tr t trr t ts - - PowerPoint PPT Presentation

■♥tr♦ t♦ ❈♦♥t❡♠♣♦r❛r② ▼❛t❤

❈♦♠♣❧❡♠❡♥ts✱ ❯♥✐♦♥s✱ ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ♦❢ ❊✈❡♥ts

❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❯❑

◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡

❆♥♥♦✉♥❝❡♠❡♥t✿ ❨♦✉ ❤❛✈❡ ❤♦♠❡✇♦r❦ ❞✉❡ t♦♥✐❣❤t✳ ❍❡r❡ ✐s ❛ ❧✐st ♦❢ ✶✷✸ ❜❡❛❞s s♦rt❡❞ ❜② ❝♦❧♦r ❛♥❞ s❤❛♣❡✳ ❙✉♣♣♦s❡ ✇❡ ❞r❛✇ ❛ ❜❡❛❞✳ △ △ △

• ❚♦t❛❧

❘❡❞ ✶ ✽ ✾

• r❡❡♥

✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸

◆✉♠❜❡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❡✳✑

◆✉♠❜❡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) = ✽✵ ✶✷✸ .

❄✭✷✳✶✮ ◆✉♠❜❡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✐♦♥✳

◆✉♠❜❡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) = ✺✻ ✶✷✸ .

❈♦♠♣❧❡♠❡♥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).

❈♦♠♣❧❡♠❡♥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).

❈♦♠♣❧❡♠❡♥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).

❄✭✷✳✷✮ ❈♦♠♣❧❡♠❡♥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)❄

❄✭✷✳✷✮ ❈♦♠♣❧❡♠❡♥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✿ = ✶· ✼ ✼ − ✹ ✼ = ✼ ✼ − ✹ ✼

❈♦♠♣❧❡♠❡♥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✿ = ✶· ✼ ✼ − ✹ ✼ = ✼ ✼ − ✹ ✼ = ✼−✹ ✼ = ✸ ✼ .

❄✭✷✳✸✮ ◆✉♠❜❡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)✳

◆✉♠❜❡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) = ✹✸ ✶✷✸ .

◆✉♠❜❡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♦ P( ¯ E) = ✶−P(E) = ✶− ✽✵ ✶✷✸ = ✶✷✸ ✶✷✸ − ✽✵ ✶✷✸ = ✹✸ ✶✷✸ .

❯♥✐♦♥s ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ✭❖r ❛♥❞ ❆♥❞✮

▲❡t E ❛♥❞ F ❜❡ t✇♦ ❡✈❡♥ts ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡✳

◮ ❚❤❡ ❡✈❡♥t t❤❛t ❤❛s ♦✉t❝♦♠❡s ✐♥ E✱ ♦r ✐♥ F✱ ♦r ✐♥ ❜♦t❤✱ ✐s

❝❛❧❧❡❞ t❤❡ ✉♥✐♦♥ ♦❢ E ❛♥❞ F ❛♥❞ ✐s ❞❡♥♦t❡❞ E F✳

◮ ❚❤❡ ❡✈❡♥t t❤❛t ❤❛s ♦✉t❝♦♠❡s ✐♥ ❜♦t❤ E ❛♥❞ F✱ ✐s ❝❛❧❧❡❞

t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ E ❛♥❞ F ❛♥❞ ✐s ❞❡♥♦t❡❞ E F✳

❯♥✐♦♥s ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ✭❖r ❛♥❞ ❆♥❞✮

▲❡t E ❛♥❞ F ❜❡ t✇♦ ❡✈❡♥ts ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡✳

◮ ❚❤❡ ❡✈❡♥t t❤❛t ❤❛s ♦✉t❝♦♠❡s ✐♥ E✱ ♦r ✐♥ F✱ ♦r ✐♥ ❜♦t❤✱ ✐s

❝❛❧❧❡❞ t❤❡ ✉♥✐♦♥ ♦❢ E ❛♥❞ F ❛♥❞ ✐s ❞❡♥♦t❡❞ E F✳

◮ ❚❤❡ ❡✈❡♥t t❤❛t ❤❛s ♦✉t❝♦♠❡s ✐♥ ❜♦t❤ E ❛♥❞ F✱ ✐s ❝❛❧❧❡❞

t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ E ❛♥❞ F ❛♥❞ ✐s ❞❡♥♦t❡❞ E F✳

◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡ ✭❖r✮

△ △ △

• ❚♦t❛❧

❘❡❞ ✶ ✽ ✾

• r❡❡♥

✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❚♦ ✜♥❞ P(E F)✱ ✇❡ ♥❡❡❞ t♦ ❝♦✉♥t t❤❡ ❜❡❛❞s ✇❤✐❝❤ ❛r❡ ❣r❡❡♥✱ ♦r sq✉❛r❡✱ ♦r ❜♦t❤✳

△ △ △

• ❚♦t❛❧

❘❡❞ ✶ ✽ ✾

• r❡❡♥

✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❚♦ ✜♥❞ P(E F)✱ ✇❡ ♥❡❡❞ t♦ ❝♦✉♥t t❤❡ ❜❡❛❞s ✇❤✐❝❤ ❛r❡ ❣r❡❡♥✱ ♦r sq✉❛r❡✱ ♦r ❜♦t❤✳

◮ ❚❤❡ ✻✹ ❣r❡❡♥ ❜❡❛❞s✱

❡✈❡♥ t❤♦✉❣❤ t❤❡② ❛r❡ tr✐❛♥❣✉❧❛r

◮ ❚❤❡ ✽ sq✉❛r❡ ❜❡❛❞s✱ ❡✈❡♥

t❤♦✉❣❤ t❤❡② ❛r❡ r❡❞

◮ ❚❤❡ ✸✷ sq✉❛r❡ ❜❡❛❞s✱

❡✈❡♥ t❤♦✉❣❤ t❤❡② ❛r❡ ❜❧✉❡

◮ ❚❤❡ ✶✻ ❣r❡❡♥✱ ❛♥❞

sq✉❛r❡ ❜❡❛❞s

△ △ △

• ❚♦t❛❧

❘❡❞ ✶ ✽ ✾

• r❡❡♥

✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❚♦ ✜♥❞ P(E F)✱ ✇❡ ♥❡❡❞ t♦ ❝♦✉♥t t❤❡ ❜❡❛❞s ✇❤✐❝❤ ❛r❡ ❣r❡❡♥✱ ♦r sq✉❛r❡✱ ♦r ❜♦t❤✳

◮ ❚❤❡ ✻✹ ❣r❡❡♥ ❜❡❛❞s✱

❡✈❡♥ t❤♦✉❣❤ t❤❡② ❛r❡ tr✐❛♥❣✉❧❛r

◮ ❚❤❡ ✽ sq✉❛r❡ ❜❡❛❞s✱ ❡✈❡♥

t❤♦✉❣❤ t❤❡② ❛r❡ r❡❞

◮ ❚❤❡ ✸✷ sq✉❛r❡ ❜❡❛❞s✱

❡✈❡♥ t❤♦✉❣❤ t❤❡② ❛r❡ ❜❧✉❡

◮ ❚❤❡ ✶✻ ❣r❡❡♥✱ ❛♥❞

sq✉❛r❡ ❜❡❛❞s

△ △ △

• ❚♦t❛❧

❘❡❞ ✶ ✽ ✾

• r❡❡♥

✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❚♦ ✜♥❞ P(E F)✱ ✇❡ ♥❡❡❞ t♦ ❝♦✉♥t t❤❡ ❜❡❛❞s ✇❤✐❝❤ ❛r❡ ❣r❡❡♥✱ ♦r sq✉❛r❡✱ ♦r ❜♦t❤✳

◮ ❚❤❡ ✻✹ ❣r❡❡♥ ❜❡❛❞s✱

❡✈❡♥ t❤♦✉❣❤ t❤❡② ❛r❡ tr✐❛♥❣✉❧❛r

◮ ❚❤❡ ✽ sq✉❛r❡ ❜❡❛❞s✱ ❡✈❡♥

t❤♦✉❣❤ t❤❡② ❛r❡ r❡❞

◮ ❚❤❡ ✸✷ sq✉❛r❡ ❜❡❛❞s✱

❡✈❡♥ t❤♦✉❣❤ t❤❡② ❛r❡ ❜❧✉❡

◮ ❚❤❡ ✶✻ ❣r❡❡♥✱ ❛♥❞

sq✉❛r❡ ❜❡❛❞s

△ △ △

• ❚♦t❛❧

❘❡❞ ✶ ✽ ✾

• r❡❡♥

✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❚♦ ✜♥❞ P(E F)✱ ✇❡ ♥❡❡❞ t♦ ❝♦✉♥t t❤❡ ❜❡❛❞s ✇❤✐❝❤ ❛r❡ ❣r❡❡♥✱ ♦r sq✉❛r❡✱ ♦r ❜♦t❤✳

◮ ❚❤❡ ✻✹ ❣r❡❡♥ ❜❡❛❞s✱

❡✈❡♥ t❤♦✉❣❤ t❤❡② ❛r❡ tr✐❛♥❣✉❧❛r

◮ ❚❤❡ ✽ sq✉❛r❡ ❜❡❛❞s✱ ❡✈❡♥

t❤♦✉❣❤ t❤❡② ❛r❡ r❡❞

◮ ❚❤❡ ✸✷ sq✉❛r❡ ❜❡❛❞s✱

❡✈❡♥ t❤♦✉❣❤ t❤❡② ❛r❡ ❜❧✉❡

◮ ❚❤❡ ✶✻ ❣r❡❡♥ ❛♥❞

sq✉❛r❡ ❜❡❛❞s✳

◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡

△ △ △

• ❚♦t❛❧

❘❡❞ ✶ ✽ ✾

• r❡❡♥

✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❚♦ ✜♥❞ P(E F)✱ ✇❡ ♥❡❡❞ t♦ ❝♦✉♥t t❤❡ ❜❡❛❞s ✇❤✐❝❤ ❛r❡ ❣r❡❡♥✱ ♦r sq✉❛r❡✱ ♦r ❜♦t❤✳ ✻✹+✶✻+✽+✸✷ = ✶✷✵ ❜❡❛❞s ✐♥ E

• F.

❍❡♥❝❡ P(E

• F) = ✶✷✵

✶✷✸.

❄✭✷✳✹✮ ◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡ ✭❆♥❞✮

△ △ △

• ❚♦t❛❧

❘❡❞ ✶ ✽ ✾

• r❡❡♥

✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❋✐♥❞ P(E F)✳

◆✉♠❜❡rs ✐♥ ❛ ❚❛❜❧❡ ✭❆♥❞✮

△ △ △

• ❚♦t❛❧

❘❡❞ ✶ ✽ ✾

• r❡❡♥

✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❚❤❡ ❡✈❡♥t E F ✐s ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥ ❛♥❞ sq✉❛r❡✿✑ ✶✻ ❜❡❛❞s ✐♥ E

• F.

❍❡♥❝❡ P(E

• F) =

✶✻ ✶✷✸ .

❯♥✐♦♥s ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ❋♦r♠✉❧❛

❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❡✈❡♥t E ♦r ❡✈❡♥t F ♦❝❝✉rr✐♥❣ ✭♦r ❜♦t❤✮ ✐s P(E

• F) = P(E)+P(F)−P(E
• F).

❚❤✐s ❢♦r♠✉❧❛ ✐s ✉s❡❢✉❧ ❢♦r ❝❤❡❝❦✐♥❣ ②♦✉r ❛♥s✇❡rs✱ ❛♥❞ ✐❢ ②♦✉ ❞♦ ♥♦t ❦♥♦✇ ✇❤❛t ♦✉t❝♦♠❡s ❛r❡ ✐♥ t❤❡ ❡✈❡♥ts✳

❯♥✐♦♥s ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ❋♦r♠✉❧❛

△ △ △

• ❚♦t❛❧

❘❡❞ ✶ ✽ ✾

• r❡❡♥

✻✹ ✶✻ ✽✵ ❇❧✉❡ ✷ ✸✷ ✸✹ ❚♦t❛❧ ✻✼ ✺✻ ✶✷✸ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥✱✑ ❛♥❞ F ❜❡ t❤❡ ❡✈❡♥t ✏❚❤❡ ❜❡❛❞ ✐s ❛ sq✉❛r❡✳✑ ❚❤❡ ❡✈❡♥t E F ✐s ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥ ♦r sq✉❛r❡✳✑ ❚❤❡ ❡✈❡♥t E F ✐s ✏❚❤❡ ❜❡❛❞ ✐s ❣r❡❡♥ ❛♥❞ sq✉❛r❡✳✑ P(E) = ✽✵ ✶✷✸ P(F) = ✺✻ ✶✷✸ P(E

• F)

= ✶✻ ✶✷✸

❯♥✐♦♥s ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ❋♦r♠✉❧❛

P(E) = ✽✵ ✶✷✸ P(F) = ✺✻ ✶✷✸ P(E

• F)

= ✶✻ ✶✷✸ ❚❤✉s P(E

• F)

= P(E)+P(F)−P(E

• F)

= ✽✵ ✶✷✸ + ✺✻ ✶✷✸ − ✶✻ ✶✷✸ = ✽✵+✺✻−✶✻ ✶✷✸ = ✶✷✵ ✶✷✸

P(E

• F)

= P(E)+P(F)−P(E

• F)

= ✽✵ ✶✷✸ + ✺✻ ✶✷✸ − ✶✻ ✶✷✸ = ✽✵+✺✻−✶✻ ✶✷✸ = ✶✷✵ ✶✷✸ ■❢ ✇❡ ❥✉st ❛❞❞ t❤❡ ♥✉♠❜❡r ♦❢ ❣r❡❡♥ ❜❡❛❞s ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ sq✉❛r❡ ❜❡❛❞s✱ ✽✵+✺✻ = ✶✸✻, ✇❡ ❣❡t ❛♥ ❛♥s✇❡r t❤❛t ✐s t♦♦ ❜✐❣✳ ■t ❞♦✉❜❧❡ ❝♦✉♥ts t❤❡ ✶✻ ❜❡❛❞s t❤❛t ❛r❡ ❣r❡❡♥ ❛♥❞ sq✉❛r❡✱ s♦ t❤❛t ✐s ✇❤② t❤❡ ❢♦r♠✉❧❛ s✉❜tr❛❝ts t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥✳ ■♥ t❤❡ ♥✉♠❡r❛t♦r✱ ✐t s✉❜tr❛❝ts t❤❡ ♥✉♠❜❡r ♦❢ ❜❡❛❞s t❤❛t ❛r❡ ❣r❡❡♥ ❛♥❞ sq✉❛r❡✿ ✶✸✻−✶✻ = ✶✷✵, t❤❡ ❝♦rr❡❝t ♥✉♠❡r❛t♦r ❛♥❞ t❤❡ ❛❝t✉❛❧ ♥✉♠❜❡r ♦❢ ❜❡❛❞s t❤❛t ❛r❡ ❣r❡❡♥ ❛♥❞ sq✉❛r❡✳

❊♥❞

❨♦✉ ❤❛✈❡ ❤♦♠❡✇♦r❦ ❞✉❡ t♦♥✐❣❤t✳