Bài giảng Chương 1: Xác suất môn Xác suất thống kê | Đại học Duy Tân
Bài giảng Chương 1: Xác suất môn Xác suất thống kê | Đại học Duy Tân. Tài liệu được sưu tầm giúp bạn tham khảo, ôn tập và đạt kết quả cao. Mời bạn đọc đón xem.
Môn: Xác xuất thống kê (STA 151) 144 tài liệu
Trường: Đại học Duy Tân 1.9 K tài liệu
Tác giả:
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❳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! Ckn = 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 $➢❝ ♥❤➙♥✿ ❈!♥❣ ✈✐➺❝ ❆ ✤,:❝ /❤0❝ ❤✐➺♥ ;✉❛ m ❣✐❛✐ ✤♦↕♥ ❧✐➯♥ /✐➳♣✳
✯ ●✐❛✐ ✤♦↕♥ ✶✿ ❝) n1 ❝→❝❤
✯ ●✐❛✐ ✤♦↕♥ ✷✿ ❝) n2 ❝→❝❤
✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳
✯ ●✐❛✐ ✤♦↕♥ m✿ ❝) nm ❝→❝❤
❙8 ❝→❝❤ /❤0❝ ❤✐➺♥ ❝!♥❣ ✈✐➺❝ ❆✿ n1 ∗ n2 ∗ ... ∗ nm ❚!♥ ❚❤➜% ❚& ✹✴✹✽ ❱➼ ❞0 ✶
▼!" ♥❤%♠ ❝% ✶✵ ❤*❝ +✐♥❤✱ ".♦♥❣ ✤% ❝% ✻ ♥❛♠ ✈➔ ✹ ♥7✳ ❍:✐ ❝% ❜❛♦ ♥❤✐➯✉ ❝→❝❤✿
❛✮ ❳➳♣ "❤➔♥❤ ✶ ❤➔♥❣ ❞*❝✳
❜✮ ❈❤*♥ ♠!" ♥❤%♠ ❝% ✹ ❤*❝ +✐♥❤ ✤➲✉ ❝% ♥❛♠ ✈➔ ♥7 ✈G✐ +H ❧JK♥❣ ❦❤→❝ ♥❤❛✉✳
❝✮ ❈❤*♥ ♠!" ♥❤%♠ ❝% ✹ ❤*❝ +✐♥❤ ".♦♥❣ ✤% ❝% ➼" ♥❤➜" ♠!" ♥7✳
✣→♣ #$✿ ❛✮ ✶✵✦ ❜✮ C1C3 + C3C1 ❝✮ C4 − C4 6 4 6 4 10 6 ❱➼ ❞0 ✷
▼!" ❜✐➸♥ +H ①❡ ✤JK❝ ❝➜✉ "↕♦ "S ✺ ❝❤7 +H✳ ❍:✐ ❝% ❜❛♦ ♥❤✐➯✉ ❜✐➸♥ +H ❝%✿
❛✮ ❈→❝ ❝❤7 +H ❦❤→❝ ♥❤❛✉✳
❜✮ ➑" ♥❤➜" ✷ ❝❤7 +H ❣✐H♥❣ ♥❤❛✉✳
❝✮ ❈% ✤W♥❣ ✸ ❝❤7 +H ❣✐H♥❣ ♥❤❛✉✳ ✣→♣ #$✿ ❛✮ A5 ❜✮ 105 − A5 10 10
❝✮ ✣-.❝ /➼♥❤ 3✉❛ ✸ ❣✐❛✐ ✤♦↕♥✿
✲ ●✤ ✶✿ ❈❤>♥ ✸ ✈@ /A➼ /A♦♥❣ ✺ ✈@ /A➼✿ C35
✲ ●✤ ✷✿ ❈❤>♥ ✶ ❝❤D #$ ❣✐$♥❣ ♥❤❛✉ ❝❤♦ ✸ ✈@ /A➼ /A➯♥✿ C110
✲ ●✤ ✸✿ ❝❤>♥ ✷ ❝❤D #$ ❝F♥ ❧↕✐ ✭❦❤J♥❣ ♣❤➙♥ ❜✐➺/✮✿ 92
❱➟②✱ /❤❡♦ 3✉② /➢❝ ♥❤➙♥✿ C3C1 29 ❝→❝❤✳ 5 10 ❚!♥ ❚❤➜% ❚& ✺✴✹✽
✷✳ ❑❤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❛ ❆✱ ❦➼ ❤✐➺✉ ¯
A ❤♦➦❝ Ac✱ ①→❝ ✤X♥❤✿ ¯
A = 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♥❣✳ ❱➼ ❞# ✻
❈❤$♥ ♥❣➝✉ ♥❤✐➯♥ ♠H6 ❝❤D &A 6^ ✵ ✤➳♥ ✾✳
❚❛ ❝! Ω = {0, 1, 2, ..., 9}✳
❳➨6 ✸ ❜✐➳♥ ❝A A = {0, 2, 4}✱ B = {4, 5, 6} ✈➔ C = {7, 8, 9}✳ ▲>❝ ✤!✿
• A ∩ B = {4}, A ∪ B = {0, 2, 4, 5, 6}, ¯ A = {1, 3, 5, 6, 7, 8, 9}
• ❆ ✈➔ ❈ ①✉♥❣ ❦❤➢❝ ✈➻ A ∩ C = ∅ ♥❤-♥❣ ❆ ✈➔ ❈ ❦❤F♥❣ ♣❤↔✐ ❧➔ ❤❛✐ ❜✐➳♥ ❝A ✤A✐ ♥❤❛✉✳ ❚!♥ ❚❤➜% ❚& ✾✴✹✽ ❱➼ ❞# ✼
❈! ✷ ①↕ %❤'✱ ♠*✐ ♥❣./✐ ❜➢♥ ✶ ✈✐➯♥ ✤↕♥ ✈➔♦ ♠8❝ %✐➯✉✳ ●=✐ ❆ ✈➔ ❇ %.@♥❣ A♥❣ ❧➔ ❝→❝ ❜✐➳♥
❝E✿ ✏♥❣./✐ %❤A ♥❤➜% ✈➔ %❤A ❤❛✐ ❜➢♥ %JK♥❣ ♠8❝ %✐➯✉✑ %.@♥❣ A♥❣✳ ❑❤✐ ✤! %❛ ❝! ❜✐➸✉ ❞✐➵♥
❝→❝ ❜✐➳♥ ❝E ♥❤. Q❛✉✿
✲ ❈! ✤↕♥ %JK♥❣ ✤➼❝❤✿ A ∪ B
✲ ❈! ✤K♥❣ ✶ ✈✐➯♥ ✤↕♥ %JK♥❣ ✤➼❝❤✿ A ¯ B ∪ ¯ AB
✲ ❈❤➾ ❝! ♥❣./✐ %❤A ♥❤➜% ❜➢♥ %JK♥❣✿ A ¯ B
✲ ❈! ♥❤✐➲✉ ♥❤➜% ♠V% ✈✐➯♥ ✤↕♥ %JK♥❣ ✤➼❝❤✿ ¯ A ¯ B ∪ A ¯ B ∪ ¯ AB ❤♦➦❝ AB
▲✉➟( ❉❡✲▼♦.❣❛♥ ✈➲ ♣❤6 ✤8♥❤✿ ∪Ai = ∩ ¯ Ai ∩Ai = ∪ ¯ Ai ❚!♥ ❚❤➜% ❚& ✶✵✴✹✽
✸✳ ✣8♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ (➼♥❤ ❝❤➜( ❝6❛ ①→❝ C✉➜(
✣8♥❤ ♥❣❤➽❛ ✭(❤❡♦ ❤➺ (✐➯♥ ✤➲✮✿ ❈❤♦ #$%&❝ ♠)# #❤➼ ♥❣❤✐➺♠ ✈&✐ ❦❤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 A P (A =1 n) = n) 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) = C4 ✳ 10
❙? #5ER♥❣ ❤=♣ #❤✉➟♥ ❧=✐✿ n(A) = C1C3 + C2C2 + C3C1✳ 6 4 6 4 6 4
❳→❝ U✉➜# ❝➛♥ #➻♠✿ n(A) C1C3 + C2C2 + C3C1 P (A) = = 6 4 6 4 6 4 n(Ω) C410 ❚!♥ ❚❤➜% ❚& ✶✸✴✹✽ ❱➼ ❞5 ✶✵
▼-& ✤♦➔♥ &➔✉ ❝0 ✸ &♦❛✳ ❈0 ✶✺ ❦❤→❝❤ ❧➯♥ ♥❣➝✉ ♥❤✐➯♥ ✸ &♦❛ &➔✉✳ ❇✐➳& ♠N✐ &♦❛ ✤➲✉ ❝❤P❛
✤AB❝ ✶✺ ❦❤→❝❤✳ ❚➼♥❤ ①→❝ #✉➜&✿
❛✳ ❚♦❛ ✶ ❝0 ✹ ❦❤→❝❤✳
❜✳ ❈0 ✷ &♦❛ ♠N✐ &♦❛ ❝0 ✻ ❦❤→❝❤✳ ●✐↔✐✳
❛✳ ●K✐ ❆ ❧➔ ❜✐➳♥ ❝? ✧#♦❛ ✶ ❝6 ✹ ❦❤→❝❤✧✳ ❚❛ ❝6✿ n(Ω) = 315✳
●✐→ #5H n(A) ✤E=❝ #➼♥❤ #❤[♥❣ .✉❛ ✷ ❣✐❛✐ ✤♦↕♥✿
✲ ●✤ ✶✿ ❈❤K♥ ✹ ❦❤→❝❤✿ C4 ❝→❝❤ 15
✲ ●✤ ✷✿ ❳➳♣ ✶✶ ❦❤→❝❤ ❝^♥ ❧↕✐✿ 211 ❝→❝❤✳
❙✉② 5❛✿ n(A) = C4 211 15
❳→❝ U✉➜# ❝➛♥ #➻♠✿ n(A) C4 211 P (A) = = 15 n(Ω) 315 ❚!♥ ❚❤➜% ❚& ✶✹✴✹✽
❜✳ ●K✐ ❇ ❧➔ ❜✐➳♥ ❝? ✧❝6 ✷ #♦❛ ♠`✐ #♦❛ ❝6 ✻ ❦❤→❝❤✧✳
●✐→ #5H n(B) ✤E=❝ #➼♥❤ #❤[♥❣ .✉❛ ❝→❝ ❣✐❛✐ ✤♦↕♥✿
✲ ●✤ ✶✿ ❈❤K♥ ✷ #♦❛✿ C2 ❝→❝❤ 3
✲ ●✤ ✷✿ ❈❤K♥ ✻ ❦❤→❝❤ ❝❤♦ ♠`✐ #♦❛✿ C6 C6 ❝→❝❤✳ 15 9
✲ ●✤ ✸✿ ❳➳♣ ✸ ❦❤→❝❤ ❝^♥ ❧↕✐ 13 = 1 ❝→❝❤
❙✉② 5❛✿ n(B) = C2C6 C6 3 15 9
❳→❝ U✉➜# ❝➛♥ #➻♠✿ n(B) C2C6 C6 P (B) = = 3 15 9 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) = C4 C4C4 12 8 4 ❛✳ n(A) = C4C4C4 ❜✳ n(B) = 3!C3C3C3 9 8 4 9 6 3 ❚!♥ ❚❤➜% ❚& ✶✺✴✹✽ ❚➼♥❤ ❝❤➜&
✐✮ 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 (AiAj) iX + P (AiAjAk) i...... (−1)n−1P (A1A2...An) ❚!♥ ❚❤➜% ❚& ✶✻✴✹✽
❑❤✐ ❝→❝ ❜✐➳♥ ❝4 ①✉♥❣ ❦❤➢❝ ✤-✐ ♠0& %❤➻
P (A1 ∪ A2 ∪ ... ∪ An) = P (A1) + ... + P (An) ◆5✐ 6✐➯♥❣✿ P (A) + P ( ¯ A) = 1 ❱➼ ❞3 ✶✷
▼!" ❧$♣ ❝' ✷✵ *✐♥❤ ✈✐➯♥✱ "1♦♥❣ ✤' ❝' ✶✵ *✐♥❤ ✈✐➯♥ ❜✐➳" "✐➳♥❣ ❆♥❤✱ ✶✷ *✐♥❤ ✈✐➯♥ ❜✐➳" "✐➳♥❣
9❤→♣ ✈➔ ✼ *✐♥❤ ✈✐➯♥ ❜✐➳" ❝↔ ❤❛✐ "❤? "✐➳♥❣✳ ❈❤B♥ ♥❣➝✉ ♥❤✐➯♥ ♠!" *✐♥❤ ✈✐➯♥✳ ❚➻♠ ①→❝ *✉➜"
✤➸ *✐♥❤ ✈✐➯♥ ✤' ❜✐➳" ➼" ♥❤➜" ♠!" ♥❣♦↕✐ ♥❣M✳ ●✐↔✐✳
●9✐ ❆✱ ❇ ❧➔ ❜✐➳♥ ❝4 ❝❤9♥ ✤@A❝ B✐♥❤ ✈✐➯♥ ❜✐➳% %✐➳♥❣ ❆♥❤✱ C❤→♣✳
▲G❝ ✤5✱ ①→❝ B✉➜% ❝➛♥ %➻♠✿
P (A ∪ B) = P (A) + P (B) − P (AB) = 10/20 + 12/20 − 7/20 = 3/4 ❚!♥ ❚❤➜% ❚& ✶✼✴✹✽ ❱➼ ❞3 ✶✸
▼!" ❧N ❤➔♥❣ ❝' ✶✺ "❤✐➳" ❜P✱ "1♦♥❣ ✤' ❝' ✻ "❤✐➳" ❞♦ ♥❤➔ ♠→② ❳ *↔♥ ①✉➜" ✈➔ ✾ "❤✐➳" ❜P ❞♦
♥❤➔ ♠→② ❨ *↔♥ ①✉➜"✳ ◆❣XY✐ "❛ ❝❤B♥ ♥❣➝✉ ♥❤✐➯♥ ✹ "❤✐➳" ❜P ✤➸ ❦✐➸♠ "1❛✳ ❚➼♥❤ ①→❝ *✉➜"✿
❛✳ ❈↔ ✹ "❤✐➳" ❜P ✤X]❝ ❝❤B♥ ❞♦ ❝^♥❣ ♥❤➔ ♠→② *↔♥ ①✉➜"
❜✳ ❈' ➼" ♥❤➜" ♠!" "❤✐➳" ❜P ✤X]❝ ❝❤B♥ ❞♦ ♥❤➔ ♠→② ❳ *↔♥ ①✉➜"✳
●✐↔✐✳ ❛✳ ●9✐ AX, AY ❧➔ ❝→❝ ❜✐➳♥ ❝4 ❝→❝ %❤✐➳% ❜M ✤@A❝ ❝❤9♥ ❞♦ ♥❤➔ ♠→② ❳✱ ❨ B↔♥ ①✉➜%✳
❚❛ ❝5 AX, AY ①✉♥❣ ❦❤➢❝✳
❳→❝ B✉➜% ❝↔ ✹ %❤✐➳% ❜M ✤@A❝ ❝❤9♥ ❞♦ ❝X♥❣ ♥❤➔ ♠→② B↔♥ ①✉➜%✿ C4 C4 P (A 6 9
X ∪ AY ) = P (AX ) + P (AY ) = + C4 C4 15 15
❜✳ ●9✐ ❇ ❧➔ ❜✐➳♥ ❝4 ❝5 ➼% ♥❤➜% ♠/% %❤✐➳% ❜M ✤@A❝ ❝❤9♥ ❞♦ ♥❤➔ ♠→② ❳ B↔♥ ①✉➜%✳ ❚❛ ❝5✿ C4 P (B) = 1 − P ( ¯ B) = 1 − 9 C415 ❚!♥ ❚❤➜% ❚& ✶✽✴✹✽ ❱➼ ❞# ✶✹
▼!" ✤!✐ ❜&♥❣ ❜➔♥ ❝+❛ ✶ ✤.♥ ✈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 ( ¯ A ¯
B) = 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 ( ¯ A|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 ( ¯
B|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 ( ¯ A) = 1 − C4 /C4 12 15
❜✳ P (B|A) = P (AB)/P (A) = P (B)/P (A)✱ 0D♦♥❣ ✤= P (B) = C2C2 /C4 ✳ 3 12 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 (A1A2...An−1) > 0✳ ❑❤✐ ✤2✿
P (A1A2...An) = P (A1).P (A2|A1).P (A3|A1A2)....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 (A ¯ B) = P (A)P ( ¯
B|A) = P (A)[1 − P (B|A)] = 0, 7[1 − 0, 4] = 0, 42 ❝✳ P (A ¯ B ∪ ¯ AB) = P (A ¯ B) + P ( ¯ AB) = . . . ❚!♥ ❚❤➜% ❚& ✷✻✴✹✽ ❱➼ ❞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 ( ¯ A1A2) = P ( ¯ A1)P (A2| ¯ A1) = 7/9 ∗ 2/8 ❜✳ P ( ¯ A ¯ ¯ 1A2A3) = P ( ¯ A1)P ( ¯ A2| ¯ A1)P (A3| ¯ A1A2) = 7/9 ∗ 6/8 ∗ 2/7 ❝✳ P (A1 ∪ ¯ A1A2) = P (A1) + P ( ¯ A1A2) = 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 B ¯ i Ai Ai 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 ( ¯ A1 ∪ ¯ A2 ∪ ¯
A3) = 1 − P (A1A2A3) = 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 (A ¯ ¯ 1A2 ∪ ¯ A1A2) = P (A1A2) + P ( ¯ A1A2)
= 0, 1 ∗ [1 − P (A2)] + 0, 9P (A2) = 0, 1 + 0, 8P (A2) = 0, 26✳
❙✉② @❛✿ P (A2) = 0, 2✳ ❜✳ P (A P (A P (A 1(A1 ∪ A2)) 1) 1|A1 ∪ A2) = = P (A1 ∪ A2) P (A1 ∪ A2)
P (A1 ∪ A2) = P (A1) + P (A2) − P (A1A2) = 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, i = 1, n ✤CD❝ ❣!✐ ❧➔ ✤➛② ✤F ♥➳✉ .❤1❛ ♠➣♥ ✷ ✤✐➲✉ ❦✐➺♥✿ n ✐✮ H S i ∩ Hj = ∅, ∀i 6= j ✐✐✮ 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, ¯
A1} ❧➔ ♥❤:♠ ✤➛② ✤>✳ ❉♦ ✤:✿
P (A2) = P (A1)P (A2|A1) + P ( ¯ A1)P (A2| ¯
A1) = 2/3 ∗ 1/2 + 1/3 ∗ 2/2 = 2/3
❜✳ ❚❛ ❝: P (A1) = P (A2) = 2/3✱ ❝➛♥ /➼♥❤ P (A3)✳ ▼➦/ ❦❤→❝✱ P (A3) = 1 − P ( ¯
A3) = 1 − P (A1A2) = 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 (H P (H P (H i).P (A|Hi) i).P (A|Hi) i|A) = = , i = 1, n P (A) n 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 (H 0, 85 ∗ (1 − 1/7) P (H A)P (A|HA) A|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
❚❤❡♦ ❝G♥❣ :❤H❝ ❇❛②❡5✿ P (H 0, 4 ∗ 0, 02 P (H 1)P (A|H1) 1|A) = = = 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 (H 0, 3 ∗ 0, 6 P (H 1)P (A|H1) 1|A) = = = 9/16 P (A) 0, 32 ❜✳ P (H 0, 3 ∗ (1 − 0, 6) P (H 1)P ( ¯ A|H1) 1| ¯ A) = = = 3/17 P ( ¯ A) 1 − 0, 32 ❚!♥ ❚❤➜% ❚& ✸✾✴✹✽ ❱➼ ❞# ✷✽
▼!" ♠→② ❜❛② ❜➢♥ ✤!❝ ❧➟♣ ✷ 0✉↔ "➯♥ ❧4❛ ✈➔♦ ♠!" ♠8❝ "✐➯✉✳ ❳→❝ <✉➜" 0✉↔ "❤? ♥❤➜" ✈➔
"❤? ✷ "@A♥❣ ❧➔ ✵✱✻ ✈➔ ✵✱✼✳ ◆➳✉ ❝I ✶ 0✉↔ "@A♥❣ "❤➻ ♠8❝ "✐➯✉ ❜L "✐➯✉ ❞✐➺" ✈O✐ ①→❝ <✉➜" ✵✱✼
✈➔ ♥➳✉ ❝I ✷ 0✉↔ "@A♥❣ "❤➻ ①→❝ <✉➜" ♥➔② ❧➔ ✵✱✾✳
❛✮ ❚➼♥❤ ①→❝ <✉➜" ❝I ➼" ♥❤➜" ✶ 0✉↔ "@A♥❣✳
❜✮ ❚➼♥❤ ①→❝ <✉➜" ♠8❝ "✐➯✉ ❜L "✐➯✉ ❞✐➺"✳
❝✮ ❇✐➳" ♠8❝ "✐➯✉ ❜L "✐➯✉ ❞✐➺"✱ "➼♥❤ ①→❝ <✉➜" ❝↔ ✷ 0✉↔ ✤➲✉ "@A♥❣✳
●✐↔✐✳ ●!✐ Ai ❧➔ ❜✐➳♥ ❝) *✉↔ -➯♥ ❧/❛ -❤2 i -34♥❣ ♠7❝ -✐➯✉✱ i = 1, 2✳ ❚❛ ❝; A1, A2 ✤=❝ ❧➟♣✳
❛✳ P (A1 ∪ A2) = P (A1) + P (A2) − P (A1A2) = 0, 6 + 0, 7 − 0, 6 ∗ 0, 7 = 0, 88
❜✳ ●!✐ Hi ❧➔ ❜✐➳♥ ❝) ❝; i *✉↔ -➯♥ ❧/❛ -34♥❣✱ i = 0, 1, 2✱ ✈➔ A ❧➔ ❜✐➳♥ ❝) ♠7❝ -✐➯✉ ❜A -✐➯✉
❞✐➺-✳ ❚❛ ❝; {H0, H1, H2} ❧➔ ♥❤;♠ ✤➛② ✤F✳ P (H ¯ ¯ 0) = P ( ¯
A1A2) = 0, 4 ∗ 0, 3 = 0, 12; P (H1) = P (A1A2 ∪ ¯
A1A2) = 0, 6 ∗ 0, 3 + 0, 4 ∗
0, 7 = 0, 46; P (H2) = P (A1A2) = 0, 6 ∗ 0, 7 = 0, 42✳
❚❤❡♦ ❝I♥❣ -❤2❝ ①→❝ L✉➜- -♦➔♥ ♣❤➛♥✿
P (A) = P (H0)P (A|H0) + P (H1)P (A|H1) + P (H2)P (A|H2)
= 0, 12 ∗ 0 + 0, 46 ∗ 0, 7 + 0, 42 ∗ 0, 9 = 0, 7 ❝✳ P (H 0, 42 ∗ 0, 9 P (H 2)P (A|H2) 2|A) = = = 0, 54 P (A) 0, 7 ❚!♥ ❚❤➜% ❚& ✹✵✴✹✽ ❱➼ ❞# ✷✾
◆❣WX✐ "❛ ❞Y♥❣ ♠!" "❤✐➳" ❜L ✤➸ ❦✐➸♠ "@❛ ♠!" ❧♦↕✐ <↔♥ ♣❤➞♠ ♥❤➡♠ ①→❝ ✤L♥❤ <↔♥ ♣❤➞♠ ❝I
✤↕" ②➯✉ ❝➛✉ ❦❤`♥❣✳ ❇✐➳" @➡♥❣ <↔♥ ♣❤➞♠ ❝I "➾ ❧➺ ♣❤➳ ♣❤➞♠ ❧➔ ♣✭✪✮✳ ❚❤✐➳" ❜L ❝I ❦❤↔ ♥➠♥❣
♣❤→" ❤✐➺♥ ✤A♥❣ <↔♥ ♣❤➞♠ ❧➔ ♣❤➳ ♣❤➞♠ ✈O✐ ①→❝ <✉➜" α ✈➔ ♣❤→" ❤✐➺♥ ✤A♥❣ <↔♥ ♣❤➞♠ ✤↕"
❝❤➜" ❧We♥❣ ✈O✐ ①→❝ <✉➜" β✳ ❑✐➸♠ "@❛ ♥❣➝✉ ♥❤✐➯♥ ♠!" <↔♥ ♣❤➞♠✱ "➻♠ ①→❝ <✉➜" <❛♦ ❝❤♦
<↔♥ ♣❤➞♠ ♥➔②✿
❛✳ ✣We❝ ❦➳" ❧✉➟♥ ❧➔ ♣❤➳ ♣❤➞♠✳
❜✳ ✣We❝ ❦➳" ❧✉➟♥ ❧➔ ✤↕" ❝❤➜" ❧We♥❣ "❤➻ ❧↕✐ ❧➔ ♣❤➳ ♣❤➞♠✳
❝✳ ✣We❝ ❦➳" ❧✉➟♥ ✤A♥❣ ✈O✐ "❤j❝ ❝❤➜" ❝k❛ ♥I✳
●+✐ ,✳ ●!✐ H1, H2 ❧➔ ❜✐➳♥ ❝) L↔♥ ♣❤➞♠ ✤PQ❝ ❝❤!♥ ❧➔ ❝❤➼♥❤ ♣❤➞♠✱ ♣❤➳ ♣❤➞♠✳ ●!✐ A
❧➔ ❜✐➳♥ ❝) ✤PQ❝ ❦➳- ❧✉➟♥ ❧➔ ♣❤➳ ♣❤➞♠✳ ❚❛ ❝; {H1, H2} ❧➔ ♥❤;♠ ✤➛② ✤F✱ P (H2) = p, P (A|H2) = α, P ( ¯ A|H1) = β✳
❛✳ P (A) = P (H1)P (A|H1) + P (H2)P (A|H2) = (1 − p)(1 − β) + pα ❜✳ P (H1|A) ❝✳ P (AH2) + P ( ¯ AH1) ❚!♥ ❚❤➜% ❚& ✹✶✴✹✽
✽✳ ❉➣② ♣❤➨♣ 3❤4 ❇❡7♥♦✉❧❧✐ ✣=♥❤ ♥❣❤➽❛
❉➣② n ♣❤➨♣ -❤/ ✤PQ❝ ❣!✐ ❧➔ ❞➣② n ♣❤➨♣ -❤/ ❇❡3♥♦✉❧❧✐ ♥➳✉ -❤X❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ L❛✉✿
✲ ❈→❝ ♣❤➨♣ -❤/ ✤=❝ ❧➟♣
✲ ❚3♦♥❣ ♠\✐ ♣❤➨♣ -❤/ ❝❤➾ ①↔② 3❛ ♠=- -3♦♥❣ ❤❛✐ ❜✐➳♥ ❝)✱ ❦➼ ❤✐➺✉ A ✭-❤➔♥❤ ❝I♥❣✮ ✈➔ ¯ A ✭-❤➜- ❜↕✐✮✳
✲ ❳→❝ L✉➜- ①✉➜- ❤✐➺♥ ❜✐➳♥ ❝) ❆ ❧➔ p = P (A) ❦❤I♥❣ ✤c✐ -3♦♥❣ ❝→❝ ♣❤➨♣ -❤/✳
❈B♥❣ 3❤C❝ ①→❝ G✉➜3 ♥❤= 3❤C❝
❑➼ ❤✐➺✉ pn(k) ❧➔ ①→❝ L✉➜- ❝; ✤4♥❣ k ❧➛♥ ①✉➜- ❤✐➺♥ ❜✐➳♥ ❝) ❆ -3♦♥❣ ❞➣② ♥ ♣❤➨♣ -❤/
❇❡3♥♦✉❧❧✐✳ ❑❤✐ ✤;✿
pn(k) = Cknpk(1 − p)n−k, k = 0, n.
❈I♥❣ -❤2❝ -3➯♥ ❝e♥ ✤PQ❝ ❣!✐ ❧➔ ❝I♥❣ -❤2❝ ❇❡3♥♦✉❧❧✐✳ ❚!♥ ❚❤➜% ❚& ✹✷✴✹✽ ❱➼ ❞# ✸✵
❳→❝ #✉➜& ✤➸ ✶ *✉↔ &,-♥❣ ✤❡♠ ➜♣ ♥3 ,❛ ❣➔ ❝♦♥ ❧➔ ✵✳✽✺✳ ✣❡♠ ➜♣ ✶✵ *✉↔ &,-♥❣✳ ❚➼♥❤ ①→❝
#✉➜& ✤➸ ❝A ✤B♥❣ ✽ *✉↔ ♥3 ,❛ ❣➔ ❝♦♥✳
●✐↔✐✳ ❚❛ ❝# ♠% ❤➻♥❤ ❞➣② ♣❤➨♣ .❤/ ❇❡2♥♦✉❧❧✐ ✈8✐ n = 10; p = 0, 85✳
❳→❝ <✉➜. ❝# ✤?♥❣ ✽ B✉↔ ♥D 2❛ ❣➔ ❝♦♥✿ p10(8) = C8 0, 8580, 152 10 ❱➼ ❞# ✸✶
❚➼♥ ❤✐➺✉ &❤E♥❣ &✐♥ ✤FG❝ ♣❤→& ✤✐ ✸ ❧➛♥ ✤J❝ ❧➟♣ ♥❤❛✉✳ ❳→❝ #✉➜& &❤✉ ✤FG❝ ♠L✐ ❧➛♥ ❧➔ ✵✳✹✳
❛✮ ❚➻♠ ①→❝ #✉➜& ✤➸ ♥❣✉P♥ &❤✉ ♥❤➟♥ ✤FG❝ &❤E♥❣ &✐♥ ✤B♥❣ ✷ ❧➛♥✳
❜✮ ❚➻♠ ①→❝ #✉➜& ✤➸ ♥❣✉P♥ &❤✉ ♥❤➟♥ ✤FG❝ &❤E♥❣ &✐♥ ✤A✳
❝✮ ◆➳✉ ♠✉U♥ ①→❝ #✉➜& &❤✉ ✤FG❝ &✐♥ ≥ ✵✳✾✾ &❤➻ ♣❤↔✐ ♣❤→& ✤✐ ➼& ♥❤➜& ❜❛♦ ♥❤✐➯✉ ❧➛♥✳ ❚!♥ ❚❤➜% ❚& ✹✸✴✹✽ ●✐↔✐✳
❚❛ ❝# ♠% ❤➻♥❤ ❞➣② ♣❤➨♣ .❤/ ❇❡2♥♦✉❧❧✐ ✈8✐ n = 3; p = 0, 4✳
❛✳ ❳→❝ <✉➜. ✤➸ ♥❣✉H♥ .❤✉ ♥❤➟♥ ✤JK❝ .❤%♥❣ .✐♥ ✤?♥❣ ✷ ❧➛♥✿ p3(2) = C20, 420, 61 3
❜✳ ❳→❝ <✉➜. ✤➸ ♥❣✉H♥ .❤✉ ♥❤➟♥ ✤JK❝ .❤%♥❣ .✐♥ ✤#✿
p3(k ≥ 1) = 1 − p3(0) = 1 − C00, 400, 63 3
❝✳ ●P✐ n ❧➔ pn(k ≥ 1) ≥ 0, 99 ⇔ 1 − pn(0) ≥ 0, 99 ln(0, 01)
⇔ C0n0, 400, 6n ≤ 0, 01 ⇔ n ≥ = 9, 015 ln(0, 6) ❱➟②✱ n = 10✳ ❚!♥ ❚❤➜% ❚& ✹✹✴✹✽ ❱➼ ❞# ✸✷
❚,➯♥ ❣✐→ ❝A ✹ ❝➙② #B♥❣ ❧♦↕✐ ✶ ✈➔ ✻ ❝➙② #B♥❣ ❧♦↕✐ ✷✳ ▼J& ①↕ &❤^ ❝❤_♥ ♥❣➝✉ ♥❤✐➯♥ ✶ ❝➙②
#B♥❣ ✈➔ ❜➢♥ ✺ ♣❤→& ✤J❝ ❧➟♣✳ ❇✐➳& ①→❝ #✉➜& ❜➢♥ &,B♥❣ ❦❤✐ #d ❞f♥❣ #B♥❣ ❧♦↕✐ ✶ ✈➔ ✷ ❧➛♥
❧FG& ❧➔ ✵✱✻ ✈➔ ✵✱✸✳ ❚➼♥❤ ①→❝ #✉➜& ❝A ✤B♥❣ ✷ ✈✐➯♥ ✤↕♥ &,B♥❣ ✤➼❝❤✳
●✐↔✐✳ ●P✐ Hi ❧➔ ❜✐➳♥ ❝Q ①↕ .❤Y ❝❤P♥ ✤JK❝ <?♥❣ ❧♦↕✐ i, i = 1, 2 ✈➔ ❆ ❧➔ ❜✐➳♥ ❝Q ①↕ .❤Y
❜➢♥ .2?♥❣ ✷ ✈✐➯♥✳ ❚❛ ❝# {H1, H2} ❧➔ ♥❤#♠ ✤➛② ✤Y✳
❚❤❡♦ ❝%♥❣ .❤]❝ ①→❝ <✉➜. .♦➔♥ ♣❤➛♥✿
P (A) = P (H1)P (A|H1) + P (H2)P (A|H2) .2♦♥❣ ✤#✿
P (H1) = 4/10 = 0, 4; P (H2) = 6/10 = 0, 6
P (A|H1) = p5(2) = C20, 620, 43; P (A|H 0, 320, 73 5 2) = p5(2) = C 2 5 ❚!♥ ❚❤➜% ❚& ✹✺✴✹✽ ❱➼ ❞# ✸✸
❚!♦♥❣ ♥➠♠ ❤(❝ ✈+❛ -✉❛✱ 0 1!23♥❣ ✤↕✐ ❤(❝✱ 1➾ ❧➺ :✐♥❤ ✈✐➯♥ 1❤✐ 1!2<1 ♠=♥ ❚♦→♥ ❧➔ ✸✹✪✱ 1❤✐
1!2<1 ♠=♥ ▲D ❧➔ ✷✵✪✱ ✈➔ 1!♦♥❣ :G ❝→❝ :✐♥❤ ✈✐➯♥ 1!2<1 ♠=♥ ❚♦→♥✱ ❝H ✺✵✪ :✐♥❤ ✈✐➯♥ 1!2<1
♠=♥ ▲D✳ K❤↔✐ ❝❤(♥ ❜❛♦ ♥❤✐➯✉ :✐♥❤ ✈✐➯♥ ❝N❛ 1!23♥❣ ♥➔② :❛♦ ❝❤♦✱ ✈P✐ ①→❝ :✉➜1 ❦❤=♥❣ ❜➨
❤U♥ ✾✾✪✱ 1!♦♥❣ :G ✤H ❝H ➼1 ♥❤➜1 ♠X1 :✐♥❤ ✈✐➯♥ ✤➟✉ ❝↔ ❤❛✐ ♠=♥ ❚♦→♥ ✈➔ ▲D✳
●✐↔✐✳ ●!✐ ❚✱ ▲ ❧➔ ❝→❝ ❜✐➳♥ ❝- .✐♥❤ 0❤✐ 01230 ♠5♥ ❚♦→♥✱ ▲7 028♥❣ :♥❣✳ ❚❛ ❝=✿
P (T ) = 0, 34; P (L) = 0, 2; P (L|T ) = 0, 5
❳→❝ .✉➜0 0❤✐ ✤➟✉ ❝↔ ✷ ♠5♥✿ P ( ¯ T ¯
L) = 1 − P (T ∪ L) = 1 − [P (T ) + P (L) − P (T L)]
= 1 − P (T ) − P (L) + P (T )P (L|T ) = 1 − 0, 34 − 0, 2 + 0, 34 ∗ 0, 5 = 0, 63
●!✐ n ❧➔ .- .✐♥❤ ✈✐➯♥ ❝➛♥ ❦❤↔♦ .→0✳ ❚❛ ❝= ♠5 ❤➻♥❤ ❇❡1♥♦✉❧❧✐ ✈M✐ n ♣❤➨♣ 0❤P ✈➔ ①→❝ .✉➜0
p = 0, 63✳ ❚❤❡♦ ❣✐↔ 0❤✐➳0✿
pn(k ≥ 1) = 1 − pn(0) ≥ 0, 99 ⇔ (1 − 0, 63)n ≤ 0, 01 ⇔ n ≥ 4, 63 ⇒ n = 5 ❚!♥ ❚❤➜% ❚& ✹✻✴✹✽
❙* ❧➛♥ ❝/ ❦❤↔ ♥➠♥❣ ❧4♥ ♥❤➜6
❇➔✐ 6♦→♥✿ ❚➻♠ k .❛♦ ❝❤♦ ①→❝ .✉➜0 pn(k) ✤↕0 ❣✐→ 01S ❧M♥ ♥❤➜0✳
❑➳6 >✉↔✿ ✣➦0 q = 1 − p ✈➔ ❣!✐ K ❧➔ ❣✐→ 01S ❝➛♥ 0➻♠✳ ❑❤✐ ✤= K ❧➔ .- ♥❣✉②➯♥ 0❤X❛ ✤✐➲✉ ❦✐➺♥✿
np − q ≤ K ≤ np − q + 1 ❱➼ ❞# ✸✹
▼X1 ①20♥❣ ❞➺1 ❝H ✺✵ ♠→② ❞➺1✳ ❳→❝ :✉➜1 ♠]✐ ♠→② ❞➺1 ❜^ ❤_♥❣ 1!♦♥❣ ♠]✐ ❝❛ ❧➔♠ ✈✐➺❝ ❧➔
✵✳✶✳ ❚➻♠ :G ♠→② ❤_♥❣ ✈P✐ ❦❤↔ ♥➠♥❣ ❧P♥ ♥❤➜1 1!♦♥❣ ✶ ❝❛ ❧➔♠ ✈✐➺❝✳
●✐↔✐✳ ●!✐ K ❧➔ .- ♠→② ❤X♥❣ ✈M✐ ❦❤↔ ♥➠♥❣ ❧M♥ ♥❤➜0✳ ❚❛ ❝= ♠5 ❤➻♥❤ ❇❡1♥♦✉❧❧✐ ✈M✐ n = 50; p = 0, 1✳
▲↕✐ ❝=✿ q = 1 − p = 0, 9✳ ●✐→ 01S K 0❤X❛ ✤✐➲✉ ❦✐➺♥✿
np − q ≤ K ≤ np − q + 1 ⇔ 4, 1 ≤ K ≤ 5, 1 ❱➟②✱ K = 5✳ ❚!♥ ❚❤➜% ❚& ✹✼✴✹✽
❤" ❧"❝ ✈➲ ♠( ♣❤*♥❣
❚❛ ❦➼ ❤✐➺✉ R ❧➔ ❣✐→ 01S ♥❣➝✉ ♥❤✐➯♥ ♥➡♠ 01♦♥❣ ✤♦↕♥ ❬✵✱✶❪ ✤23❝ ♣❤→0 .✐♥❤ ❜d✐ ❤➔♠ 1❛♥❞♦♠
01♦♥❣ ♠→② 0➼♥❤✳
▼B ♣❤D♥❣ ❣✐❡♦ ✤G♥❣ ①✉
✲ g❤→0 .✐♥❤ ❣✐→ 01S ❝h❛ R
✲ ◆➳✉ R < 0.5 0❤➻ ❝❤♦ ❦➳0 j✉↔ ①✉➜0 ❤✐➺♥ ♠➦0 ✓.➜♣✔✱ ♥❣23❝ ❧↕✐ 0❤➻ ❝❤♦ ❦➳0 j✉↔ ♠➦0 ✓♥❣P❛✔✳
▼B ♣❤D♥❣ ❣✐❡♦ ❝♦♥ ①I❝ ①➢❝
✲ g❤→0 .✐♥❤ ❣✐→ 01S ❝h❛ R ✲ ◆➳✉
R ♥➡♠ 01♦♥❣ ❦❤♦↔♥❣ hi−1; i
0❤➻ ❝❤♦ ①✉➜0 ❤✐➺♥ ❦➳0 j✉↔ ❧➔ ✓♠➦0 ✐✔✱ ✈M✐ i = 6 6
1, 2, ..., 6 028♥❣ :♥❣✳ ❚!♥ ❚❤➜% ❚& ✹✽✴✹✽
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