥❣ ❛♥❞ )❤❡ ❙❝✐❡♥❝❡5✱ ✽)❤ ❊❞✐)✐♦♥✱ ❇V♦♦❦5✴❈♦❧❡✱ ❈❡♥❣❛❣❡ ▲❡❛V♥✐♥❣✳ ✷✳ ▲➯ ❱➠♥ ❉b♥❣ ✭✷✵✶✻✮✱ ●✐→♦ )V➻♥❤ ❳→❝ 5✉➜) )❤G♥❣ ❦➯✱ ◆❳❇ ❚❤I♥❣ )✐♥ ✈➔ )V✉②➲♥ )❤I♥❣✳ ✸✳ ▲✐♥❦✿ ❤))♣5✿✴✴5✐)❡5✳❣♦♦❣❧❡✳❝♦♠✴5✐)❡✴)✉5♣❞♥✴❧)❤✉②❡) ✯ >❤➛♥ ♠➲♠✿ ❊①❝❡❧✱ ❘✱ ●❡♦❣❡❜V❛✳ ❚!♥ ❚❤➜% ❚& ✷✴✹✽ ❈❤34♥❣ ✶✿ ❳→❝ *✉➜- ✶✳ ❑✐➳♥ 8❤C❝ ✈➲ 8E ❤F♣ ✲ ❙G ❤♦→♥ ✈Q ❝k❛ ♠l) )➟♣ ♥ ♣❤➛♥ )o✿ Pn = n! ✲ ❙G ❝→❝❤ ❝❤p♥ k ♣❤➛♥ )o ✭❦❤I♥❣ )❤q )r✮ )V♦♥❣ )➟♣ ♥ ♣❤➛♥ )o✿ n! Cnk = k!(n − k)! ✲ ❙G ❝→❝❤ ❝❤p♥ k ♣❤➛♥ )o ✭❝s )❤q )r✮ )V♦♥❣ )➟♣ n ♣❤➛♥ )o✿ n! Akn = (n − k)! ❚!♥ ❚❤➜% ❚& ✸✴✹✽
✲ ◗✉② $➢❝ ❝'♥❣✿ ❈!♥❣ ✈✐➺❝ ❆ ❝) m ♣❤,-♥❣ →♥ /❤0❝ ❤✐➺♥✳ ✯ 3❤,-♥❣ →♥ ✶✿ ❝) n1 ❝→❝❤ ✯ 3❤,-♥❣ →♥ ✷✿ ❝) n2 ❝→❝❤ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✯ 3❤,-♥❣ →♥ m✿ ❝) nm ❝→❝❤ ❙8 ❝→❝❤ /❤0❝ ❤✐➺♥ ❝!♥❣ ✈✐➺❝ ❆✿ n1 + n2 + ... + nm ✲ ◗✉② $➢❝ ♥❤➙♥✿ ❈!♥❣ ✈✐➺❝ ❆ ✤,:❝ /❤0❝ ❤✐➺♥ ;✉❛ m ❣✐❛✐ ✤♦↕♥ ❧✐➯♥ /✐➳♣✳ ✯ ●✐❛✐ ✤♦↕♥ ✶✿ ❝) n1 ❝→❝❤ ✯ ●✐❛✐ ✤♦↕♥ ✷✿ ❝) n2 ❝→❝❤ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✯ ●✐❛✐ ✤♦↕♥ m✿ ❝) nm ❝→❝❤ ❙8 ❝→❝❤ /❤0❝ ❤✐➺♥ ❝!♥❣ ✈✐➺❝ ❆✿ n1 ∗ n2 ∗ ... ∗ nm ❚!♥ ❚❤➜% ❚& ✹✴✹✽ ❱➼ ❞0 ✶ ▼!" ♥❤%♠ ❝% ✶✵ ❤*❝ +✐♥❤✱ ".♦♥❣ ✤% ❝% ✻ ♥❛♠ ✈➔ ✹ ♥7✳ ❍:✐ ❝% ❜❛♦ ♥❤✐➯✉ ❝→❝❤✿ ❛✮ ❳➳♣ "❤➔♥❤ ✶ ❤➔♥❣ ❞*❝✳ ❜✮ ❈❤*♥ ♠!" ♥❤%♠ ❝% ✹ ❤*❝ +✐♥❤ ✤➲✉ ❝% ♥❛♠ ✈➔ ♥7 ✈G✐ +H ❧JK♥❣ ❦❤→❝ ♥❤❛✉✳ ❝✮ ❈❤*♥ ♠!" ♥❤%♠ ❝% ✹ ❤*❝ +✐♥❤ ".♦♥❣ ✤% ❝% ➼" ♥❤➜" ♠!" ♥7✳ ✣→♣ #$✿ ❛✮ ✶✵✦ ❜✮ C61 C43 + C63 C41 ❝✮ C10 4 − C64 ❱➼ ❞0 ✷ ▼!" ❜✐➸♥ +H ①❡ ✤JK❝ ❝➜✉ "↕♦ "S ✺ ❝❤7 +H✳ ❍:✐ ❝% ❜❛♦ ♥❤✐➯✉ ❜✐➸♥ +H ❝%✿ ❛✮ ❈→❝ ❝❤7 +H ❦❤→❝ ♥❤❛✉✳ ❜✮ ➑" ♥❤➜" ✷ ❝❤7 +H ❣✐H♥❣ ♥❤❛✉✳ ❝✮ ❈% ✤W♥❣ ✸ ❝❤7 +H ❣✐H♥❣ ♥❤❛✉✳ ✣→♣ #$✿ ❛✮ A510 ❜✮ 105 − A510 ❝✮ ✣-.❝ /➼♥❤ 3✉❛ ✸ ❣✐❛✐ ✤♦↕♥✿ ✲ ●✤ ✶✿ ❈❤>♥ ✸ ✈@ /A➼ /A♦♥❣ ✺ ✈@ /A➼✿ C53 ✲ ●✤ ✷✿ ❈❤>♥ ✶ ❝❤D #$ ❣✐$♥❣ ♥❤❛✉ ❝❤♦ ✸ ✈@ /A➼ /A➯♥✿ C10 1 ✲ ●✤ ✸✿ ❝❤>♥ ✷ ❝❤D #$ ❝F♥ ❧↕✐ ✭❦❤J♥❣ ♣❤➙♥ ❜✐➺/✮✿ 92 ❱➟②✱ /❤❡♦ 3✉② /➢❝ ♥❤➙♥✿ C53 C10 2 ❝→❝❤✳ 1 9 ❚!♥ ❚❤➜% ❚& ✺✴✹✽ ✷✳ ❑❤5♥❣ ❣✐❛♥ ♠➝✉ ✈➔ ❜✐➳♥ ❝> ❛✳ ✣@♥❤ ♥❣❤➽❛ ✲ ❚❤➼ ♥❣❤✐➺♠ ♥❣➝✉ ♥❤✐➯♥✿ /❤➼ ♥❣❤✐➺♠ ♠➔ /❛ ❝) /❤➸ ❧➦♣ ❧↕✐ ♥❤✐➲✉ ❧➛♥ /L♦♥❣ ❝M♥❣ ✤✐➲✉ ❦✐➺♥ ♥❤,♥❣ ❦➳/ ;✉↔ ❦❤!♥❣ /❤➸ ❞0 ✤♦→♥ /L,Q❝✳ ✲ ❑❤]♥❣ ❣✐❛♥ ♠➝✉ ❝R❛ /❤➼ ♥❣❤✐➺♠✿ ❚➟♣ /➜/ ❝↔ ❝→❝ ❦➳/ ;✉↔✱ ❦➼ ❤✐➺✉ Ω✳ ✲ ❇✐➳♥ ❝H ✿ ♠W/ /➟♣ ❝♦♥ ❜➜/ ❦➻ ❝R❛ ❦❤!♥❣ ❣✐❛♥ ♠➝✉✳ ❑➼ ❤✐➺✉✿ ❆✱❇✱❈✱ ✳✳✳ ❱➲ ♠➦/ /❤0❝ /➳✱ ❜✐➳♥ ❝8 ✲ ❝→❝ ❤✐➺♥ /,:♥❣ ❝) /❤➸ ①↔② L❛ ❤♦➦❝ ❦❤!♥❣ ①↔② L❛ /L♦♥❣ /❤➼ ♥❣❤✐➺♠ ♥❣➝✉ ♥❤✐➯♥✳ ✲ ❇✐➳♥ ❝H +_ ❝➜♣✿ ❜✐➳♥ ❝8 ❝❤➾ ❣a♠ ✶ ♣❤➛♥ /b✳ ✲ ❇✐➳♥ ❝8 A ✤,:❝ ①❡♠ ❧➔ ①↔② .❛ ♥➳✉ ❝) ➼/ ♥❤➜/ ♠W/ ❦➳/ ;✉↔ /L♦♥❣ A ①✉➜/ ❤✐➺♥✳ ❱➼ ❞0 ✸ ✲ ●✐❡♦ ✤d♥❣ ①✉ ✈G✐ ✷ ♠➦" +➜♣✱ ♥❣f❛✿ Ω = {S, N }✳ ✲ ❈❤*♥ ♥❣➝✉ ♥❤✐➯♥ ✶ ❝❤7 +H "S ✵ ✤➳♥ ✾✿ Ω = {0, 1, 2, ..., 8, 9}✳ ✲ ●✐❡♦ ♥❣➝✉ ♥❤✐➯♥ ✷ ❝♦♥ ①W❝ ①➢❝✿ Ω = {(i, j) : i, j = 1, 6} ❚!♥ ❚❤➜% ❚& ✻✴✹✽
❱➼ ❞# ✹ ❈! ✸ ❤$❝ &✐♥❤ ❆✱ ❇✱ ❈ ✤-.❝ ①➳♣ ♥❣➝✉ ♥❤✐➯♥ 6❤➔♥❤ ✶ ❤➔♥❣ ❞$❝✳ ❑❤✐ ✤!✿ Ω = {ABC, ACB, BAC, BCA, CAB, CBA}✳ ▲>❝ ✤!✱ 6❛ ❝! ❜✐➳♥ ❝A✿ ✲ ❆ ✤C♥❣ ❣✐D❛✿ {BAC, CAB} ✲ ❆ ❦❤F♥❣ ✤C♥❣ ❣✐D❛✿ {ABC, ACB, BCA, CBA} ❱➼ ❞# ✺ ▼H6 ①↕ 6❤J ❜➢♥ ♥❣➝✉ ♥❤✐➯♥ ✈➔♦ ✶ 6➜♠ ❜✐❛ ❝❤♦ ✤➳♥ ❦❤✐ 6P>♥❣ ✤➼❝❤✳ ❑➼ ❤✐➺✉✿ ✶ ✲ 6P>♥❣✱ ✵ ✲6P➟6✳ ❑❤✐ ✤!✿ Ω = {1, 01, 001, 0001, 00001, ....} ▲>❝ ✤!✱ 6❛ ❝! ❜✐➳♥ ❝A✿ ✲ ❜➢♥ 6P>♥❣ ✤➼❝❤ ❦❤F♥❣ U✉→ ✸ ❧➛♥✿ A = {1, 01, 001} ✲ 6❤Y❝ ❤✐➺♥ ➼6 ♥❤➜6 ✷ ❧➛♥ ❜➢♥✿ B = {01, 001, 0001, ....} ❚!♥ ❚❤➜% ❚& ✼✴✹✽ ❈→❝ ❜✐➳♥ ❝- ✤➦❝ ❜✐➺1✿ ✲ ❇✐➳♥ ❝A ❦❤F♥❣ 6❤➸ ✭∅✮✿ ❧➔ ❜✐➳♥ ❝+ ❦❤.♥❣ 0❤➸ ①↔② 5❛ ✲ ❇✐➳♥ ❝A ❝❤➢❝ ❝❤➢♥ ✭Ω✮✿ ❧➔ ❜✐➳♥ ❝+ ❧✉.♥ ①↔② 5❛ ❈❤➥♥❣ ❤↕♥✱ ❣✐❡♦ ♥❣➝✉ ♥❤✐➯♥ ✷ ❝♦♥ ①A❝ ①➢❝✳ ▲A❝ ✤F✿ ✲ ❜✐➳♥ ❝+ ✧0H♥❣ I+ ❝❤➜♠ ①✉➜0 ❤✐➺♥ ❧M♥ ❤N♥ ✶✷✧ ❧➔ ❜✐➳♥ ❝+ ❦❤.♥❣ 0❤➸✳ ✲ ❜✐➳♥ ❝+ ✧0H♥❣ I+ ❝❤➜♠ ①✉➜0 ❤✐➺♥ ❧M♥ ❤N♥ ✶✧ ❧➔ ❜✐➳♥ ❝+ ❝❤➢❝ ❝❤➢♥✳ ❜✳ ❈→❝ ♣❤➨♣ 1♦→♥ 18➯♥ ❜✐➳♥ ❝- ❈❤♦ ❤❛✐ ❜✐➳♥ ❝+ ❆ ✈➔ ❇✳ ❑❤✐ ✤F✿ ✲ ❇✐➳♥ ❝A ✤A✐ ❝T❛ ❆✱ ❦➼ ❤✐➺✉ Ā ❤♦➦❝ Ac ✱ ①→❝ ✤X♥❤✿ Ā = Ω\A ✭①↔② 5❛ ♥➳✉ ❆ ❦❤.♥❣ ①↔② 5❛✮ ✲ ●✐❛♦ ❝T❛ ❆ ✈➔ ❇✱ ❦➼ ❤✐➺✉ A ∩ B, AB ✱ ①→❝ ✤X♥❤ AB = {x : x ∈ A, x ∈ B} ✭①↔② 5❛ ♥➳✉ ❆ ✈➔ ❇ ✤Y♥❣ 0❤Z✐ ①↔② 5❛✮✳ ✲ ❍.♣ ❝T❛ ❆ ✈➔ ❇✱ ❦➼ ❤✐➺✉ A ∪ B ✱ ①→❝ ✤X♥❤ A ∪ B = {x : x ∈ A ❤♦➦❝ x ∈ B} ✭①↔② 5❛ ♥➳✉ ➼0 ♥❤➜0 ❆ ❤♦➦❝ ❇ ①↔② 5❛✮ ✲ ❍❛✐ ❜✐➳♥ ❝+ ❆ ✈➔ ❇ ✤\]❝ ❣^✐ ❧➔ ①✉♥❣ ❦❤➢❝ ♥➳✉ ❝❤A♥❣ ❦❤.♥❣ ✤Y♥❣ 0❤Z✐ ①↔② 5❛✳ ❚!♥ ❚❤➜% ❚& ✽✴✹✽ ◆❤➟♥ ①➨1 ✲ ❍❛✐ ❜✐➳♥ ❝+ ❆ ✈➔ ❇ ①✉♥❣ ❦❤➢❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ AB = ∅✳ ✲ ❍❛✐ ❜✐➳♥ ❝+ ✤+✐ ♥❤❛✉ 0❤➻ ①✉♥❣ ❦❤➢❝ ✈M✐ ♥❤❛✉✱ ♥❤\♥❣ ✤✐➲✉ ♥❣\]❝ ❧↕✐ ♥F✐ ❝❤✉♥❣ ❧➔ ❦❤.♥❣ ✤A♥❣✳ ❱➼ ❞# ✻ ❈❤$♥ ♥❣➝✉ ♥❤✐➯♥ ♠H6 ❝❤D &A 6^ ✵ ✤➳♥ ✾✳ ❚❛ ❝! Ω = {0, 1, 2, ..., 9}✳ ❳➨6 ✸ ❜✐➳♥ ❝A A = {0, 2, 4}✱ B = {4, 5, 6} ✈➔ C = {7, 8, 9}✳ ▲>❝ ✤!✿ • A ∩ B = {4}, A ∪ B = {0, 2, 4, 5, 6}, Ā = {1, 3, 5, 6, 7, 8, 9} • ❆ ✈➔ ❈ ①✉♥❣ ❦❤➢❝ ✈➻ A ∩ C = ∅ ♥❤-♥❣ ❆ ✈➔ ❈ ❦❤F♥❣ ♣❤↔✐ ❧➔ ❤❛✐ ❜✐➳♥ ❝A ✤A✐ ♥❤❛✉✳ ❚!♥ ❚❤➜% ❚& ✾✴✹✽
❱➼ ❞# ✼ ❈! ✷ ①↕ %❤'✱ ♠*✐ ♥❣./✐ ❜➢♥ ✶ ✈✐➯♥ ✤↕♥ ✈➔♦ ♠8❝ %✐➯✉✳ ●=✐ ❆ ✈➔ ❇ %.@♥❣ A♥❣ ❧➔ ❝→❝ ❜✐➳♥ ❝E✿ ✏♥❣./✐ %❤A ♥❤➜% ✈➔ %❤A ❤❛✐ ❜➢♥ %JK♥❣ ♠8❝ %✐➯✉✑ %.@♥❣ A♥❣✳ ❑❤✐ ✤! %❛ ❝! ❜✐➸✉ ❞✐➵♥ ❝→❝ ❜✐➳♥ ❝E ♥❤. Q❛✉✿ ✲ ❈! ✤↕♥ %JK♥❣ ✤➼❝❤✿ A ∪ B ✲ ❈! ✤K♥❣ ✶ ✈✐➯♥ ✤↕♥ %JK♥❣ ✤➼❝❤✿ AB̄ ∪ ĀB ✲ ❈❤➾ ❝! ♥❣./✐ %❤A ♥❤➜% ❜➢♥ %JK♥❣✿ AB̄ ✲ ❈! ♥❤✐➲✉ ♥❤➜% ♠V% ✈✐➯♥ ✤↕♥ %JK♥❣ ✤➼❝❤✿ ĀB̄ ∪ AB̄ ∪ ĀB ❤♦➦❝ AB ▲✉➟( ❉❡✲▼♦.❣❛♥ ✈➲ ♣❤6 ✤8♥❤✿ ∪Ai = ∩Āi ∩Ai = ∪Āi ❚!♥ ❚❤➜% ❚& ✶✵✴✹✽ ✸✳ ✣8♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ (➼♥❤ ❝❤➜( ❝6❛ ①→❝ C✉➜( ✣8♥❤ ♥❣❤➽❛ ✭(❤❡♦ ❤➺ (✐➯♥ ✤➲✮✿ ❈❤♦ #$%&❝ ♠)# #❤➼ ♥❣❤✐➺♠ ✈&✐ ❦❤1♥❣ ❣✐❛♥ ♠➝✉ Ω✳ ❑❤✐ ✤8✱ ①→❝ C✉➜( ❝:❛ ♠)# ❜✐➳♥ ❝= ❆✱ ❦➼ ❤✐➺✉ P (A)✱ ❧➔ A= ✤♦ ❦❤↔ ♥➠♥❣ ①↔② $❛ ❜✐➳♥ ❝= A✳ F♥❣ ✈&✐ ♠G✐ ❜✐➳♥ ❝= ❆ #❛ ✤➦# #%I♥❣ J♥❣ ✈&✐ ❣✐→ #$L P (A) #❤M❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ A❛✉✿ ✭✐✮ ❱&✐ ♠S✐ ❜✐➳♥ ❝= A, P (A) ≥ 0 ✭✐✐✮ P (Ω) = 1 ✭✐✐✐✮ ◆➳✉ A1 , A2 , ... ❧➔ ❝→❝ ❜✐➳♥ ❝= ✤1✐ ♠)# ①✉♥❣ ❦❤➢❝ #❤➻ +∞ X P (∪∞ n=1 An ) = P (An ) n=1 ▲X❝ ✤8 P (A) ✤%Y❝ ❣S✐ ❧➔ ①→❝ Q✉➜% ❝:❛ ❜✐➳♥ ❝= ❆✳ ◆❤➟♥ ①➨(✿ ❚$♦♥❣ ♠)# A= #$%[♥❣ ❤Y♣✱ #]② ✈➔♦ ❦❤1♥❣ ❣✐❛♥ ♠➝✉ ❝^♥❣ ♥❤% ❝→❝ #❤✐➳# ❧➟♣ ✧#%I♥❣ J♥❣✧ a ♠➔ #❛ A➩ #❤✉ ✤%Y❝ ♠)# ✈➔✐ ✤L♥❤ ♥❣❤➽❛ ①→❝ A✉➜# ❦❤→❝✳ ❚!♥ ❚❤➜% ❚& ✶✶✴✹✽ ✣8♥❤ ♥❣❤➽❛ ✭K✉❛♥ ✤✐➸♠ (❤N♥❣ ❦➯✮✿ ❳➨# ❜✐➳♥ ❝= A✳ ❚❤g❝ ❤✐➺♥ ♣❤➨♣ #❤h n ❧➛♥ #❤➻ ❝8 m ❧➛♥ ①✉➜# ❤✐➺♥ ❜✐➳♥ ❝= A✳ ❑❤✐ ✤8 #➾ A= fn = m/n ✤%Y❝ ❣S✐ ❧➔ %➛♥ Q✉➜% ①✉➜# ❤✐➺♥ A✳ ❑❤✐ A= ♣❤➨♣ #❤h #➠♥❣ ❧➯♥ ✈1 ❤↕♥✱ #➛♥ A✉➜# fn A➩ #✐➳♥ ✤➳♥ ♠)# ❤➡♥❣ A= ①→❝ ✤L♥❤✳ ❍➡♥❣ A= ♥➔② ✤%Y❝ ❣S✐ ❧➔ ①→❝ Q✉➜% ❝:❛ ❜✐➳♥ ❝= A✳ ❱➻ #❤➳✱ #$♦♥❣ #❤g❝ #➳ ❦❤✐ A= ♣❤➨♣ #❤h n ❧&♥✱ #❛ ❝8 #❤➸ ①❡♠ #➛♥ A✉➜# fn ♥❤% ❧➔ ①→❝ A✉➜# ❝:❛ ❤✐➺♥ #%Y♥❣ ❆✳ ❱➼ ❞# ✽ ❈! ✶✵✵✵ ♥❣./✐ ❝! %J✐➺✉ ❝❤A♥❣ ❆ ✤➳♥ ❝@ Q[ ② %➳ ✤➸ ❦❤→♠ ❜➺♥❤✳ ❑➳% ^✉↔ ❝! %❤➜② ✼✵✵ ♠➢❝ ❜➺♥❤ ❳✳ ❚❛ ❝! f = 700/1000 = 70%✳ ❉♦ ✤!✱ %❛ ❝! ❝@ Q[ ❞d ✤♦→♥ ♥➳✉ ♠V% ♥❣./✐ ❝! %J✐➺✉ ❝❤A♥❣ ❆ %❤➻ ①→❝ Q✉➜% ♠➢❝ ❜➺♥❤ ❳ ①➜♣ ①➾ ✼✵✪✳ ❚!♥ ❚❤➜% ❚& ✶✷✴✹✽
✣!♥❤ ♥❣❤➽❛ ✭(✉❛♥ ✤✐➸♠ ❝/ ✤✐➸♥✮✿ ❈❤♦ #❤➼ ♥❣❤✐➺♠ ✈+✐ n(Ω) ❦➳# .✉↔ ✤2♥❣ ❦❤↔ ♥➠♥❣✱ #5♦♥❣ ✤6 ❝6 ♥✭❆✮ ❦➳# .✉↔ #❤✉➟♥ ❧=✐ ❝❤♦ ❜✐➳♥ ❝? ❆✳ ❑❤✐ ✤6✱ ①→❝ #✉➜& ❝B❛ ❜✐➳♥ ❝? ❆✱ ❦➼ ❤✐➺✉ D✭❆✮✱ ✤E=❝ ①→❝ ✤H♥❤✿ n(A) P (A) = n(Ω) ❱➼ ❞5 ✾ ❚(♦♥❣ ♠-& ❤-♣ ❝0 ✻ ❜✐ ✤❡♥ ✈➔ ✹ ❜✐ &(➢♥❣✳ ▲➜② ♥❣➝✉ ♥❤✐➯♥ ✹ ✈✐➯♥✳ ❚➻♠ ①→❝ #✉➜& ❧➜② ✤AB❝ ❝↔ ❤❛✐ ❧♦↕✐ ❜✐✳ ●✐↔✐✳ ●K✐ ❆ ❧➔ ❜✐➳♥ ❝? ✧❧➜② ✤E=❝ ❝↔ ❤❛✐ ❧♦↕✐ ❜✐✧✳ ❙? #5ER♥❣ ❤=♣ ✤2♥❣ ❦❤↔ ♥➠♥❣✿ n(Ω) = C10 4 ✳ ❙? #5ER♥❣ ❤=♣ #❤✉➟♥ ❧=✐✿ n(A) = C6 C4 + C62 C42 + C63 C41 ✳ 1 3 ❳→❝ U✉➜# ❝➛♥ #➻♠✿ n(A) C C + C6 C4 + C63 C41 1 3 2 2 P (A) = = 6 4 4 n(Ω) C10 ❚!♥ ❚❤➜% ❚& ✶✸✴✹✽ ❱➼ ❞5 ✶✵ ▼-& ✤♦➔♥ &➔✉ ❝0 ✸ &♦❛✳ ❈0 ✶✺ ❦❤→❝❤ ❧➯♥ ♥❣➝✉ ♥❤✐➯♥ ✸ &♦❛ &➔✉✳ ❇✐➳& ♠N✐ &♦❛ ✤➲✉ ❝❤P❛ ✤AB❝ ✶✺ ❦❤→❝❤✳ ❚➼♥❤ ①→❝ #✉➜&✿ ❛✳ ❚♦❛ ✶ ❝0 ✹ ❦❤→❝❤✳ ❜✳ ❈0 ✷ &♦❛ ♠N✐ &♦❛ ❝0 ✻ ❦❤→❝❤✳ ●✐↔✐✳ ❛✳ ●K✐ ❆ ❧➔ ❜✐➳♥ ❝? ✧#♦❛ ✶ ❝6 ✹ ❦❤→❝❤✧✳ ❚❛ ❝6✿ n(Ω) = 315 ✳ ●✐→ #5H n(A) ✤E=❝ #➼♥❤ #❤[♥❣ .✉❛ ✷ ❣✐❛✐ ✤♦↕♥✿ ✲ ●✤ ✶✿ ❈❤K♥ ✹ ❦❤→❝❤✿ C154 ❝→❝❤ ✲ ●✤ ✷✿ ❳➳♣ ✶✶ ❦❤→❝❤ ❝^♥ ❧↕✐✿ 211 ❝→❝❤✳ ❙✉② 5❛✿ n(A) = C154 211 ❳→❝ U✉➜# ❝➛♥ #➻♠✿ n(A) C 4 211 P (A) = = 1515 n(Ω) 3 ❚!♥ ❚❤➜% ❚& ✶✹✴✹✽ ❜✳ ●K✐ ❇ ❧➔ ❜✐➳♥ ❝? ✧❝6 ✷ #♦❛ ♠`✐ #♦❛ ❝6 ✻ ❦❤→❝❤✧✳ ●✐→ #5H n(B) ✤E=❝ #➼♥❤ #❤[♥❣ .✉❛ ❝→❝ ❣✐❛✐ ✤♦↕♥✿ ✲ ●✤ ✶✿ ❈❤K♥ ✷ #♦❛✿ C32 ❝→❝❤ ✲ ●✤ ✷✿ ❈❤K♥ ✻ ❦❤→❝❤ ❝❤♦ ♠`✐ #♦❛✿ C156 C 6 ❝→❝❤✳ 9 ✲ ●✤ ✸✿ ❳➳♣ ✸ ❦❤→❝❤ ❝^♥ ❧↕✐ 13 = 1 ❝→❝❤ ❙✉② 5❛✿ n(B) = C32 C15 6 C6 9 ❳→❝ U✉➜# ❝➛♥ #➻♠✿ n(B) C 2C 6 C 63 15 9 P (B) = = n(Ω) 315 ❱➼ ❞5 ✶✶ ▼-& (T ❝❛♠ ❣U♠ ✶✷ V✉↔✱ &(♦♥❣ ✤0 ❝0 ✸ V✉↔ ❤X♥❣✳ ❈❤✐❛ ✤➲✉ ✶✷ V✉↔ ♥➔② ❝❤♦ ✸ ♥❣AY✐✱ ♠N✐ ♥❣AY✐ ✹ V✉↔✳ ❚➼♥❤ ①→❝ #✉➜&✿ ❛✳ ◆❣AY✐ &❤P ♥❤➜& ❦❤[♥❣ ❝0 V✉↔ ❤X♥❣✳ ❜✳ ▼N✐ ♥❣AY✐ ✤➲✉ ❝0 V✉↔ ❤X♥❣✳ ●!✐ #✳ ❚❛ ❝#✿ n(Ω) = C124 C84 C44 ❛✳ n(A) = C94 C84 C44 ❜✳ n(B) = 3!C93 C63 C33 ❚!♥ ❚❤➜% ❚& ✶✺✴✹✽
❚➼♥❤ ❝❤➜& ✐✮ P (∅) = 0, P (Ω) = 1 ✐✐✮ 0 ≤ P (A) ≤ 1, ∀A ✐✐✐✮ ◆➳✉ A ⊂ B %❤➻ P (A) ≤ P (B) ✐✈✮ ❈*♥❣ %❤-❝ ❝/♥❣✿ P (A ∪ B) = P (A) + P (B) − P (AB) P (A ∪ B ∪ C) = P (A) + P (B) + P (C) − P (AB) − P (BC) − P (AC) + P (ABC) P (A1 ∪ A2 ∪ ... ∪ An ) = P (A1 ) + ... + P (An ) X − P (Ai Aj ) i
❱➼ ❞# ✶✹ ▼!" ✤!✐ ❜&♥❣ ❜➔♥ ❝+❛ ✶ ✤.♥ ✈0 ❣1♠ ✷ ✈➟♥ ✤!♥❣ ✈✐➯♥ ❆ ✈➔ ❇✳ ❳→❝ ;✉➜" ❆✱ ❇ ✈?@" A✉❛ ✈B♥❣ ❜↔♥❣ ❧➛♥ ❧?@" ❧➔ ✵✱✼ ✈➔ ✵✱✺✳ ❉♦ ↔♥❤ ❤?L♥❣ "➙♠ ❧N ♥➯♥ ①→❝ ;✉➜" ❝↔ ❤❛✐ ♥❣?P✐ ✤➲✉ ✈?@" A✉❛ ✈B♥❣ ❜↔♥❣ ❧➔ ✵✱✹✳ ❚➼♥❤ ①→❝ ;✉➜" ❝↔ ❤❛✐ ✈➟♥ ✤!♥❣ ✈✐➯♥ ✤➲✉ ❦❤V♥❣ ✈?@" A✉❛ ✈B♥❣ ❜↔♥❣✳ ●✐↔✐✳ ●!✐ ❆✱ ❇ ❧➔ ❝→❝ ❜✐➳♥ ❝- ✈➟♥ ✤1♥❣ ✈✐➯♥ ❆✱ ❇ ✈456 7✉❛ ✈:♥❣ ❜↔♥❣✳ ❚❛ ❝>✿ P (A) = 0, 7; P (B) = 0, 5; P (AB) = 0, 4 ❳→❝ A✉➜6 ❝↔ ❤❛✐ ✈➟♥ ✤1♥❣ ✈✐➯♥ ✤➲✉ ❦❤F♥❣ ✈456 7✉❛ ✈:♥❣ ❜↔♥❣✿ P (ĀB̄) = 1 − P (A ∪ B) = 1 − [P (A) + P (B) − P (AB)] = 1 − (0, 7 + 0, 5 − 0, 4) = 0, 2 ❚!♥ ❚❤➜% ❚& ✶✾✴✹✽ ✹✳ ❳→❝ -✉➜0 ❝1 ✤✐➲✉ ❦✐➺♥ ▼16 ❧F ❤➔♥❣ 6✐✈✐ ✈H✐ ❝→❝ A- ❧✐➺✉ A❛✉✿ ▲● ❙❖◆❨ ❚-6 ✶✺ ✶✷ ❇R ❤S♥❣ ✺ ✽ ❈❤!♥ ♥❣➝✉ ♥❤✐➯♥ ✶ 6✐✈✐ 6❤➻ ✤45❝ 6✐✈✐ ▲●✳ ❱➟②✱ ❦❤↔ ♥➠♥❣ ♥> ❜R ❤S♥❣ ❧➔ ❜❛♦ ♥❤✐➯✉❄ ●7✐ 8✳ ✲ ❑❤✐ ❦❤F♥❣ ❝> 6❤F♥❣ 6✐♥✱ ❞➵ 6❤➜② ①→❝ A✉➜6 ♥➔② ❧➔ ✶✸✴✹✵✳ ✲ ❱➻ ❜✐➳6 ❝❤!♥ ✤45❝ 6✐✈✐ ▲● ♥➯♥ 6❛ ✧6❤✉ ❤➭♣✧ ♣❤↕♠ ✈✐ 7✉❛♥ A→6 ✈➔ ❝❤➾ 6➼♥❤ ✤➳♥ ❧♦↕✐ 6✐✈✐ ▲●✳ ▲m❝ ✤>✱ ❦❤↔ ♥➠♥❣ ❝❤!♥ ✤45❝ 6✐✈✐ ❤S♥❣ ✈H✐ ✧✤✐➲✉ ❦✐➺♥✧ ❧➔ 6✐✈✐ ▲● ❜➡♥❣ ✺✴✷✵❂✶✴✹✳ ❇■➌❚ ✣■➋❯ ❑■➏◆ ❳❷❨ ❘❆ → ❳⑩❈ ❙❯❻❚ ❚❍❆❨ ✣✃■ ❚!♥ ❚❤➜% ❚& ✷✵✴✹✽ ✣M♥❤ ♥❣❤➽❛✿ ❈❤♦ ❤❛✐ ❜✐➳♥ ❝- ❆ ✈➔ ❇ ✈H✐ P (B) > 0✳ ❑❤✐ ✤>✱ ①→❝ A✉➜6 ❝> ✤✐➲✉ ❦✐➺♥ ❝p❛ ❆ ✈H✐ ✤✐➲✉ ❦✐➺♥ ❜✐➳♥ ❝- ❇ ✤➣ ①↔② r❛✱ ❦➼ ❤✐➺✉ P (A|B)✱ ①→❝ ✤R♥❤ ♥❤4 A❛✉✿ P (AB) P (A|B) = P (B) ❚➼♥❤ ❝❤➜0✿ ✐✳ 0 ≤ P (A|B) ≤ 1 ✐✐✳ P (Ā|B) = 1 − P (A|B) ✐✐✐✳ P (A ∪ B|C) = P (A|C) + P (B|C) − P (AB|C) ❚!♥ ❚❤➜% ❚& ✷✶✴✹✽
❱➼ ❞# ✶✺ ▼!" ❝$♥❣ "② ✤➜✉ "❤➛✉ ✷ ❞/ →♥ ❆ ✈➔ ❇✳ ❳→❝ 7✉➜" "❤➢♥❣ "❤➛✉ ❞/ →♥ ❆ ✈➔ ❇ "9:♥❣ ;♥❣ ❧➔ ✵✱✻ ✈➔ ✵✱✼✳ ❳→❝ 7✉➜" "❤➢♥❣ "❤➛✉ ✤A♥❣ "❤B✐ ❝↔ ✷ ❞/ →♥ ❧➔ ✵✱✺✳ ❚➼♥❤ ①→❝ 7✉➜"✿ ❛✮ ❈$♥❣ "② "❤➢♥❣ "❤➛✉ ❞/ →♥ ❆ ❜✐➳" ✤➣ "❤➢♥❣ "❤➛✉ ❞/ →♥ ❇✳ ❜✮ ❈$♥❣ "② ❦❤$♥❣ "❤➢♥❣ "❤➛✉ ❞/ →♥ ❇ ❜✐➳" ✤➣ "❤➢♥❣ "❤➛✉ ❞/ →♥ ❆✳ ●✐↔✐✳ ●!✐ ❆✱ ❇ ❧➔ ❝→❝ ❜✐➳♥ ❝- ❝.♥❣ 0② 0❤➢♥❣ ❞5 →♥ ❆✱ ❇✳ ❚❤❡♦ ❣✐↔ 0❤✐➳0✿ P (A) = 0, 6; P (B) = 0, 7; P (AB) = 0, 5 ❛✳ P (A|B) = P (AB)/P (B) = 5/7 ❜✳ P (B̄|A) = 1 − P (B|A) = 1 − P (AB)/P (A) = 1 − 5/6 = 1/6✳ ❚!♥ ❚❤➜% ❚& ✷✷✴✹✽ ❇➻♥❤ ❧✉➟♥✳ ❚❛ ❝=✿ P (A|B) = 5/7 > P (A) = 0, 6✳ ◆❤? ✈➟②✱ ✈✐➺❝ ❜✐➳0 ❜✐➳♥ ❝- ❇ ①↔② D❛ ✤➣ ❧➔♠ 0➠♥❣ ①→❝ I✉➜0 ①↔② D❛ ❜✐➳♥ ❝- ❆✱ ❤❛② ♥=✐ ❝→❝❤ ❦❤→❝✱ ✈✐➺❝ 0❤➢♥❣ ❞5 →♥ ❇ ❧➔ ❝= ❧M✐ ❝❤♦ N✉→ 0D➻♥❤ ✤➜✉ 0❤➛✉ ❞5 →♥ ❆✳ ◆❤➟♥ ①➨4 ❚D♦♥❣ ♠Q0 I- 0D?R♥❣ ❤M♣✱ ①→❝ I✉➜0 ❝= ✤✐➲✉ ❦✐➺♥ P (A|B) ❝= 0❤➸ ✤?M❝ 0➼♥❤ ❞5❛ ✈➔♦ ✈✐➺❝ ✧✤➳♠ 0D5❝ N✉❛♥✧✱ 0X❝ ❧➔ 0❛ 0❤✉ ❤➭♣ 0➟♣ ❝→❝ ❦➳0 N✉↔ I❛✉ ❦❤✐ ❜✐➳♥ ❝- ❇ ①↔② D❛ ✈➔ ✤➳♠ ❧↕✐ I- ❦➳0 N✉↔ ✤[♥❣ ❦❤↔ ♥➠♥❣✱ I- ❦➳0 N✉↔ 0❤✉➟♥ ❧M✐ ✤➸ 0➼♥❤ ①→❝ I✉➜0 ❝= ✤✐➲✉ ❦✐➺♥✳ ❈❤➥♥❣ ❤↕♥✱ 0❛ N✉❛② ❧↕✐ ✈\✐ ✈➼ ❞] ✈➲ 0✐✈✐ ❙♦♥② ✈➔ ▲●✳ ❱➻ ❜✐➳0 ❝❤!♥ ✤?M❝ 0✐✈✐ ▲● ♥➯♥ 0➟♣ ❦➳0 N✉↔ 0❤✉ ❤➭♣ ✈➲ ✷✵ 0✐✈✐ ▲●✳ ❈❤➾ ❝= ✺ 0✐✈✐ ▲● ❜g ❤h♥❣✱ ♥➯♥ ①→❝ I✉➜0 ❝❤!♥ ✤?M❝ 0✐✈✐ ❤h♥❣ ❦❤✐ ❜✐➳0 ✤= ❧➔ 0✐✈✐ ❜➡♥❣ ✺✴✷✵❂✶✴✹✳ ❚!♥ ❚❤➜% ❚& ✷✸✴✹✽ ❱➼ ❞# ✶✻ ❈Q ✶✺ "❤❛♥❤ ❘❆▼✱ "T♦♥❣ ✤Q ❝Q ✸ "❤❛♥❤ ❜W ❤X♥❣✳ ❈❤Y♥ ♥❣➝✉ ♥❤✐➯♥ ✹ "❤❛♥❤✳ ❛✳ ❚➼♥❤ ①→❝ 7✉➜" ❝Q ➼" ♥❤➜" ✶ "❤❛♥❤ ❜W ❤X♥❣ ✤9]❝ ❝❤Y♥✳ ❜✳ ❇✐➳" ❝❤Y♥ ✤9]❝ ➼" ♥❤➜" ✶ "❤❛♥❤ ❤X♥❣✱ "➼♥❤ ①→❝ 7✉➜" ❝❤Y♥ ✤^♥❣ ✷ "❤❛♥❤ ❜W ❤X♥❣✳ ●✐↔✐✳ ●!✐ ❆✱ ❇ ❧➔ ❝→❝ ❜✐➳♥ ❝- ❝❤!♥ ✤?M❝ ➼0 ♥❤➜0 ✶ 0❤❛♥❤ ❤h♥❣✱ ❝❤!♥ ✤?M❝ ✤n♥❣ ✷ 0❤❛♥❤ ❤h♥❣✳ ❛✳ P (A) = 1 − P (Ā) = 1 − C12 4 /C 4 15 ❜✳ P (B|A) = P (AB)/P (A) = P (B)/P (A)✱ 0D♦♥❣ ✤= P (B) = C32 C12 2 /C 4 ✳ 15 ❚!♥ ❚❤➜% ❚& ✷✹✴✹✽
✺✳ ❈#♥❣ &❤(❝ ♥❤➙♥ ①→❝ -✉➜& ❈❤♦ ❤❛✐ ❜✐➳♥ ❝) ❆ ✈➔ ❇ ✈.✐ P (A) > 0✳ ❑❤✐ ✤2✿ P (AB) P (B|A) = ⇒ P (AB) = P (A)P (B|A) P (A) ❚56♥❣ 89✱ 8❛ ;➩ ❝2✿ P (AB) = P (A)P (B|A) = P (B)P (A|B) ❚1♥❣ 2✉→&✿ ❈❤♦ n ❜✐➳♥ ❝) Ai ✈.✐ P (A1 A2 ...An−1 ) > 0✳ ❑❤✐ ✤2✿ P (A1 A2 ...An ) = P (A1 ).P (A2 |A1 ).P (A3 |A1 A2 )....P (An |A1 ...An−1 ) ❚!♥ ❚❤➜% ❚& ✷✺✴✹✽ ❱➼ ❞7 ✶✼ ▼!" ❝$♥❣ "② ✤➜✉ "❤➛✉ ✷ ❞/ →♥ ❆ ✈➔ ❇✳ ❳→❝ 7✉➜" "❤➢♥❣ "❤➛✉ ❧➛♥ ❧:;" ❧➔ ✵✳✼ ✈➔ ✵✳✹✳ ◆➳✉ ❞/ →♥ ❆ ✤➣ "❤➢♥❣ "❤➛✉ "❤➻ ①→❝ 7✉➜" "❤➢♥❣ "✐➳♣ ❞/ →♥ ❇ ❧➔ ✵✳✹✳ ❚➼♥❤ ①→❝ 7✉➜"✿ ❛✳ ❈$♥❣ "② "❤➢♥❣ ➼" ♥❤➜" ♠!" ❞/ →♥✳ ❜✳ ❈$♥❣ "② ❝❤➾ "❤➢♥❣ ❞/ →♥ ❆✳ ❝✳ ❚❤➢♥❣ ✤N♥❣ ✶ ❞/ →♥✳ ●✐↔✐✳ ●>✐ ❆✱ ❇ ❧➔ ❜✐➳♥ ❝) ❝@♥❣ 8② 8❤➢♥❣ ❞9 →♥ ❆✱ ❇✳ ❚❛ ❝2✿ P (A) = 0, 7; P (B) = 0, 4, P (B|A) = 0, 4 ❛✳ P (A ∪ B) = P (A) + P (B) − P (AB) = P (A) + P (B) − P (A)P (B|A) = 0, 7 + 0, 4 − 0, 7 ∗ 0, 4 = 0, 82 ❜✳ P (AB̄) = P (A)P (B̄|A) = P (A)[1 − P (B|A)] = 0, 7[1 − 0, 4] = 0, 42 ❝✳ P (AB̄ ∪ ĀB) = P (AB̄) + P (ĀB) = . . . ❚!♥ ❚❤➜% ❚& ✷✻✴✹✽ ❱➼ ❞7 ✶✽ ▼!" "❤P ❦❤♦ ❝S ♠!" ❝❤T♠ ❝❤➻❛ ❦❤S❛ ❣U♠ ✾ ❝❤✐➳❝✱ ❜➲ ♥❣♦➔✐ ❝❤N♥❣ ❣✐Y♥❣ ❤➺" ♥❤❛✉ ♥❤:♥❣ "[♦♥❣ ✤S ❝❤➾ ❝S ✤N♥❣ ✷ ❝❤✐➳❝ ♠\ ✤:;❝ ❦❤♦✳ ❆♥❤ "❛ "❤] ♥❣➝✉ ♥❤✐➯♥ "`♥❣ ❝❤➻❛ ❝❤♦ ✤➳♥ ❦❤✐ ♠\ ✤:;❝ ❦❤♦ "❤➻ ❞`♥❣✳ ❚➼♥❤ ①→❝ 7✉➜" ✈✐➺❝ ❧➔♠ ♥➔②✿ ❛✳ ❉`♥❣ \ ❧➛♥ "❤] "❤b ✷✳ ❜✳ ❉`♥❣ \ ❧➛♥ "❤] "❤b ✸✳ ❝✳ ❙❛✉ ♥❤✐➲✉ ♥❤➜" ❤❛✐ ❧➛♥ "❤]✳ ●✐↔✐✳ ●>✐ Ai ❧➔ ❜✐➳♥ ❝) ❧➛♥ 8❤F i ❝❤>♥ ✤G♥❣ ❝❤➻❛ ❦❤2❛✱ i = 1, 2, 3, ... ❛✳ P (Ā1 A2 ) = P (Ā1 )P (A2 |Ā1 ) = 7/9 ∗ 2/8 ❜✳ P (Ā1 Ā2 A3 ) = P (Ā1 )P (Ā2 |Ā1 )P (A3 |Ā1 Ā2 ) = 7/9 ∗ 6/8 ∗ 2/7 ❝✳ P (A1 ∪ Ā1 A2 ) = P (A1 ) + P (Ā1 A2 ) = 2/9 + 7/9 ∗ 2/8 ❚!♥ ❚❤➜% ❚& ✷✼✴✹✽
P (AB) = P (A).P (B) A1 , ..., An \ Y P( Ak ) = P (Ak ), ∀I ⊂ {1, 2, ..., n} k∈I k∈I ⇔ P (A|B) = P (A) P (B|A) = P (B) n A1 , ..., An n B1 , B2 , ..., Bn Bi Ai Āi n A1 , ..., An Ai i i = 1, 2, 3 A1 , A2 , A3 P (A1 ) = 0, 1; P (A2 ) = 0, 2; P (A3 ) = 0, 3 P (Ā1 ∪ Ā2 ∪ Ā3 ) = 1 − P (A1 A2 A3 ) = 1 − P (A1 )P (A2 )P (A3 ) = 0, 994 P (A) = 0, 1; P (B) = 0, 1; P (C) = 0, 05 P (AB ∪ C) = P (AB) + P (C) − P (ABC) = P (A)P (B) + P (C) − P (A)P (B)P (C)
❱➼ ❞# ✷✶ ▼!" "❤✐➳" ❜' ❝) ✷ ❜! ♣❤➟♥ ❤♦↕" ✤!♥❣ ✤!❝ ❧➟♣✳ ❳→❝ 6✉➜" ❜! ♣❤➟♥ "❤9 ♥❤➜" ❜' ❤:♥❣ ❧➔ ✵✱✶✳ ❳→❝ 6✉➜" ❝) ✤?♥❣ ✶ ❜! ♣❤➟♥ ❜' ❤:♥❣ ❧➔ ✵✱✷✻✳ ❛✳ ❚➼♥❤ ①→❝ 6✉➜" ❜! ♣❤➟♥ "❤9 ✷ ❜' ❤:♥❣✳ ❜✳ ❇✐➳" ❝) ➼" ♥❤➜" ✶ ❜! ♣❤➟♥ ❤:♥❣✱ "➼♥❤ ①→❝ 6✉➜" ❜! ♣❤➟♥ ✶ ❤:♥❣✳ ●✐↔✐✳ ●!✐ Ai ❧➔ ❜✐➳♥ ❝) ❜* ♣❤➟♥ .❤/ i ❜0 ❤1♥❣✱ i = 1, 2✳ ❚❛ ❝7 A1 , A2 ✤*❝ ❧➟♣✳ ❛✳ ❚❤❡♦ ❣✐↔ .❤✐➳.✿ P (A1 Ā2 ∪ Ā1 A2 ) = P (A1 Ā2 ) + P (Ā1 A2 ) = 0, 1 ∗ [1 − P (A2 )] + 0, 9P (A2 ) = 0, 1 + 0, 8P (A2 ) = 0, 26✳ ❙✉② @❛✿ P (A2 ) = 0, 2✳ P (A1 (A1 ∪ A2 )) P (A1 ) ❜✳ P (A1 |A1 ∪ A2 ) = = P (A1 ∪ A2 ) P (A1 ∪ A2 ) P (A1 ∪ A2 ) = P (A1 ) + P (A2 ) − P (A1 A2 ) = 0, 1 + 0, 2 − 0, 1 ∗ 0, 2 = 0, 28 0, 1 5 ⇒ P (A1 |A1 ∪ A2 ) = = 0, 28 14 ❚!♥ ❚❤➜% ❚& ✸✶✴✹✽ ✼✳ ❈,♥❣ /❤1❝ ①→❝ 5✉➜/ /♦➔♥ ♣❤➛♥ ✈➔ ❝,♥❣ /❤1❝ ❇❛②❡5 ✣'♥❤ ♥❣❤➽❛✿ ◆❤7♠ ❜✐➳♥ ❝) Hi,ni = 1, n ✤CD❝ ❣!✐ ❧➔ ✤➛② ✤F ♥➳✉ .❤1❛ ♠➣♥ ✷ ✤✐➲✉ ❦✐➺♥✿ ✐✮ Hi ∩ Hj = ∅, ∀i 6= j ✐✐✮ S Hi = Ω i=1 ❱➼ ❞# ✷✷ ❛✳ {A, A }, {∅, Ω} ❧➔ ❝→❝ ♥❤)♠ ❜✐➳♥ ❝J ✤➛② ✤M✳ ❜✳ ▼!" ❤!♣ ❝) ✸ ❜✐ ✤: ✈➔ ✷ ❜✐ ①❛♥❤✳ ▲➜② ♥❣➝✉ ♥❤✐➯♥ ✷ ✈✐➯♥ ❜✐✳ ●T✐ Hi ❧➔ ❜✐➳♥ ❝J ❧➜② ✤UV❝ i ❜✐ ①❛♥❤✱ i = 0, 1, 2✳ ❚❛ ❝) {H0, H1, H2} ❧➔ ♥❤)♠ ✤➛② ✤M✳ ✣B♥❤ ❧D✿ ✭❈,♥❣ /❤1❝ ①→❝ 5✉➜/ ✤➛② ✤H ✮ ❈❤♦ Hi , i = 1, n ❧➔ ♠*. ♥❤7♠ ❜✐➳♥ ❝) ✤➛② ✤F ✈➔ P (Hi ) > 0✳ ❑❤✐ ✤7 ✈P✐ ❜✐➳♥ ❝) ❆ ❜➜. ❦➻ ❧✐➯♥ U✉❛♥ ✤➳♥ ♣❤➨♣ .❤W✱ .❛ ❧✉X♥ ❝7✿ n X P (A) = P (Hi ).P (A|Hi ) i=1 ❚!♥ ❚❤➜% ❚& ✸✷✴✹✽ ❈X♥❣ .❤/❝ .@➯♥ ✤CD❝ ❣!✐ ❧➔ ❝W♥❣ "❤9❝ ①→❝ 6✉➜" ✤➛② ✤M ✭ ❤❛② ❝W♥❣ "❤9❝ ①→❝ 6✉➜" "♦➔♥ ♣❤➛♥✮✳ ◆7✐ @✐➯♥❣✱ .❛ ❝7 P (B) = P (A).P (B|A) + P (A).(B|A) ✈P✐ 0 < P (A) < 1 ✳ ❱➼ ❞# ✷✸ ❈) ✷ ❧W 6↔♥ ♣❤➞♠✳ ▲W ✶ ❝) ✺✵ 6↔♥ ♣❤➞♠ "\♦♥❣ ✤) ❝) ✷✵ 6↔♥ ♣❤➞♠ ①➜✉✳ ▲W ✷ ❝) ✹✵ 6↔♥ ♣❤➞♠✱ "\♦♥❣ ✤) ❝) ✶✺ 6↔♥ ♣❤➞♠ ①➜✉✳ ▲➜② ♥❣➝✉ ♥❤✐➯♥ ♠!" ❧W ✈➔ "^ ✤) ❧➜② \❛ ✶ 6↔♥ ♣❤➞♠✳ ❚➻♠ ①→❝ 6✉➜" ✤➸ 6↔♥ ♣❤➞♠ ❧➜② \❛ ❧➔ 6↔♥ ♣❤➞♠ "J"✳ ●✐↔✐✳ ●!✐ Hi ❧➔ ❝❤!♥ ❧X Z↔♥ ♣❤➞♠ i, i = 1, 2✳ ●!✐ ❆ ❧➔ ❜✐➳♥ ❝) Z↔♥ ♣❤➞♠ ❧➜② @❛ ❧➔ Z↔♥ ♣❤➞♠ .).✳ ❚❛ ❝7 {H1 , H2 } ❧➔ ♥❤7♠ ✤➛② ✤F✳ ❚❤❡♦ ❝X♥❣ .❤/❝ ①→❝ Z✉➜. .♦➔♥ ♣❤➛♥✿ P (A) = P (H1 )P (A|H1 ) + P (H2 )P (A|H2 ) = 1/2 ∗ 30/50 + 1/2 ∗ 25/40 = 49/80 ❚!♥ ❚❤➜% ❚& ✸✸✴✹✽
❱➼ ❞# ✷✹ ▼!" ♥❤%♠ ❝% ✸ ♥❣*+✐ ♥❤*♥❣ ❝❤➾ ❝% ✷ ✈➨ ①❡♠ ❜%♥❣ ✤→✳ ✣➸ ❝❤✐❛ ✈➨ ❤: ❧➔♠ ♥❤* =❛✉✿ ▲➜② ✸ ♣❤✐➳✉✱ ✷ ♣❤✐➳✉ ❣❤✐ =F ✶ ✈➔ ✶ ♣❤✐➳✉ ❣❤✐ =F ✵✳ ❙❛✉ ✤% "❤❛② ♣❤✐➯♥ ♥❤❛✉ ❜F❝ ♥❣➝✉ ♥❤✐➯♥ ❧➛♥ ❧*M" ❦❤O♥❣ ❤♦➔♥ ❧↕✐✳ ❆✐ ✤*M❝ ♣❤✐➳✉ ❣❤✐ =F ✶ "❤➻ ✤*M❝ ✈➨✳ ❛✳ ❚➼♥❤ ①→❝ =✉➜" ♥❣*+✐ "❤V ✷ ✤*M❝ ✈➨✳ ❜✳ ❍X✐ ✈✐➺❝ ❜F❝ ♣❤✐➳✉ ✤% ❝% ❝O♥❣ ❜➡♥❣ ❤❛② ❦❤O♥❣ ❄ ●✐↔✐✳ ●!✐ Ai ❧➔ ❜✐➳♥ ❝) ♥❣+,✐ -./ /❤1 i ✤+3❝ ✈➨✱i = 1, 2, 3✳ ❛✳ ❚❛ ❝: {A1 , Ā1 } ❧➔ ♥❤:♠ ✤➛② ✤>✳ ❉♦ ✤:✿ P (A2 ) = P (A1 )P (A2 |A1 ) + P (Ā1 )P (A2 |Ā1 ) = 2/3 ∗ 1/2 + 1/3 ∗ 2/2 = 2/3 ❜✳ ❚❛ ❝: P (A1 ) = P (A2 ) = 2/3✱ ❝➛♥ /➼♥❤ P (A3 )✳ ▼➦/ ❦❤→❝✱ P (A3 ) = 1 − P (Ā3 ) = 1 − P (A1 A2 ) = 1 − P (A1 )P (A2 |A1 ) = 1 − 2/3 ∗ 1/2 = 2/3 ❱➟②✱ ✈✐➺❝ ❧➔♠ /-➯♥ ❧➔ ❝K♥❣ ❜➡♥❣✳ ❚!♥ ❚❤➜% ❚& ✸✹✴✹✽ * ♥❣❤➽❛ ❝1♥❣ 2❤3❝ ①→❝ 6✉➜2 2♦➔♥ ♣❤➛♥ ❈K♥❣ /❤1❝ ①→❝ O✉➜/ /♦➔♥ ♣❤➛♥ ❣✐.♣ /❛ /➼♥❤ ①→❝ O✉➜/ ①↔② -❛ ❝>❛ ♠T/ ❜✐➳♥ ❝) ❞V❛ ✈➔♦ ♠T/ ♥❤:♠ ✤➛② ✤> ❝→❝ ❣✐↔ /❤✐➳/ ❝❤✐ ♣❤)✐ ♥:✳ ✣>♥❤ ❧@ ✭❈1♥❣ 2❤3❝ ❇❛②❡6✮ ❈❤♦ ❆ ❧➔ ♠T/ ❜✐➳♥ ❝) ✈➔ P (A) > 0✱ {H1 , ..., Hn } ❧➔ ♠T/ ♥❤:♠ ❜✐➳♥ ❝) ✤➛② ✤>✳ ▲.❝ ✤:✱ /❛ ❝: ❝K♥❣ /❤1❝✿ P (Hi ).P (A|Hi ) P (Hi ).P (A|Hi ) P (Hi |A) = = n , i = 1, n P (A) P P (Hj ).P (A|Hj ) j=1 ❚!♥ ❚❤➜% ❚& ✸✺✴✹✽ ❱➼ ❞# ✷✺ ▼!" "\↕♠ ❝❤➾ ♣❤→" ❤❛✐ "➼♥ ❤✐➺✉ ❆ ✈➔ ❇ ✈^✐ ①→❝ =✉➜" "*_♥❣ V♥❣ ✵✱✽✺ ✈➔ ✵✱✶✺✳ ❉♦ ❝% ♥❤✐➵✉ "\➯♥ ✤*+♥❣ "\✉②➲♥ ♥➯♥ ✶✴✼ "➼♥ ❤✐➺✉ ❆ ❜g ♠➨♦ ✈➔ "❤✉ ✤*M❝ ♥❤* "➼♥ ❤✐➺✉ ❇❀ ❝i♥ ✶✴✽ "➼♥ ❤✐➺✉ ❇ ❜g ♠➨♦ ✈➔ "❤✉ ✤*M❝ ♥❤* ❆✳ ❛✳ ❚➻♠ ①→❝ =✉➜" "❤✉ ✤*M❝ "➼♥ ❤✐➺✉ ❆✳ ❜✳ ●✐↔ =l ✤➣ "❤✉ ✤*M❝ "➼♥ ❤✐➺✉ ❆✳ ❚➻♠ ①→❝ =✉➜" "❤✉ ✤*M❝ ✤n♥❣ "➼♥ ❤✐➺✉ ❧n❝ ♣❤→"✳ ●✐↔✐✳ ●!✐ HA , HB ❧➔ ❜✐➳♥ ❝) /-↕♠ ♣❤→/ /➼♥ ❤✐➺✉ ❆✱ ❇✳ ●!✐ ❆ ❧➔ ❜✐➳♥ ❝) /-↕♠ /❤✉ ✤+3❝ /➼♥ ❤✐➺✉ ❆✳ ❛✳ ❚❛ ❝:✿ {HA , HB } ❧➔ ♥❤:♠ ❜✐➳♥ ❝) ✤➛② ✤>✳ ❚❤❡♦ ❝K♥❣ /❤1❝ ①→❝ O✉➜/ /♦➔♥ ♣❤➛♥✿ P (A) = P (HA )P (A|HA ) + P (HB )P (A|HB ) = 0, 85 ∗ (1 − 1/7) + 0, 15 ∗ 1/8 = 0, 747 ❜✳ ❚❤❡♦ ❝K♥❣ /❤1❝ ❇❛②❡O✿ P (HA )P (A|HA ) 0, 85 ∗ (1 − 1/7) P (HA |A) = = = 0, 975 P (A) 0, 747 ❚!♥ ❚❤➜% ❚& ✸✻✴✹✽
❱➼ ❞# ✷✻ ▼!" ❝$❛ ❤➔♥❣ ❜→♥ ❜,♥❣ ✤➧♥ ❝/♥❣ ❧♦↕✐ ❞♦ ✸ ❝6 78 7↔♥ ①✉➜" ❝✉♥❣ ❝➜♣✳ ❈6 78 ■✱ ■■✱ ■■■ ❝✉♥❣ ❝➜♣ ❧BC♥❣ ❤➔♥❣ "B6♥❣ D♥❣ ❧➔ ✹✵✪✱ ✸✺✪✱ ✷✺✪✳ ❇✐➳" "➾ ❧➺ ❜,♥❣ ❤N♥❣ ❞♦ ❝6 78 ■✱ ■■✱ ■■■ 7↔♥ ①✉➜" ❧➛♥ ❧BC" ❧➔ ✷✪✱ ✷✪✱ ✸✪✳ ❚❛ ♠✉❛ ♥❣➝✉ ♥❤✐➯♥ ✶ ❜,♥❣ ❝U❛ ❝$❛ ❤➔♥❣✳ ●✐↔ 7$ ❜,♥❣ ♠✉❛ ❜W ❤N♥❣✳ ❍N✐ ❜,♥❣ "❛ ♠✉❛ ❝, ❦❤↔ ♥➠♥❣ ❞♦ ❝6 78 ♥➔♦ 7↔♥ ①✉➜" ♥❤➜" ❄ ●✐↔✐✳ ●!✐ Hi ❧➔ ❜✐➳♥ ❝) ❜*♥❣ ✤-.❝ ♠✉❛ ❞♦ ❝4 56 i 5↔♥ ①✉➜:✱ i = 1, 2, 3✳ ●!✐ ❆ ❧➔ ❜✐➳♥ ❝) ❜*♥❣ ✤-.❝ ♠✉❛ ❜> ❤@♥❣✳ ❚❛ ❝*✿ {H1 , H2 , H3 } ❧➔ ♥❤*♠ ❜✐➳♥ ❝) ✤➛② ✤E✳ ❚❤❡♦ ❝G♥❣ :❤H❝ ①→❝ 5✉➜: :♦➔♥ ♣❤➛♥✿ P (A) = P (H1 )P (A|H1 ) + P (H2 )P (A|H2 ) + P (H3 )P (A|H3 ) = 0, 4 ∗ 0, 02 + 0, 35 ∗ 0, 02 + 0, 25 ∗ 0, 03 = 0.0225 P (H )P (A|H ) 0, 4 ∗ 0, 02 ❚❤❡♦ ❝G♥❣ :❤H❝ ❇❛②❡5✿ P (H1 |A) = 1 1 = = 0.356 P (A) 0.0225 ❚-4♥❣ :L✿ P (H2 |A) = 0.311, P (H3 |A) = 0.333 ❱➻ P (H1 |A) > P (H3 |A) > P (H2 |A) ♥➯♥ ❦❤↔ ♥➠♥❣ ❜*♥❣ ✤-.❝ ♠✉❛ ❞♦ ❝4 56 ✶ 5↔♥ ①✉➜: ❧➔ ❧S♥ ♥❤➜:✳ ❚!♥ ❚❤➜% ❚& ✸✼✴✹✽ * ♥❣❤➽❛ ❝1♥❣ 2❤3❝ ❇❛②❡7 ❈→❝ ①→❝ 5✉➜: P (H1 ), ..., P (Hn ) ✤-.❝ ①→❝ ✤>♥❤ :U-S❝ ❦❤✐ ♣❤➨♣ :❤W :✐➳♥ ❤➔♥❤ ✈➔ ❞♦ ✤♦ ❝❤Y♥❣ ✤-.❝ ❣!✐ ❧➔ ❝→❝ ①→❝ 7✉➜" "✐➯♥ ♥❣❤✐➺♠✳ ❈→❝ ①→❝ 5✉➜: P (H1 |A), ..., P (Hn |A) ✤-.❝ ①→❝ ✤>♥❤ 5❛✉ ❦❤✐ ♣❤➨♣ :❤W ✤-.❝ :✐➳♥ ❤➔♥❤ ✈➔ ❜✐➳♥ ❝) ❆ ✤➣ ①↔② U❛ ✈➔ ❞♦ ❞♦ ❝❤Y♥❣ ✤-.❝ ❣!✐ ❧➔ ❝→❝ ①→❝ 7✉➜" ❤➟✉ ♥❣❤✐➺♠✳ ❱➻ :❤➳ ❝G♥❣ :❤H❝ ❇❛②❡5 ❝[♥ ✤-.❝ ❣!✐ ❧➔ ❝]♥❣ "❤D❝ "➼♥❤ ①→❝ 7✉➜" ❤➟✉ ♥❣❤✐➺♠✳ ✯ ❈G♥❣ :❤H❝ ❇❛②❡5 ❝❤♦ ♣❤➨♣ ✤→♥❤ ❣✐→ ❧↕✐ ①→❝ 5✉➜: ①↔② U❛ ❝E❛ ❝→❝ ❣✐↔ :❤✐➳: 5❛✉ ❦❤✐ ✤➣ ❜✐➳: ❦➳: ^✉↔ ❝E❛ ♣❤➨♣ :❤W ❧➔ ❜✐➳♥ ❝) ❆ ✤➣ ①↔② U❛✳ ❚!♥ ❚❤➜% ❚& ✸✽✴✹✽ ❱➼ ❞# ✷✼ ❚➾ ❧➺ ♥❣B_✐ ❞➙♥ 8 "➾♥❤ ❆ ♥❣❤✐➺♥ "❤✉b❝ ❧→ ❧➔ ✸✵✪✱ "c♦♥❣ 7b ♥❣B_✐ ♥❣❤✐➺♥ "❤✉b❝ "➾ ❧➺ ♥❣B_✐ ❜W ❜➺♥❤ ♣❤d✐ ❧➔ ✻✵✪✳ ❚c♦♥❣ 7b ♥❤f♥❣ ♥❣B_✐ ❦❤]♥❣ ♥❣❤✐➺♥ "❤✉b❝ ❝, ✷✵✪ ❜W ❜➺♥❤ ♣❤d✐✳ ❈❤g♥ ♥❣➝✉ ♥❤✐➯♥ ♠!" ♥❣B_✐✳ ❚➼♥❤ ①→❝ 7✉➜" ♥❣B_✐ ✤, ♥❣❤✐➺♥ "❤✉b❝ "c♦♥❣ ❤❛✐ "cB_♥❣ ❤C♣✳ ❛✮ ❇✐➳" ♥❣B_✐ ✤, ❜W ❜➺♥❤ ♣❤d✐✳ ❜✮ ❇✐➳" ♥❣B_✐ ✤, ❦❤]♥❣ ❜W ❜➺♥❤ ♣❤d✐✳ ●✐↔✐✳ ●!✐ H1 , H2 ❧➔ ❝→❝ ❜✐➳♥ ❝) ♥❣-_✐ ✤-.❝ ❝❤!♥ ❤Y: ✈➔ ❦❤G♥❣ ❤Y: :❤✉)❝ ❧→✳ ●!✐ ❆ ❧➔ ❜✐➳♥ ❝) ♥❣-_✐ ✤-.❝ ❝❤!♥ ❜> ❜➺♥❤ ♣❤a✐✳ ❚❛ ❝*✿ {H1 , H2 } ❧➔ ♥❤*♠ ✤➛② ✤E✳ ❚❤❡♦ ❝G♥❣ :❤H❝ ①→❝ 5✉➜: :♦➔♥ ♣❤➛♥✿ P (A) = P (H1 )P (A|H1 ) + P (H2 )P (A|H2 ) = 0, 3 ∗ 0, 6 + 0, 7 ∗ 0, 2 = 0, 32 P (H1 )P (A|H1 ) 0, 3 ∗ 0, 6 ❛✳ P (H1 |A) = = = 9/16 P (A) 0, 32 P (H1 )P (Ā|H1 ) 0, 3 ∗ (1 − 0, 6) ❜✳ P (H1 |Ā) = = = 3/17 P (Ā) 1 − 0, 32 ❚!♥ ❚❤➜% ❚& ✸✾✴✹✽
❱➼ ❞# ✷✽ ▼!" ♠→② ❜❛② ❜➢♥ ✤!❝ ❧➟♣ ✷ 0✉↔ "➯♥ ❧4❛ ✈➔♦ ♠!" ♠8❝ "✐➯✉✳ ❳→❝
❱➼ ❞# ✸✵ ❳→❝ #✉➜& ✤➸ ✶ *✉↔ &,-♥❣ ✤❡♠ ➜♣ ♥3 ,❛ ❣➔ ❝♦♥ ❧➔ ✵✳✽✺✳ ✣❡♠ ➜♣ ✶✵ *✉↔ &,-♥❣✳ ❚➼♥❤ ①→❝ #✉➜& ✤➸ ❝A ✤B♥❣ ✽ *✉↔ ♥3 ,❛ ❣➔ ❝♦♥✳ ●✐↔✐✳ ❚❛ ❝# ♠% ❤➻♥❤ ❞➣② ♣❤➨♣ .❤/ ❇❡2♥♦✉❧❧✐ ✈8✐ n = 10; p = 0, 85✳ ❳→❝
❱➼ ❞# ✸✸ ❚!♦♥❣ ♥➠♠ ❤(❝ ✈+❛ -✉❛✱ 0 1!23♥❣ ✤↕✐ ❤(❝✱ 1➾ ❧➺ :✐♥❤ ✈✐➯♥ 1❤✐ 1!2