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tr t trr t ts s trsts ts rtt tts

  1. ■♥tr♦ t♦ ❈♦♥t❡♠♣♦r❛r② ▼❛t❤ ❈♦♠♣❧❡♠❡♥ts✱ ❯♥✐♦♥s✱ ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ♦❢ ❊✈❡♥ts ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❯❑

  2. ◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡ ❆♥♥♦✉♥❝❡♠❡♥t ✿ ❨♦✉ ❤❛✈❡ ❤♦♠❡✇♦r❦ ❞✉❡ t♦♥✐❣❤t✳ ❍❡r❡ ✐s ❛ ❧✐st ♦❢ ✶✷✸ ❜❡❛❞s s♦rt❡❞ ❜② ❝♦❧♦r ❛♥❞ s❤❛♣❡✳ ❙✉♣♣♦s❡ ✇❡ ❞r❛✇ ❛ ❜❡❛❞✳ ❚♦t❛❧ △ △ △ � � � ❘❡❞ ✶ ✽ ✾ ●r❡❡♥ ✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸

  3. ◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡ ❆♥♥♦✉♥❝❡♠❡♥t ✿ ❨♦✉ ❤❛✈❡ ❤♦♠❡✇♦r❦ ❞✉❡ t♦♥✐❣❤t✳ ❍❡r❡ ✐s ❛ ❧✐st ♦❢ ✶✷✸ ❜❡❛❞s s♦rt❡❞ ❜② ❝♦❧♦r ❛♥❞ s❤❛♣❡✳ ❙✉♣♣♦s❡ ✇❡ ❞r❛✇ ❛ ❜❡❛❞✳ △ ❚♦t❛❧ △ △ � � � ❘❡❞ ✶ ✽ ✾ ●r❡❡♥ ✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑

  4. ◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡ ❆♥♥♦✉♥❝❡♠❡♥t ✿ ❨♦✉ ❤❛✈❡ ❤♦♠❡✇♦r❦ ❞✉❡ t♦♥✐❣❤t✳ ❍❡r❡ ✐s ❛ ❧✐st ♦❢ ✶✷✸ ❜❡❛❞s s♦rt❡❞ ❜② ❝♦❧♦r ❛♥❞ s❤❛♣❡✳ ❙✉♣♣♦s❡ ✇❡ ❞r❛✇ ❛ ❜❡❛❞✳ △ ❚♦t❛❧ △ △ � � � ❘❡❞ ✶ ✽ ✾ ●r❡❡♥ ✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ✻✹ + ✶✻ = ✽✵ ❜❡❛❞s ❛r❡ ❣r❡❡♥ ♦✉t ♦❢ ✶✷✸ t♦t❛❧✱ s♦ ✽✵ P ( E ) = ✶✷✸ .

  5. ❄✭✷✳✶✮ ◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡ ✭❊✈❡♥t ❋✮ ❍❡r❡ ✐s ❛ ❧✐st ♦❢ ✶✷✸ ❜❡❛❞s s♦rt❡❞ ❜② ❝♦❧♦r ❛♥❞ s❤❛♣❡✳ ❙✉♣♣♦s❡ ✇❡ ❞r❛✇ ❛ ❜❡❛❞✳ △ △ △ � � � ❚♦t❛❧ ❘❡❞ ✶ ✽ ✾ ●r❡❡♥ ✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❲❤❛t ✐s P ( F ) ✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞r❛✇✐♥❣ ❛ sq✉❛r❡ ❜❡❛❞❄ ❚②♣❡ ❛♥❞ s❡♥❞ ❛ ❢r❛❝t✐♦♥ ✳

  6. ◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡ ✭❊✈❡♥t ❋✮ ❍❡r❡ ✐s ❛ ❧✐st ♦❢ ✶✷✸ ❜❡❛❞s s♦rt❡❞ ❜② ❝♦❧♦r ❛♥❞ s❤❛♣❡✳ ❙✉♣♣♦s❡ ✇❡ ❞r❛✇ ❛ ❜❡❛❞✳ △ ❚♦t❛❧ △ △ � � � ❘❡❞ ✶ ✽ ✾ ●r❡❡♥ ✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❚❤❡r❡ ❛r❡ ✽ + ✶✻ + ✸✷ = ✺✻ sq✉❛r❡ ❜❡❛❞s ♦✉t ♦❢ ✶✷✸ t♦t❛❧✱ s♦ ✺✻ P ( F ) = ✶✷✸ .

  7. ❈♦♠♣❧❡♠❡♥ts ◮ ❚❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ ❛♥ ❡✈❡♥t E ✐s t❤❡ ❡✈❡♥t ✏ E ❞♦❡s ♥♦t ❤❛♣♣❡♥✳✑ ◮ ❚❤❡ ♥♦t❛t✐♦♥ ¯ E ♠❡❛♥s t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ E ✳ ◮ ■❢ ✇❡ ❦♥♦✇ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ E ✱ P ( E ) ✱ t❤❡♥ P ( ¯ E ) = ✶ − P ( E ) .

  8. ❈♦♠♣❧❡♠❡♥ts ◮ ❚❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ ❛♥ ❡✈❡♥t E ✐s t❤❡ ❡✈❡♥t ✏ E ❞♦❡s ♥♦t ❤❛♣♣❡♥✳✑ ◮ ❚❤❡ ♥♦t❛t✐♦♥ ¯ E ♠❡❛♥s t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ E ✳ ◮ ■❢ ✇❡ ❦♥♦✇ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ E ✱ P ( E ) ✱ t❤❡♥ P ( ¯ E ) = ✶ − P ( E ) .

  9. ❈♦♠♣❧❡♠❡♥ts ◮ ❚❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ ❛♥ ❡✈❡♥t E ✐s t❤❡ ❡✈❡♥t ✏ E ❞♦❡s ♥♦t ❤❛♣♣❡♥✳✑ ◮ ❚❤❡ ♥♦t❛t✐♦♥ ¯ E ♠❡❛♥s t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ E ✳ ◮ ■❢ ✇❡ ❦♥♦✇ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ E ✱ P ( E ) ✱ t❤❡♥ P ( ¯ E ) = ✶ − P ( E ) .

  10. ❄✭✷✳✷✮ ❈♦♠♣❧❡♠❡♥ts Pr❛❝t✐❝❡ ◮ ❚❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ ❛♥ ❡✈❡♥t E ✐s t❤❡ ❡✈❡♥t ✏ E ❞♦❡s ♥♦t ❤❛♣♣❡♥✳✑ ◮ ❚❤❡ ♥♦t❛t✐♦♥ ¯ E ♠❡❛♥s t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ E ✳ ◮ ■❢ ✇❡ ❦♥♦✇ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ E ✱ P ( E ) ✱ t❤❡♥ P ( ¯ E ) = ✶ − P ( E ) . ❆♥ ❡✈❡♥t E ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡ ✇✐t❤ s❡✈❡♥ ♦❜❥❡❝ts ❤❛s ♣r♦❜❛❜✐❧✐t② ✹ / ✼ ♦❢ ♦❝❝✉rr✐♥❣✳ ❲❤❛t ❢r❛❝t✐♦♥ ✐s P ( ¯ E ) ❄

  11. ❄✭✷✳✷✮ ❈♦♠♣❧❡♠❡♥ts Pr❛❝t✐❝❡ ❆♥ ❡✈❡♥t E ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡ ✇✐t❤ s❡✈❡♥ ♦❜❥❡❝ts ❤❛s ♣r♦❜❛❜✐❧✐t② ✹ / ✼ ♦❢ ♦❝❝✉rr✐♥❣✳ ❲❤❛t ❢r❛❝t✐♦♥ ✐s P ( ¯ E ) ❄ P ( ¯ E ) = ✶ − P ( E ) = ✶ − ✹ ✼ ❚♦ ❛❞❞ ♦r s✉❜tr❛❝t ❛ ✇❤♦❧❡ ♥✉♠❜❡r ❛♥❞ ❛ ❢r❛❝t✐♦♥✱ ✉s❡ ❛ ❝♦♠♠♦♥ ❞❡♥♦♠✐♥❛t♦r✿ ✶ · ✼ ✼ − ✹ = ✼ ✼ ✼ − ✹ = ✼

  12. ❈♦♠♣❧❡♠❡♥ts Pr❛❝t✐❝❡ ❆♥ ❡✈❡♥t E ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡ ✇✐t❤ s❡✈❡♥ ♦❜❥❡❝ts ❤❛s ♣r♦❜❛❜✐❧✐t② ✹ / ✼ ♦❢ ♦❝❝✉rr✐♥❣✳ ❲❤❛t ❢r❛❝t✐♦♥ ✐s P ( ¯ E ) ❄ P ( ¯ E ) = ✶ − P ( E ) = ✶ − ✹ ✼ ❚♦ ❛❞❞ ♦r s✉❜tr❛❝t ❛ ✇❤♦❧❡ ♥✉♠❜❡r ❛♥❞ ❛ ❢r❛❝t✐♦♥✱ ✉s❡ ❛ ❝♦♠♠♦♥ ❞❡♥♦♠✐♥❛t♦r✿ ✶ · ✼ ✼ − ✹ = ✼ ✼ ✼ − ✹ = ✼ ✼ − ✹ = ✸ = ✼ . ✼

  13. ❄✭✷✳✸✮ ◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡ ✭❈♦♠♣❧❡♠❡♥t✮ ❍❡r❡ ✐s ❛ ❧✐st ♦❢ ✶✷✸ ❜❡❛❞s s♦rt❡❞ ❜② ❝♦❧♦r ❛♥❞ s❤❛♣❡✳ ❙✉♣♣♦s❡ ✇❡ ❞r❛✇ ❛ ❜❡❛❞✳ △ △ △ � � � ❚♦t❛❧ ❘❡❞ ✶ ✽ ✾ ●r❡❡♥ ✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❋✐♥❞ P ( ¯ E ) ✳

  14. ◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡ ✭❈♦♠♣❧❡♠❡♥t✮ △ ❚♦t❛❧ △ △ � � � ❘❡❞ ✶ ✽ ✾ ●r❡❡♥ ✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❚❤❡♥ ¯ E ✐s t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ♥♦t ❣r❡❡♥✱✑ s♦ ✐t ✐s r❡❞ ♦r ❜❧✉❡✿ ✾ + ✸✹ = ✹✸ ❜❡❛❞s ❛r❡ ♥♦t ❣r❡❡♥✳ ✹✸ P ( ¯ E ) = ✶✷✸ .

  15. ◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡ ✭❈♦♠♣❧❡♠❡♥t✮ △ △ △ � � � ❚♦t❛❧ ❘❡❞ ✶ ✽ ✾ ●r❡❡♥ ✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❯s✐♥❣ t❤❡ ❝♦♠♣❧❡♠❡♥t✱ ✇❡ s❡❡ t❤❛t ✽✵ ❜❡❛❞s ❛r❡ ❣r❡❡♥ ♦✉t ♦❢ ✶✷✸ t♦t❛❧✱ s♦ E ) = ✶ − P ( E ) = ✶ − ✽✵ ✶✷✸ = ✶✷✸ ✶✷✸ − ✽✵ ✹✸ P ( ¯ ✶✷✸ = ✶✷✸ .

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