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■♥tr♦ t♦ ❈♦♥t❡♠♣♦r❛r② ▼❛t❤

❯♥✐♦♥s ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ♦❢ ■♥t❡r✈❛❧s

❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❯❑

❆♥♥♦✉♥❝❡♠❡♥t

◮ ❨♦✉ ❤❛✈❡ ❛ ❤♦♠❡✇♦r❦ ❛ss✐❣♥♠❡♥t ❞✉❡ ♥❡①t ▼♦♥❞❛②✳ ◮ ▼✐♥✐✲❡①❛♠ ✷ ✐s ♥❡①t ❲❡❞♥❡s❞❛②✳

❯♥✐♦♥s ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ♦❢ ■♥t❡r✈❛❧s

■♥t❡r✈❛❧ ❡✈❡♥ts ♦♥ t❤❡ r❡❛❧ ❧✐♥❡ ❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ t♦ ❢♦r♠ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ❡✈❡♥ts ✉s✐♥❣ ✉♥✐♦♥s ❛♥❞ ✐♥t❡rs❡❝t✐♦♥s t♦ r❡♣r❡s❡♥t ♦✉t❝♦♠❡s ✐♥ ♦♥❡ ❡✈❡♥t ♦r t❤❡ ♦t❤❡r ✭✉♥✐♦♥s✮ ♦r ♦✉t❝♦♠❡s ✐♥ t❤❡ ♦✈❡r❧❛♣ ♦❢ ❜♦t❤ ❡✈❡♥ts ✭✐♥t❡rs❡❝t✐♦♥s✮✳

❯♥✐♦♥s ❛♥❞ ■♥t❡rs❡❝t✐♦♥s ♦❢ ■♥t❡r✈❛❧s

▲❡t Ω ❜❡ ❛♥ ✐♥t❡r✈❛❧ ♦❢ r❡❛❧ ♥✉♠❜❡rs✱ ❛♥❞ ❧❡t E ❛♥❞ F ❜❡ ❡✈❡♥t ✐♥t❡r✈❛❧s ✐♥ Ω✳

◮ ❆ r❡❛❧ ♥✉♠❜❡r ♦✉t❝♦♠❡ ✐s ✐♥ E ✐❢ ✐t ✐s ❜❡t✇❡❡♥ t❤❡

❡♥❞♣♦✐♥ts ♦❢ E✳

◮ ❚❤❡ ❡✈❡♥t E F ✐s t❤❡ ✉♥✐♦♥ ♦❢ E ❛♥❞ F✳ ■t ✐s t❤❡ s❡t ♦❢

r❡❛❧ ♥✉♠❜❡r ♦✉t❝♦♠❡s ✐♥ E ♦r ✐♥ F ✭♦r ✐♥ ❜♦t❤✮✳

◮ ❚❤❡ ✉♥✐♦♥ ❝♦✉❧❞ ❜❡ ❛ ❧❛r❣❡r ✐♥t❡r✈❛❧✱ ♦r t✇♦ s❡♣❛r❛t❡

✐♥t❡r✈❛❧s✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r E ❛♥❞ F ♦✈❡r❧❛♣✳

◮ ❚❤❡ ❡✈❡♥t E F ✐s t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ E ❛♥❞ F✳ ■t ✐s

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

◮ ■❢ E ❛♥❞ F ♦✈❡r❧❛♣✱ t❤❡♥ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ✐s t❤❡ ✐♥t❡r✈❛❧

❢♦r♠❡❞ ❜② t❤❡ ♦✈❡r❧❛♣✳

❯♥✐♦♥ ♦❢ ❚✇♦ ❖✈❡r❧❛♣♣✐♥❣ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✺] ❛♥❞ F = [✶✸,✶✻]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ♦r ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿

◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✷ ✐s ✐♥ E F✱ ❜❡❝❛✉s❡ ✶✷ ✐s ✐♥ E

✭❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✮✳

◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✺.✺ ✐s ✐♥ E F✱ ❜❡❝❛✉s❡ ✶✺.✺ ✐s ✐♥ F

✭❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✮✳

◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✹ ✐s ✐♥ E F✱ ❜❡❝❛✉s❡ ✶✹ ✐s ✐♥ E ❛♥❞ ✐♥

F ✭❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠✮✳

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

E ♥♦r ✐♥ F✳

❯♥✐♦♥ ♦❢ ❚✇♦ ❖✈❡r❧❛♣♣✐♥❣ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✺] ❛♥❞ F = [✶✸,✶✻]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ♦r ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿

❯♥✐♦♥ ♦❢ ❚✇♦ ❖✈❡r❧❛♣♣✐♥❣ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✺] ❛♥❞ F = [✶✸,✶✻]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ♦r ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿

❯♥✐♦♥ ♦❢ ❚✇♦ ❖✈❡r❧❛♣♣✐♥❣ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✺] ❛♥❞ F = [✶✸,✶✻]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ♦r ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿ ❚❤✉s✱ E F ✐s ❛❝t✉❛❧❧② t❤❡ ✐♥t❡r✈❛❧ [✶✶,✶✻]✳ ■ts ❧❡♥❣t❤ ✐s ✶✻−✶✶ = ✺. ❚❤✉s✱ P(E

• F) = Length of E F

Length of Ω = ✶✻−✶✶ ✶✼−✶✵ = ✺ ✼

❯♥✐♦♥ ♦❢ ❚✇♦ ❙❡♣❛r❛t❡ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✸] ❛♥❞ F = [✶✹,✶✼]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✸✱ ♦r ❜❡t✇❡❡♥ ✶✹ ❛♥❞ ✶✼✿

◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✷ ✐s ✐♥ E F✱ ❜❡❝❛✉s❡ ✶✷ ✐s ✐♥ E✳ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✺.✺ ✐s ✐♥ E F✱ ❜❡❝❛✉s❡ ✶✺.✺ ✐s ✐♥ F✳ ◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✸.✺ ✐s ♥♦t ✐♥ E F✱ ❜❡❝❛✉s❡ ✶✸.✺ ✐s ♥♦t

✐♥ E ♥♦r ✐♥ F✳

❯♥✐♦♥ ♦❢ ❚✇♦ ❙❡♣❛r❛t❡ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✸] ❛♥❞ F = [✶✹,✶✼]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✸✱ ♦r ❜❡t✇❡❡♥ ✶✹ ❛♥❞ ✶✼✿

❯♥✐♦♥ ♦❢ ❚✇♦ ❙❡♣❛r❛t❡ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✸] ❛♥❞ F = [✶✹,✶✼]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✸✱ ♦r ❜❡t✇❡❡♥ ✶✹ ❛♥❞ ✶✼✿

❯♥✐♦♥ ♦❢ ❚✇♦ ❙❡♣❛r❛t❡ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✸] ❛♥❞ F = [✶✹,✶✼]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✸✱ ♦r ❜❡t✇❡❡♥ ✶✹ ❛♥❞ ✶✼✿ ❚❤✉s✱ E F ✐s ❛❝t✉❛❧❧② t✇♦ ✐♥t❡r✈❛❧s✳ ■t ❤❛s ❛ t♦t❛❧ ❧❡♥❣t❤ ❢♦✉♥❞ ❜② ❛❞❞✐♥❣ ✭❝♦♠❜✐♥✐♥❣✮ t❤❡ ❧❡♥❣t❤s ♦❢ E ❛♥❞ F✿ ▲❡♥❣t❤ ♦❢ E : ✸−✶ = ✷ ▲❡♥❣t❤ ♦❢ F : ✼−✹ = ✸ ❚♦t❛❧ ❧❡♥❣t❤ ♦❢ E

• F : ✷+✸ = ✺.

❍❡♥❝❡ P(E

• F) = Total length of E F

Length of Ω = ✺ ✼ .

■♥t❡rs❡❝t✐♦♥ ♦❢ ❚✇♦ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✺] ❛♥❞ F = [✶✸,✶✻]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ❛♥❞ ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿

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

❜✉t ♥♦t ✐♥ F✳

◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✺.✺ ✐s ♥♦t ✐♥ E F✱ ❜❡❝❛✉s❡ ✶✺.✺ ✐s ✐♥

F✱ ❜✉t ♥♦t ✐♥ E✳

◮ ❚❤❡ r❡❛❧ ♥✉♠❜❡r ✶✹ ✐s ✐♥ E F✱ ❜❡❝❛✉s❡ ✶✹ ✐s ✐♥ E ❛♥❞ ✐♥

F ✭❜♦t❤ ♦❢ t❤❡♠✮✳

■♥t❡rs❡❝t✐♦♥ ♦❢ ❚✇♦ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✺] ❛♥❞ F = [✶✸,✶✻]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ❛♥❞ ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿

■♥t❡rs❡❝t✐♦♥ ♦❢ ❚✇♦ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✺] ❛♥❞ F = [✶✸,✶✻]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ❛♥❞ ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿

■♥t❡rs❡❝t✐♦♥ ♦❢ ❚✇♦ ■♥t❡r✈❛❧s

▲❡t Ω = [✶✵,✶✼], E = [✶✶,✶✺] ❛♥❞ F = [✶✸,✶✻]✳ ❚❤❡ ❡✈❡♥t E F ❝♦♥s✐sts ♦❢ r❡❛❧ ♥✉♠❜❡rs ❜❡t✇❡❡♥ ✶✶ ❛♥❞ ✶✺✱ ❛♥❞ ❜❡t✇❡❡♥ ✶✸ ❛♥❞ ✶✻✿ ❚❤✉s✱ E F ✐s ❛❝t✉❛❧❧② t❤❡ ✐♥t❡r✈❛❧ [✶✸,✶✺]✳ ■ts ❧❡♥❣t❤ ✐s ✺−✸ = ✷. ❍❡♥❝❡ P(E

• F) = Length of E F

Length of Ω = ✷ ✼ .

❄✭✹✳✶✮ ❯♥✐♦♥✴■♥t❡rs❡❝t✐♦♥ Pr❛❝t✐❝❡ ✶

▲❡t Ω ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✽,✷✹]✱ E ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✶✷,✶✻]✱ ❛♥❞ F ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✶✺,✶✾]✿ ■❢ ✇❡ ♣✐❝❦ ❛ r❛♥❞♦♠ r❡❛❧ ♥✉♠❜❡r ❜❡t✇❡❡♥ ✽ ❛♥❞ ✷✹✱ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❡✈❡♥t E F✳ ❍✐♥ts✿ ✶✳ ■❞❡♥t✐❢② t❤❡ ❡✈❡♥t [✶✷,✶✻][✶✺,✶✾] ❛s ❛♥ ✐♥t❡r✈❛❧✳ ✷✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✉♥✐♦♥❄ ✸✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡❄ ✹✳ ❋✐♥❞ P([✶✷,✶✻][✶✺,✶✾])✳ ❚②♣❡ ❛♥❞ s❡♥❞ ❛ ❢r❛❝t✐♦♥✳

❯♥✐♦♥✴■♥t❡rs❡❝t✐♦♥ Pr❛❝t✐❝❡ ✶

▲❡t Ω ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✽,✷✹]✱ E ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✶✷,✶✻]✱ ❛♥❞ F ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✶✺,✶✾]✿ ■❢ ✇❡ ♣✐❝❦ ❛ r❛♥❞♦♠ r❡❛❧ ♥✉♠❜❡r ❜❡t✇❡❡♥ ✽ ❛♥❞ ✷✹✱ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❡✈❡♥t E F✳ ❍✐♥ts✿ ✶✳ ■❞❡♥t✐❢② t❤❡ ❡✈❡♥t [✶✷,✶✻][✶✺,✶✾] ❛s ❛♥ ✐♥t❡r✈❛❧✳ ✷✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✉♥✐♦♥❄ ✸✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡❄ ✹✳ ❋✐♥❞ P([✶✷,✶✻][✶✺,✶✾])✳ ❚②♣❡ ❛♥❞ s❡♥❞ ❛ ❢r❛❝t✐♦♥✳

❯♥✐♦♥✴■♥t❡rs❡❝t✐♦♥ Pr❛❝t✐❝❡ ✶

▲❡t Ω ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✽,✷✹]✱ E ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✶✷,✶✻]✱ ❛♥❞ F ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✶✺,✶✾]✿ ■❢ ✇❡ ♣✐❝❦ ❛ r❛♥❞♦♠ r❡❛❧ ♥✉♠❜❡r ❜❡t✇❡❡♥ ✽ ❛♥❞ ✷✹✱ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❡✈❡♥t E F✳ E

• F = [✶✷,✶✾].

P(E

• F) = ▲❡♥❣t❤ ♦❢ E F

▲❡♥❣t❤ ♦❢ Ω = ✶✾−✶✷ ✷✹−✽ = ✼ ✶✻

❄✭✹✳✷✮ ❯♥✐♦♥✴■♥t❡rs❡❝t✐♦♥ Pr❛❝t✐❝❡ ✷

▲❡t Ω ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✷✹,✹✼]✱ E ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✷✾,✸✹]✱ ❛♥❞ F ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✸✻,✹✸]✿ ■❢ ✇❡ ♣✐❝❦ ❛ r❛♥❞♦♠ r❡❛❧ ♥✉♠❜❡r ❜❡t✇❡❡♥ ✷✹ ❛♥❞ ✹✼✱ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❡✈❡♥t E F✳ ❍✐♥ts✿ ✶✳ ■❞❡♥t✐❢② t❤❡ ❡✈❡♥t [✷✾,✸✹][✸✻,✹✸] ❛s ❛ ✉♥✐♦♥ ♦❢ t✇♦ s❡♣❛r❛t❡ ✐♥t❡r✈❛❧s✳ ✷✳ ❲❤❛t ✐s t❤❡ t♦t❛❧ ❧❡♥❣t❤ ♦❢ t❤❡ ✉♥✐♦♥❄ ✸✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡❄ ✹✳ ❋✐♥❞ P([✷✾,✸✹][✸✻,✹✸])✳ ❚②♣❡ ❛♥❞ s❡♥❞ ❛ ❢r❛❝t✐♦♥✳

❯♥✐♦♥✴■♥t❡rs❡❝t✐♦♥ Pr❛❝t✐❝❡ ✷

▲❡t Ω ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✷✹,✹✼]✱ E ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✷✾,✸✹]✱ ❛♥❞ F ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✸✻,✹✸]✿ ■❢ ✇❡ ♣✐❝❦ ❛ r❛♥❞♦♠ r❡❛❧ ♥✉♠❜❡r ❜❡t✇❡❡♥ ✷✹ ❛♥❞ ✹✼✱ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❡✈❡♥t E F✳ ❍✐♥ts✿ ✶✳ ■❞❡♥t✐❢② t❤❡ ❡✈❡♥t [✷✾,✸✹][✸✻,✹✸] ❛s ❛ ✉♥✐♦♥ ♦❢ t✇♦ s❡♣❛r❛t❡ ✐♥t❡r✈❛❧s✳ ✷✳ ❲❤❛t ✐s t❤❡ t♦t❛❧ ❧❡♥❣t❤ ♦❢ t❤❡ ✉♥✐♦♥❄ ✸✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡❄ ✹✳ ❋✐♥❞ P([✷✾,✸✹][✸✻,✹✸])✳ ❚②♣❡ ❛♥❞ s❡♥❞ ❛ ❢r❛❝t✐♦♥✳

❯♥✐♦♥✴■♥t❡rs❡❝t✐♦♥ Pr❛❝t✐❝❡ ✷

▲❡t Ω ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✷✹,✹✼]✱ E ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✷✾,✸✹]✱ ❛♥❞ F ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✸✻,✹✸]✿ ■❢ ✇❡ ♣✐❝❦ ❛ r❛♥❞♦♠ r❡❛❧ ♥✉♠❜❡r ❜❡t✇❡❡♥ ✷✹ ❛♥❞ ✹✼✱ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❡✈❡♥t E F✳ E

• F = [✷✾,✸✹]
• [✸✻,✹✸]

P(E

• F)

= ❚♦t❛❧ ▲❡♥❣t❤ ♦❢ E F ▲❡♥❣t❤ ♦❢ Ω = (✸✹−✷✾)+(✹✸−✸✻) ✹✼−✷✹ = ✺+✼ ✷✸ = ✶✷ ✷✸

❄✭✹✳✸✮ ❯♥✐♦♥✴■♥t❡rs❡❝t✐♦♥ Pr❛❝t✐❝❡ ✸

▲❡t Ω ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✹✼,✼✽]✱ E ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✺✶,✻✵]✱ ❛♥❞ F ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✺✹,✻✸]✿ ■❢ ✇❡ ♣✐❝❦ ❛ r❛♥❞♦♠ r❡❛❧ ♥✉♠❜❡r ❜❡t✇❡❡♥ ✹✼ ❛♥❞ ✼✽✱ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❡✈❡♥t E F✳ ❍✐♥ts✿ ✶✳ ■❞❡♥t✐❢② t❤❡ ❡✈❡♥t [✺✶,✻✵][✺✹,✻✸] ❛s ❛ s♠❛❧❧❡r ✐♥t❡r✈❛❧✳ ✷✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥❄ ✸✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡❄ ✹✳ ❋✐♥❞ P([✺✶,✻✵][✺✹,✻✸])✳ ❚②♣❡ ❛♥❞ s❡♥❞ ❛ ❢r❛❝t✐♦♥✳

❯♥✐♦♥✴■♥t❡rs❡❝t✐♦♥ Pr❛❝t✐❝❡ ✸

▲❡t Ω ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✹✼,✼✽]✱ E ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✺✶,✻✵]✱ ❛♥❞ F ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✺✹,✻✸]✿ ■❢ ✇❡ ♣✐❝❦ ❛ r❛♥❞♦♠ r❡❛❧ ♥✉♠❜❡r ❜❡t✇❡❡♥ ✹✼ ❛♥❞ ✼✽✱ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❡✈❡♥t E F✳ ❍✐♥ts✿ ✶✳ ■❞❡♥t✐❢② t❤❡ ❡✈❡♥t [✺✶,✻✵][✺✹,✻✸] ❛s ❛ s♠❛❧❧❡r ✐♥t❡r✈❛❧✳ ✷✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥❄ ✸✳ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ s❛♠♣❧❡ s♣❛❝❡❄ ✹✳ ❋✐♥❞ P([✺✶,✻✵][✺✹,✻✸])✳ ❚②♣❡ ❛♥❞ s❡♥❞ ❛ ❢r❛❝t✐♦♥✳

❯♥✐♦♥✴■♥t❡rs❡❝t✐♦♥ Pr❛❝t✐❝❡ ✸

▲❡t Ω ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✹✼,✼✽]✱ E ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✺✶,✻✵]✱ ❛♥❞ F ❜❡ t❤❡ ✐♥t❡r✈❛❧ [✺✹,✻✸]✿ ■❢ ✇❡ ♣✐❝❦ ❛ r❛♥❞♦♠ r❡❛❧ ♥✉♠❜❡r ❜❡t✇❡❡♥ ✹✼ ❛♥❞ ✼✽✱ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ ❡✈❡♥t E F✳ E

• F = [✺✹,✻✵].

P(E

• F) = ▲❡♥❣t❤ ♦❢ E F

▲❡♥❣t❤ ♦❢ Ω = ✻✵−✺✹ ✼✽−✹✼ = ✻ ✸✶

❊♥❞

◮ ❨♦✉ ❤❛✈❡ ❛ ❤♦♠❡✇♦r❦ ❛ss✐❣♥♠❡♥t ❞✉❡ ♥❡①t ▼♦♥❞❛②✳ ◮ ▼✐♥✐✲❡①❛♠ ✷ ✐s ♥❡①t ❲❡❞♥❡s❞❛②✳