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❈❍❆P❚❊❘ ✶

❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ❛♥❞ ■♥❞❡♣❡♥❞❡♥❝❡

✶✳✶✳ ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t②

❑♥♦✇❧❡❞❣❡ t❤❛t ❛ ♣❛rt✐❝✉❧❛r ❡✈❡♥tA❤❛s ♦❝❝✉rr❡❞ ✇✐❧❧ ❝❤❛♥❣❡ ♦✉r ❛ss❡ss♠❡♥t ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ♦t❤❡r ❡✈❡♥tB✳ ■♥ s✉❝❤ ❛♥ ❡①❛♠♣❧❡✱ t❤❡ t❡r♠✐♥♦❧♦❣② ✏❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✑ ✐s ✉s❡❞✳

❆♥ ❡①♣❡r✐♠❡♥t ✐s ❝♦♥❞✉❝t❡❞ ✇✐t❤ s❛♠♣❧❡ s♣❛❝❡ Ω✱ ❣✐✈❡♥ ❡✈❡♥t B ❤❛s ♦❝❝✉r❡❞✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ❡✈❡♥tA ♦❝❝✉rs ❣✐✈❡♥ ❡✈❡♥tB ❤❛s ♦❝❝✉r❡❞✱ ✇r✐tt❡♥P(A|B)✱ ✐s t❤❡

♣r♦❜❛❜✐❧✐t② ♦❢ A r❡❧❛t✐✈❡ t♦ t❤❡ r❡❞✉❝❡❞ s❛♠♣❧❡ s♣❛❝❡ B✳ (AT

B)✐s t❤❡ s✉❜s❡t ♦❢

B ❢♦r ✇❤✐❝❤ A ✐s tr✉❡✱ s♦ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢A ❣✐✈❡♥ B s❤♦✉❧❞ ❜❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ P(AT

B)✱ s❛②P(A|B) =kP(AT

B)✳ ❙✐♠✐❧❛r❧②✱P(Ac_{|B) =kP(A}cT

B)✳ ❚❤✉s✱

P(A|B)+P(Ac|B) =khP(A\B) +P(Ac\B)i=kPh(A\B)∪(Ac\B)i=kP(B) = 1

❚❤❡r❡❢♦r❡✱k= 1

P(B)✳ ❆♥❞P(A|B) =kP(A

T_{B) =} P(ATB)

P(B) ✳

❉❡❢✐♥✐t✐♦♥ ✶✳ ❙✉♣♣♦s❡ t❤❛tA❛♥❞B❛r❡ ❡✈❡♥ts ❞❡✜♥❡❞ ♦♥ s♦♠❡ s❛♠♣❧❡ s♣❛❝❡

Ω✳ ■❢P(B)>0 t❤❡♥P(A|B) = P(_PA₍T_B)B) ✐s ❝❛❧❧❡❞ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② ♦❢A ❣✐✈❡♥B✳

❊①❛♠♣❧❡ ✷✳ ❆ ❝♦✐♥ ✐s ✢✐♣♣❡❞ t✇✐❝❡✳ ❆ss✉♠✐♥❣ t❤❛t ❛❧❧ ❢♦✉r ♣♦✐♥ts ✐♥ t❤❡ s❛♠♣❧❡ s♣❛❝❡S= (h, h),(h, t),(t, h),(t, t)❛r❡ ❡q✉❛❧❧② ❧✐❦❡❧②✱ ✇❤❛t ✐s t❤❡ ❝♦♥❞✐t✐♦♥❛❧

♣r♦❜❛❜✐❧✐t② t❤❛t ❜♦t❤ ✢✐♣s ❧❛♥❞ ♦♥ ❤❡❛❞s✱ ❣✐✈❡♥ t❤❛t ✭❛✮ t❤❡ ✜rst ✢✐♣ ❧❛♥❞s ♦♥ ❤❡❛❞s❄ ✭❜✮ ❛t ❧❡❛st ♦♥❡ ✢✐♣ ❧❛♥❞s ♦♥ ❤❡❛❞s❄

❙♦❧✉t✐♦♥✳ ▲❡t B = (h, h) ❜❡ t❤❡ ❡✈❡♥t t❤❛t ❜♦t❤ ✢✐♣s ❧❛♥❞ ♦♥ ❤❡❛❞s❀ ❧❡t

F = (h, h),(h, t) ❜❡ t❤❡ ❡✈❡♥t t❤❛t t❤❡ ✜rst ✢✐♣ ❧❛♥❞s ♦♥ ❤❡❛❞s❀ ❛♥❞ ❧❡t A = (h, h),(h, t),(t, h)❜❡ t❤❡ ❡✈❡♥t t❤❛t ❛t ❧❡❛st ♦♥❡ ✢✐♣ ❧❛♥❞s ♦♥ ❤❡❛❞s✳ ❚❤❡ ♣r♦❜❛❜✐❧✲

✐t② ❢♦r ✭❛✮ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠

P(B|F) = P(BF)

P(F) =...

❋♦r ✭❜✮✱

P(B|A) = P(BA)

P(A) =...

❋r♦♠ ❞❡✜♥✐t✐♦♥ ✶✱P(AT

B) =P(B)P(A|B)✳ ❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ♣r♦❜❛❜✐❧✲

✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❛♥ ❛r❜✐tr❛r② ♥✉♠❜❡r ♦❢ ❡✈❡♥ts✱ ✐s s♦♠❡t✐♠❡s r❡❢❡rr❡❞ t♦ ❛s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ r✉❧❡✳

P(E1E2E3...En) =P(E1)P(E2|E1)P(E3|E1E2)...P(En|E1...En❂1)

✶✳✷✳ ■◆❉❊P❊◆❉❊◆❈❊ ✹

❙♦❧✉t✐♦♥✳ ▲❡tC❛♥❞K❞❡♥♦t❡✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡ ❡✈❡♥ts t❤❛t t❤❡ st✉❞❡♥t ❛♥s✇❡rs t❤❡ q✉❡st✐♦♥ ❝♦rr❡❝t❧② ❛♥❞ t❤❡ ❡✈❡♥t t❤❛t ❤❡ ♦r s❤❡ ❛❝t✉❛❧❧② ❦♥♦✇s t❤❡ ❛♥s✇❡r✳ ◆♦✇✱

P(K|C) = P(K)P(C|K)

P(K)P(C|K) +P(Kc_)P(C|Kc₎=

p.1

p.1 + (1−p)._m1 =

mp

1 + (m−1)p ❋♦r ❡①❛♠♣❧❡✱ ✐❢ m = 5✱ p = 1

2✱ t❤❡♥ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ st✉❞❡♥t ❦♥❡✇ t❤❡

❛♥s✇❡r t♦ ❛ q✉❡st✐♦♥ ❤❡ ♦r s❤❡ ❛♥s✇❡r❡❞ ❝♦rr❡❝t❧② ✐s 5 6✳

✶✳✷✳ ■♥❞❡♣❡♥❞❡♥❝❡

❆ ❝❛r❞ ✐s s❡❧❡❝t❡❞ ❛t r❛♥❞♦♠ ❢r♦♠ ❛♥ ♦r❞✐♥❛r② ❞❡❝❦ ♦❢ ✺✷ ♣❧❛②✐♥❣ ❝❛r❞s✳ ■❢E ✐s t❤❡ ❡✈❡♥t t❤❛t t❤❡ s❡❧❡❝t❡❞ ❝❛r❞ ✐s ❛♥ ❛❝❡ ❛♥❞F ✐s t❤❡ ❡✈❡♥t t❤❛t ✐t ✐s ❛ s♣❛❞❡✱ t❤❡♥E ❛♥❞F ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳

❚✇♦ ❝♦✐♥s ❛r❡ ✢✐♣♣❡❞✱ ❛♥❞ ❛❧❧ ✹ ♦✉t❝♦♠❡s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ❡q✉❛❧❧② ❧✐❦❡❧②✳ ■❢ A ✐s t❤❡ ❡✈❡♥t t❤❛t t❤❡ ✜rst ❝♦✐♥ ❧❛♥❞s ♦♥ ❤❡❛❞s ❛♥❞B t❤❡ ❡✈❡♥t t❤❛t t❤❡ s❡❝♦♥❞ ❧❛♥❞s ♦♥ t❛✐❧s✱ t❤❡♥A ❛♥❞B ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳

❉❡❢✐♥✐t✐♦♥ ✽✳ ❊✈❡♥ts A ❛♥❞ B ❛r❡ s❛✐❞ t♦ ❜❡ ✐♥❞❡♣❡♥❞❡♥t ✐❢ P(AT_{B) =}

P(A)P(B)✳

❈♦♥s✐❞❡r ❛♥ ❡①♣❡r✐♠❡♥t ✇❤❡r❡ t✇♦ ❢❛✐r ❞✐❝❡ ❛r❡ t♦ss❡❞✳ ▲❡tE ❞❡♥♦t❡ t❤❡ ❡✈❡♥t t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ ❞✐❝❡ ✐s ✻ ❛♥❞ F1 ❞❡♥♦t❡ t❤❡ ❡✈❡♥t t❤❛t t❤❡ ✜rst ❞✐❡ ❡q✉❛❧s ✹✳

❚❤❡♥ P(ET

F1) = P((4,2)) = ₃₆1 ✇❤❡r❡❛s P(E)P(F1) = ₃₆5 1₆ = ₂₁₆5 ✳ ❍❡♥❝❡✱ E

❛♥❞F1❛r❡ ♥♦t ✐♥❞❡♣❡♥❞❡♥t✳

▲❡tD❜❡ t❤❡ ❡✈❡♥t t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ ❞✐❝❡ ❡q✉❛❧s ✼✳ D❛♥❞F1❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳

❲❤②❄

Pr♦♣♦s✐t✐♦♥ ✾✳ ■❢A❛♥❞B❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ t❤❡♥A❛♥❞Bc _{❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱} Ac _{❛♥❞B ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ❛♥❞ A}c _❛♥❞Bc _{❛r❡ ✐♥❞❡♣❡♥❞❡♥t}

Pr♦♦❢✳ ❆ss✉♠❡ t❤❛t A ❛♥❞ B ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳ ❙✐♥❝❡ A = ABS

ABc _❛♥❞ AB❛♥❞ABc _{❛r❡ ♦❜✈✐♦✉s❧② ♠✉t✉❛❧❧② ❡①❝❧✉s✐✈❡✱ t❤❡♥}

P(A) =PAB[ABc_{=P(AB) +P(AB}c₎ ♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱

P(ABc_{) =P(A)−P(AB) =P(A)−P(A)P(B) =P(A) (1−P(B)) =P(A)P(B}c₎

❛♥❞ t❤❡ r❡s✉❧t ✐s ♣r♦✈❡❞✳

◆♦t✐♦♥ ♦❢ ✐♥❞❡♣❡♥❞❡♥❝❡ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❛ ✜♥✐t❡ ♦r ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ❝♦❧❧❡❝✲ t✐♦♥ ♦❢ ❡✈❡♥ts✳

❉❡❢✐♥✐t✐♦♥ ✶✵✳A1, A2, ..., An✭♦rA1, A2, ...✮ ❛r❡ ✭♠✉t✉❛❧❧②✮ ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts

✐❢ ❢♦r ❛♥② ✜♥✐t❡ s✉❜❝♦❧❧❡❝t✐♦♥Ai₁, Ai₂, ..., Aik✱

P(Ai₁\Ai₂\...\Aik) =

k

Y

j=1

P(Aij)

❚❤✉s ✐❢A1, A2, ..., An ❛r❡ ✐♥❞❡♣❡♥❞❡♥t t❤❡♥Ai ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢Aj ❢♦ri6=j❀

✶✳✸✳ P❘❖❇▲❊▼❙ ✺

❊①❛♠♣❧❡ ✶✶✳ ❈♦♥s✐❞❡r ❛♥ ❡①♣❡r✐♠❡♥t ♦❢ ♠❛❦✐♥❣ ❧✐❝❡♥s❡ ♣❧❛t❡ ♥✉♠❜❡rs ❝♦♥s✐st ♦❢ ✸ ❞✐✛❡r❡♥t ❧❡tt❡rs ✇✐t❤ t❤❡ s❛♠♣❧❡ s♣❛❝❡Ω =abc, bac, cab, bca, acb, cba, aaa, bbb, ccc ✇❤❡r❡ ❡❛❝❤ ♦✉t❝♦♠❡ ✐s ❡q✉❛❧❧② ❧✐❦❡❧②✳ ❉❡✜♥❡ ❡✈❡♥ts Ck = c in kth_{position, k =}

1,2,3)✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛tP(Ck) = 1/3, k= 1,2,3❛♥❞P(CjT

Ck) = 1/9, j6=k❀

t❤✉sCj ❛♥❞Ck ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❢♦r ❛❧❧j6=k✳ ❍♦✇❡✈❡r✱P(C1TC2TC3) = 1/96=

1/27❛♥❞ s♦C1✱C2❛♥❞C3 ❛r❡ ♥♦t ✐♥❞❡♣❡♥❞❡♥t✳ ❚❤✐s ❡①❛♠♣❧❡ s❤♦✇s t❤❛t ♣❛✐r✇✐s❡

✐♥❞❡♣❡♥❞❡♥❝❡ ❞♦❡s ♥♦t ✐♠♣❧② ♠✉t✉❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡✳ ✶✳✸✳ Pr♦❜❧❡♠s

✭✶✮ ❈♦♥s✐❞❡r ✸ ✉r♥s✳ ❯r♥ ❆ ❝♦♥t❛✐♥s ✷ ✇❤✐t❡ ❛♥❞ ✹ r❡❞ ❜❛❧❧s✱ ✉r♥ ❇ ❝♦♥t❛✐♥s ✽ ✇❤✐t❡ ❛♥❞ ✹ r❡❞ ❜❛❧❧s✱ ❛♥❞ ✉r♥ ❈ ❝♦♥t❛✐♥s ✶ ✇❤✐t❡ ❛♥❞ ✸ r❡❞ ❜❛❧❧s✳ ■❢ ✶ ❜❛❧❧ ✐s s❡❧❡❝t❡❞ ❢r♦♠ ❡❛❝❤ ✉r♥✱ ✇❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❜❛❧❧ ❝❤♦s❡♥ ❢r♦♠ ✉r♥ ❆ ✇❛s ✇❤✐t❡ ❣✐✈❡♥ t❤❛t ❡①❛❝t❧② ✷ ✇❤✐t❡ ❜❛❧❧s ✇❡r❡ s❡❧❡❝t❡❞❄ ✭✷✮ ❆ r❡❝❡♥t ❝♦❧❧❡❣❡ ❣r❛❞✉❛t❡ ✐s ♣❧❛♥♥✐♥❣ t♦ t❛❦❡ t❤❡ ✜rst t❤r❡❡ ❛❝t✉❛r✐❛❧

❡①❛♠✐♥❛t✐♦♥s ✐♥ t❤❡ ❝♦♠✐♥❣ s✉♠♠❡r✳ ❙❤❡ ✇✐❧❧ t❛❦❡ t❤❡ ✜rst ❛❝t✉❛r✐❛❧ ❡①❛♠ ✐♥ ❏✉♥❡✳ ■❢ s❤❡ ♣❛ss❡s t❤❛t ❡①❛♠✱ t❤❡♥ s❤❡ ✇✐❧❧ t❛❦❡ t❤❡ s❡❝♦♥❞ ❡①❛♠ ✐♥ ❏✉❧②✱ ❛♥❞ ✐❢ s❤❡ ❛❧s♦ ♣❛ss❡s t❤❛t ♦♥❡✱ t❤❡♥ s❤❡ ✇✐❧❧ t❛❦❡ t❤❡ t❤✐r❞ ❡①❛♠ ✐♥ ❙❡♣t❡♠❜❡r✳ ■❢ s❤❡ ❢❛✐❧s ❛♥ ❡①❛♠✱ t❤❡♥ s❤❡ ✐s ♥♦t ❛❧❧♦✇❡❞ t♦ t❛❦❡ ❛♥② ♦t❤❡rs✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t s❤❡ ♣❛ss❡s t❤❡ ✜rst ❡①❛♠✐s ✳✾✳ ■❢ s❤❡ ♣❛ss❡s t❤❡ ✜rst ❡①❛♠✱ t❤❡♥ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② t❤❛t s❤❡ ♣❛ss❡s t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s ✳✽✱ ❛♥❞ ✐❢ s❤❡ ♣❛ss❡s ❜♦t❤ t❤❡ ✜rst ❛♥❞ t❤❡ s❡❝♦♥❞ ❡①❛♠s✱ t❤❡♥ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② t❤❛t s❤❡ ♣❛ss❡s t❤❡ t❤✐r❞ ❡①❛♠ ✐s ✳✼✳ ❲❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t s❤❡ ♣❛ss❡s ❛❧❧ t❤r❡❡ ❡①❛♠s❄

✭✸✮ ❆♥ ❡❝t♦♣✐❝ ♣r❡❣♥❛♥❝② ✐s t✇✐❝❡ ❛s ❧✐❦❡❧② t♦ ❞❡✈❡❧♦♣ ✇❤❡♥ t❤❡ ♣r❡❣♥❛♥t ✇♦♠❛♥ ✐s ❛ s♠♦❦❡r ❛s ✐t ✐s ✇❤❡♥ s❤❡ ✐s ❛ ♥♦♥s♠♦❦❡r✳ ■❢ ✸✷ ♣❡r❝❡♥t ♦❢ ✇♦♠❡♥ ♦❢ ❝❤✐❧❞❜❡❛r✐♥❣ ❛❣❡ ❛r❡ s♠♦❦❡rs✱ ✇❤❛t ♣❡r❝❡♥t❛❣❡ ♦❢ ✇♦♠❡♥ ❤❛✈✐♥❣ ❡❝t♦♣✐❝ ♣r❡❣♥❛♥❝✐❡s ❛r❡ s♠♦❦❡r

✭✹✮ ❆ t♦t❛❧ ♦❢ ✹✻ ♣❡r❝❡♥t ♦❢ t❤❡ ✈♦t❡rs ✐♥ ❛ ❝❡rt❛✐♥ ❝✐t② ❝❧❛ss✐❢② t❤❡♠s❡❧✈❡s ❛s ■♥❞❡♣❡♥❞❡♥ts✱ ✇❤❡r❡❛s ✸✵ ♣❡r❝❡♥t ❝❧❛ss✐❢② t❤❡♠s❡❧✈❡s ❛s ▲✐❜❡r❛❧s ❛♥❞ ✷✹ ♣❡r❝❡♥t s❛② t❤❛t t❤❡② ❛r❡ ❈♦♥s❡r✈❛t✐✈❡s✳ ■♥ ❛ r❡❝❡♥t ❧♦❝❛❧ ❡❧❡❝t✐♦♥✱ ✸✺ ♣❡r❝❡♥t ♦❢ t❤❡ ■♥❞❡♣❡♥❞❡♥ts✱ ✻✷ ♣❡r❝❡♥t ♦❢ t❤❡ ▲✐❜❡r❛❧s✱ ❛♥❞ ✺✽ ♣❡r❝❡♥t ♦❢ t❤❡ ❈♦♥s❡r✈❛t✐✈❡s ✈♦t❡❞✳ ❆ ✈♦t❡r ✐s ❝❤♦s❡♥ ❛t r❛♥❞♦♠s❄ ●✐✈❡♥ t❤❛t t❤✐s ♣❡rs♦♥ ✈♦t❡❞ ✐♥ t❤❡ ❧♦❝❛❧ ❡❧❡❝t✐♦♥✱ ✇❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❤❡ ♦r s❤❡ ✐s ✭❛✮ ❛♥ ■♥❞❡♣❡♥❞❡♥t❄ ✭❜✮ ❛ ▲✐❜❡r❛❧❄ ✭❝✮ ❛ ❈♦♥s❡r✈❛t✐✈❡❄ ✭❞✮ ❲❤❛t ❢r❛❝t✐♦♥ ♦❢ ✈♦t❡rs ♣❛rt✐❝✐♣❛t❡❞ ✐♥ t❤❡ ❧♦❝❛❧ ❡❧❡❝t✐♦♥❄

✭✺✮ ❈♦♥s✐❞❡r t✇♦ ❜♦①❡s✱ ♦♥❡ ❝♦♥t❛✐♥✐♥❣ ✶ ❜❧❛❝❦ ❛♥❞ ✶ ✇❤✐t❡ ♠❛r❜❧❡✱ t❤❡ ♦t❤❡r ✷ ❜❧❛❝❦ ❛♥❞ ✶ ✇❤✐t❡ ♠❛r❜❧❡✳ ❆ ❜♦① ✐s s❡❧❡❝t❡❞ ❛t r❛♥❞♦♠✱ ❛♥❞ ❛ ♠❛r❜❧❡ ✐s ❞r❛✇♥ ❢r♦♠ ✐t ❛t r❛♥❞♦♠✳ ❲❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ♠❛r❜❧❡ ✐s ❜❧❛❝❦❄ ❲❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ✜rst ❜♦① ✇❛s t❤❡ ♦♥❡ s❡❧❡❝t❡❞ ❣✐✈❡♥ t❤❛t t❤❡ ♠❛r❜❧❡ ✐s ✇❤✐t❡❄

✭✻✮ ■❢P(A|C)> P(B|C)❛♥❞ P(A|Cc_{)> P(B|C}c_{)❡✐t❤❡r ♣r♦✈❡ t❤❛tP(A)>} P(B)✳