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  1. ■♥tr♦ t♦ ❈♦♥t❡♠♣♦r❛r② ▼❛t❤ ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ❛♥❞ ■♥❞❡♣❡♥❞❡♥t ❊✈❡♥ts ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❯❑ ❖❝t♦❜❡r ✺✱ ✷✵✶✽

  2. ❆♥♥♦✉♥❝❡♠❡♥t ◮ ❨♦✉ ❤❛✈❡ ❛ ❤♦♠❡✇♦r❦ ❛ss✐❣♥♠❡♥t ❞✉❡ ♥❡①t ▼♦♥❞❛②✳

  3. ❉r❛✇✐♥❣ ❛ ❈❛r❞ ◮ ❨♦✉ ❤❛✈❡ ❛ ❤♦♠❡✇♦r❦ ❛ss✐❣♥♠❡♥t ❞✉❡ ♥❡①t ▼♦♥❞❛②✳ ❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛r❞s ❤❛s ✷ s✉✐ts ✭❆✲❇✮ ❛♥❞ ✺ r❛♥❦s ✭✶✲✺✮✳ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❇✹ ❇✺

  4. ❉r❛✇✐♥❣ ❛ ❈❛r❞ ❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛r❞s ❤❛s ✷ s✉✐ts ✭❆✲❇✮ ❛♥❞ ✺ r❛♥❦s ✭✶✲✺✮✳ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❇✹ ❇✺ ◮ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❆ ❝❛r❞ ✇✐t❤ s✉✐t ❇ ✐s ❞r❛✇♥✳✑ ◮ ▲❡t F ❜❡ t❤❡ ❡✈❡♥t ✏❆ ❝❛r❞ ✇✐t❤ r❛♥❦ ✷ ✐s ❞r❛✇♥✳✑

  5. ❉r❛✇✐♥❣ ❛ ❈❛r❞ ✇✐t❤ ❛ ❉❡s✐r❡❞ ❙✉✐t ❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛r❞s ❤❛s ✷ s✉✐ts ✭❆✲❇✮ ❛♥❞ ✺ r❛♥❦s ✭✶✲✺✮✳ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❇✹ ❇✺ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞r❛✇✐♥❣ ❛ ❝❛r❞ ✇✐t❤ s✉✐t ❇ ✐s✿ P ( E ) = ✺ s✉❝❤ ❝❛r❞s ✶✵ ❝❛r❞s t♦t❛❧ = ✺ ✶✵ = ✶ ✷ .

  6. ❉r❛✇✐♥❣ ❛ ❈❛r❞ ✇✐t❤ ❛ ❉❡s✐r❡❞ ❘❛♥❦ ❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛r❞s ❤❛s ✷ s✉✐ts ✭❆✲❇✮ ❛♥❞ ✺ r❛♥❦s ✭✶✲✺✮✳ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❇✹ ❇✺ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞r❛✇✐♥❣ ❛ ❝❛r❞ ✇✐t❤ r❛♥❦ ✷ ✐s✿ P ( F ) = ✷ s✉❝❤ ❝❛r❞s ✶✵ ❝❛r❞s t♦t❛❧ = ✷ ✶✵ = ✶ ✺ .

  7. ❉r❛✇✐♥❣ ❛ ❙♣❡❝✐✜❝ ❈❛r❞ ❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛r❞s ❤❛s ✷ s✉✐ts ✭❆✲❇✮ ❛♥❞ ✺ r❛♥❦s ✭✶✲✺✮✳ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❇✹ ❇✺ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞r❛✇✐♥❣ ❝❛r❞ ❇✷ ✭s✉✐t ❇ ❛♥❞ r❛♥❦ ✷✮ ✐s✿ F ) = ✶ s✉❝❤ ❝❛r❞ ❇✷ ✶✵ ❝❛r❞s t♦t❛❧ = ✶ � P ( E ✶✵ .

  8. ❖❜s❡r✈❛t✐♦♥ ◆♦t✐❝❡ t❤❛t P ( E ) = ✶ ✷ , P ( F ) = ✶ F ) = ✶ � ✺ , P ( E ✶✵ . ❈♦✐♥❝✐❞❡♥t❛❧❧②✱ P ( E ) · P ( F ) = ✶ ✷ · ✶ ✺ = ✶ ✶✵ ❛s ✇❡❧❧✳ ❲❤❡♥ ❝❛♥ ✇❡ ♠✉❧t✐♣❧② t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ t✇♦ ❡✈❡♥ts t♦ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥❄

  9. ❖❜s❡r✈❛t✐♦♥ ◆♦t✐❝❡ t❤❛t P ( E ) = ✶ ✷ , P ( F ) = ✶ F ) = ✶ � ✺ , P ( E ✶✵ . ❈♦✐♥❝✐❞❡♥t❛❧❧②✱ P ( E ) · P ( F ) = ✶ ✷ · ✶ ✺ = ✶ ✶✵ ❛s ✇❡❧❧✳ ❲❤❡♥ ❝❛♥ ✇❡ ♠✉❧t✐♣❧② t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ t✇♦ ❡✈❡♥ts t♦ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥❄

  10. ❉r❛✇✐♥❣ ❛ ❈❛r❞✿ ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ▲❡t ✉s ❝♦♠♣✉t❡✿ ◮ P ( F | E ) ✿ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s r❛♥❦ ✷✱ ❣✐✈❡♥ t❤❛t ✐t ❤❛s s✉✐t ❇✳ ◮ P ( E | F ) ✿ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s s✉✐t ❇✱ ❣✐✈❡♥ t❤❛t ✐t ❤❛s r❛♥❦ ✷✳ ❲❡ ✇✐❧❧ ❝♦♠♣❛r❡ t❤❡s❡ t♦ ❡❛❝❤ ♦t❤❡r✱ ❛♥❞ t♦ P ( E ) ❛♥❞ P ( F ) ✳

  11. ❉r❛✇✐♥❣ ❛ ❈❛r❞✿ P ( F | E ) ✭●✐✈❡♥ ❙✉✐t✮ ❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛r❞s ❤❛s ✷ s✉✐ts ✭❆✲❇✮ ❛♥❞ ✺ r❛♥❦s ✭✶✲✺✮✳ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❇✹ ❇✺ ▲❡t✬s ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s r❛♥❦ ✷ ❣✐✈❡♥ t❤❛t ✐t ❤❛s s✉✐t ❇✳ ❚❤❡ ❣✐✈❡♥ ❡✈❡♥t ♥❛rr♦✇s ❞♦✇♥ t❤❡ s❛♠♣❧❡ s♣❛❝❡ t♦ ❥✉st t❤❡ ✜✈❡ ❝❛r❞s ✇✐t❤ s✉✐t ❇✳

  12. ❉r❛✇✐♥❣ ❛ ❈❛r❞✿ P ( F | E ) ✭●✐✈❡♥ ❙✉✐t✮ ❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛r❞s ❤❛s ✷ s✉✐ts ✭❆✲❇✮ ❛♥❞ ✺ r❛♥❦s ✭✶✲✺✮✳ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❇✹ ❇✺ ▲❡t✬s ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s r❛♥❦ ✷ ❣✐✈❡♥ t❤❛t ✐t ❤❛s s✉✐t ❇✳ ❚❤❡ ❣✐✈❡♥ ❡✈❡♥t ♥❛rr♦✇s ❞♦✇♥ t❤❡ s❛♠♣❧❡ s♣❛❝❡ t♦ ❥✉st t❤❡ ✜✈❡ ❝❛r❞s ✇✐t❤ s✉✐t ❇✳

  13. ❉r❛✇✐♥❣ ❛ ❈❛r❞✿ P ( F | E ) ✭●✐✈❡♥ ❙✉✐t✮ ❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛r❞s ❤❛s ✷ s✉✐ts ✭❆✲❇✮ ❛♥❞ ✺ r❛♥❦s ✭✶✲✺✮✳ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❇✹ ❇✺ ▲❡t✬s ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s r❛♥❦ ✷ ❣✐✈❡♥ t❤❛t ✐t ❤❛s s✉✐t ❇✳ ❚❤❡ ❣✐✈❡♥ ❡✈❡♥t ♥❛rr♦✇s ❞♦✇♥ t❤❡ s❛♠♣❧❡ s♣❛❝❡ t♦ ❥✉st t❤❡ ✜✈❡ ❝❛r❞s ✇✐t❤ s✉✐t ❇✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s r❛♥❦ ✷ ❣✐✈❡♥ t❤❛t ✐t ❤❛s s✉✐t ❇ ✐s P ( F | E ) = ✶ ❝❛r❞ ❛♠♦♥❣ t❤♦s❡ ✇✐t❤ s✉✐t ❇ t❤❛t ❤❛s r❛♥❦ ✷ = ✶ ✺ . ✺ ❝❛r❞s ✇✐t❤ s✉✐t ❇

  14. ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ◆✉♠❡r❛t♦r ▲❡t✬s ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s r❛♥❦ ✷ ❣✐✈❡♥ t❤❛t ✐t ❤❛s s✉✐t ❇✳ ❚❤❡ ❣✐✈❡♥ ❡✈❡♥t ♥❛rr♦✇s ❞♦✇♥ t❤❡ s❛♠♣❧❡ s♣❛❝❡ t♦ ❥✉st t❤❡ ✜✈❡ ❝❛r❞s ✇✐t❤ s✉✐t ❇✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s r❛♥❦ ✷ ❣✐✈❡♥ t❤❛t ✐t ❤❛s s✉✐t ❇ ✐s P ( F | E ) = ✶ ❝❛r❞ ❛♠♦♥❣ t❤♦s❡ ✇✐t❤ s✉✐t ❇ t❤❛t ❤❛s r❛♥❦ ✷ = ✶ ✺ . ✺ ❝❛r❞s ✇✐t❤ s✉✐t ❇ ❲❡ ❝❛♥ r❡♣❤r❛s❡ t❤❡ ♥✉♠❡r❛t♦r✿ P ( F | E ) = ✶ ❝❛r❞ ✇✐t❤ s✉✐t ❇ ❛♥❞ r❛♥❦ ✷ = ✶ ✺ . ✺ ❝❛r❞s ✇✐t❤ s✉✐t ❇

  15. ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ❈♦♠♣✉t❛t✐♦♥ ▲❡t E ❛♥❞ F ❜❡ ❡✈❡♥ts ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡ Ω ✳ ❚❤❡♥ P ( F | E ) ✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❣❡tt✐♥❣ ❛♥ ♦✉t❝♦♠❡ ✐♥ ❡✈❡♥t F ❣✐✈❡♥ t❤❛t t❤❡ ♦✉t❝♦♠❡ ✐s ✐♥ ❡✈❡♥t E ✱ ✐s P ( F | E ) = ◆✉♠❜❡r ♦❢ ♦✉t❝♦♠❡s ✐♥ E � F . ◆✉♠❜❡r ♦❢ ♦✉t❝♦♠❡s ✐♥ E ❍✐♥t✿ ❚❤❡ ❣✐✈❡♥ ❡✈❡♥t ❛♣♣❡❛rs ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r✳

  16. ❉r❛✇✐♥❣ ❛ ❈❛r❞✿ P ( E | F ) ✭●✐✈❡♥ ❘❛♥❦✮ ❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛r❞s ❤❛s ✷ s✉✐ts ✭❆✲❇✮ ❛♥❞ ✺ r❛♥❦s ✭✶✲✺✮✳ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❇✹ ❇✺ ▲❡t✬s ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s s✉✐t ❇ ❣✐✈❡♥ t❤❛t ✐t ❤❛s r❛♥❦ ✷✳ ❚❤❡ ❣✐✈❡♥ ❡✈❡♥t ♥❛rr♦✇s ❞♦✇♥ t❤❡ s❛♠♣❧❡ s♣❛❝❡ t♦ ❥✉st t❤❡ t✇♦ ❝❛r❞s ✇✐t❤ r❛♥❦ ✷✳

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