SLIDE 1
■♥tr♦ t♦ ❈♦♥t❡♠♣♦r❛r② ▼❛t❤
❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t② ❛♥❞ ■♥❞❡♣❡♥❞❡♥t ❊✈❡♥ts
❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❯❑
tr t trr t t - - PowerPoint PPT Presentation
tr t trr t t - - PowerPoint PPT Presentation
tr t trr t t Prt t ts rtt tts
SLIDE 2
SLIDE 3
❉r❛✇✐♥❣ ❛ ❈❛r❞
◮ ❨♦✉ ❤❛✈❡ ❛ ❤♦♠❡✇♦r❦ ❛ss✐❣♥♠❡♥t ❞✉❡ ♥❡①t ▼♦♥❞❛②✳ ❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛r❞s ❤❛s ✷ s✉✐ts ✭❆✲❇✮ ❛♥❞ ✺ r❛♥❦s ✭✶✲✺✮✳ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❇✹ ❇✺
SLIDE 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❛✇♥✳✑
SLIDE 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❛❧ = ✺ ✶✵ = ✶ ✷.
SLIDE 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❛❧ = ✷ ✶✵ = ✶ ✺.
SLIDE 7
❉r❛✇✐♥❣ ❛ ❙♣❡❝✐✜❝ ❈❛r❞
❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛r❞s ❤❛s ✷ s✉✐ts ✭❆✲❇✮ ❛♥❞ ✺ r❛♥❦s ✭✶✲✺✮✳ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❇✹ ❇✺ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞r❛✇✐♥❣ ❝❛r❞ ❇✷ ✭s✉✐t ❇ ❛♥❞ r❛♥❦ ✷✮ ✐s✿ P(E
- F) = ✶ s✉❝❤ ❝❛r❞ ❇✷
✶✵ ❝❛r❞s t♦t❛❧ = ✶ ✶✵.
SLIDE 8
❖❜s❡r✈❛t✐♦♥
◆♦t✐❝❡ t❤❛t P(E) = ✶ ✷, P(F) = ✶ ✺, P(E
- F) = ✶
✶✵. ❈♦✐♥❝✐❞❡♥t❛❧❧②✱ P(E)·P(F) = ✶ ✷ · ✶ ✺ = ✶ ✶✵ ❛s ✇❡❧❧✳ ❲❤❡♥ ❝❛♥ ✇❡ ♠✉❧t✐♣❧② t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ t✇♦ ❡✈❡♥ts t♦ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥❄
SLIDE 9
❖❜s❡r✈❛t✐♦♥
◆♦t✐❝❡ t❤❛t P(E) = ✶ ✷, P(F) = ✶ ✺, P(E
- F) = ✶
✶✵. ❈♦✐♥❝✐❞❡♥t❛❧❧②✱ P(E)·P(F) = ✶ ✷ · ✶ ✺ = ✶ ✶✵ ❛s ✇❡❧❧✳ ❲❤❡♥ ❝❛♥ ✇❡ ♠✉❧t✐♣❧② t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ t✇♦ ❡✈❡♥ts t♦ ✜♥❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥❄
SLIDE 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)✳
SLIDE 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 ❇✳
SLIDE 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 ❇✳
SLIDE 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 ❇ = ✶ ✺.
SLIDE 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 ❇ = ✶ ✺.
SLIDE 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✳
SLIDE 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❛♥❦ ✷✳
SLIDE 17
❉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❛♥❦ ✷✳
SLIDE 18
❉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❛♥❦ ✷✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s s✉✐t ❇ ❣✐✈❡♥ t❤❛t ✐t ❤❛s r❛♥❦ ✷ ✐s P(E|F) = ✶ ❝❛r❞ ✇✐t❤ r❛♥❦ ✷ ❛♥❞ s✉✐t ❇ ✷ ❝❛r❞s ✇✐t❤ r❛♥❦ ✷ = ✶ ✷.
SLIDE 19
❖❜s❡r✈❛t✐♦♥s
◮ ◆♦t✐❝❡ t❤❛t P(E) = ✶ ✷, ❛♥❞ P(E|F) = ✶ ✷ ❛s ✇❡❧❧✳ ❑♥♦✇✐♥❣ t❤❡ ❝❛r❞✬s r❛♥❦ ❞✐❞ ♥♦t ❛✛❡❝t ✐ts ❝❤❛♥❝❡s ♦❢ ❤❛✈✐♥❣ s✉✐t ❇✳ ◮ ❙✐♠✐❧❛r❧②✱ P(F) = ✶ ✺, ❛♥❞ P(F|E) = ✶ ✺ ❛s ✇❡❧❧✳ ❑♥♦✇✐♥❣ t❤❡ ❝❛r❞✬s s✉✐t ❞✐❞ ♥♦t ❛✛❡❝t ✐ts ❝❤❛♥❝❡s ♦❢ ❤❛✈✐♥❣ r❛♥❦ ✷✳
SLIDE 20
❖❜s❡r✈❛t✐♦♥s
◮ ◆♦t✐❝❡ t❤❛t P(E) = ✶ ✷, ❛♥❞ P(E|F) = ✶ ✷ ❛s ✇❡❧❧✳ ❑♥♦✇✐♥❣ t❤❡ ❝❛r❞✬s r❛♥❦ ❞✐❞ ♥♦t ❛✛❡❝t ✐ts ❝❤❛♥❝❡s ♦❢ ❤❛✈✐♥❣ s✉✐t ❇✳ ◮ ❙✐♠✐❧❛r❧②✱ P(F) = ✶ ✺, ❛♥❞ P(F|E) = ✶ ✺ ❛s ✇❡❧❧✳ ❑♥♦✇✐♥❣ t❤❡ ❝❛r❞✬s s✉✐t ❞✐❞ ♥♦t ❛✛❡❝t ✐ts ❝❤❛♥❝❡s ♦❢ ❤❛✈✐♥❣ r❛♥❦ ✷✳
SLIDE 21
❲❛r♥✐♥❣✿ ❖r❞❡r ▼❛tt❡rs
❇❡✇❛r❡✿ P(E|F) ❛♥❞ P(F|E) ❛r❡ ♥♦t ❡q✉❛❧ ✐♥ ❣❡♥❡r❛❧✦ ◮ ■♥ t❤✐s ❡①❛♠♣❧❡✱ P(E|F) = ✶ ✷ ✇❤✐❧❡ P(F|E) = ✶ ✺. ◮ ❋♦r P(E|F)✱ t❤❡ ❣✐✈❡♥ ❡✈❡♥t r❡str✐❝t❡❞ ✉s t♦ t❤❡ ❝❛r❞s ✇✐t❤ r❛♥❦ ✷✳ ◮ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r P(F|E)✱ t❤❡ ❣✐✈❡♥ ❡✈❡♥t r❡str✐❝t❡❞ ✉s t♦ t❤❡ ❝❛r❞s ✇✐t❤ s✉✐t ❇✳
SLIDE 22
❲❛r♥✐♥❣✿ ❖r❞❡r ▼❛tt❡rs
❇❡✇❛r❡✿ P(E|F) ❛♥❞ P(F|E) ❛r❡ ♥♦t ❡q✉❛❧ ✐♥ ❣❡♥❡r❛❧✦ ◮ ■♥ t❤✐s ❡①❛♠♣❧❡✱ P(E|F) = ✶ ✷ ✇❤✐❧❡ P(F|E) = ✶ ✺. ◮ ❋♦r P(E|F)✱ t❤❡ ❣✐✈❡♥ ❡✈❡♥t r❡str✐❝t❡❞ ✉s t♦ t❤❡ ❝❛r❞s ✇✐t❤ r❛♥❦ ✷✳ ◮ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r P(F|E)✱ t❤❡ ❣✐✈❡♥ ❡✈❡♥t r❡str✐❝t❡❞ ✉s t♦ t❤❡ ❝❛r❞s ✇✐t❤ s✉✐t ❇✳
SLIDE 23
❲❛r♥✐♥❣✿ ❖r❞❡r ▼❛tt❡rs
❇❡✇❛r❡✿ P(E|F) ❛♥❞ P(F|E) ❛r❡ ♥♦t ❡q✉❛❧ ✐♥ ❣❡♥❡r❛❧✦ ◮ ■♥ t❤✐s ❡①❛♠♣❧❡✱ P(E|F) = ✶ ✷ ✇❤✐❧❡ P(F|E) = ✶ ✺. ◮ ❋♦r P(E|F)✱ t❤❡ ❣✐✈❡♥ ❡✈❡♥t r❡str✐❝t❡❞ ✉s t♦ t❤❡ ❝❛r❞s ✇✐t❤ r❛♥❦ ✷✳ ◮ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❢♦r P(F|E)✱ t❤❡ ❣✐✈❡♥ ❡✈❡♥t r❡str✐❝t❡❞ ✉s t♦ t❤❡ ❝❛r❞s ✇✐t❤ s✉✐t ❇✳
SLIDE 24
■♥❞❡♣❡♥❞❡♥t ❊✈❡♥ts
▲❡t E ❛♥❞ F ❜❡ ❡✈❡♥ts ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡ Ω✳ ❚❤❡♥ E ❛♥❞ F ❛r❡ s❛✐❞ t♦ ❜❡ ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts ✐❢ P(F|E) = P(F).
SLIDE 25
■♥❞❡♣❡♥❞❡♥t ❊✈❡♥ts
▲❡t E ❛♥❞ F ❜❡ ❡✈❡♥ts ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡ Ω✳ ❚❤❡♥ E ❛♥❞ F ❛r❡ s❛✐❞ t♦ ❜❡ ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts ✐❢ P(F|E) = P(F). ❚❤❛t ✐s✱ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t② ♦❢ F ❣✐✈❡♥ t❤❛t E ♦❝❝✉rr❡❞ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ✭r❡❣✉❧❛r✮ ♣r♦❜❛❜✐❧✐t② ♦❢ F✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ F ♦❝❝✉rr✐♥❣ ✐s t❤❡ s❛♠❡ ✇❤❡t❤❡r ♦r ♥♦t E ♦❝❝✉rs✳
SLIDE 26
❈❤❡❝❦✐♥❣ ■♥❞❡♣❡♥❞❡♥❝❡
▲❡t E ❛♥❞ F ❜❡ ❡✈❡♥ts ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡ Ω✳ ❚♦ ❝❤❡❝❦ ✐❢ E ❛♥❞ F ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts✱ ✶✳ ❈♦♠♣✉t❡ P(E)✱ P(F)✱ ❛♥❞ P(F|E)✳ ✷✳ ❈♦♠♣❛r❡ P(F|E) ❛♥❞ P(F)✳ ❆r❡ t❤❡② ❡q✉❛❧ ♦r ♥♦t❄ ◮ ■❢ P(F|E) ❛♥❞ P(F) ❛r❡ ❡q✉❛❧✱ t❤❡♥ E ❛♥❞ F ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts✳ ◮ ■❢ P(F|E) ❛♥❞ P(F) ❛r❡ ❞✐✛❡r❡♥t ♥✉♠❜❡rs✱ t❤❡♥ E ❛♥❞ F ❛r❡ ♥♦t ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts✳
SLIDE 27
❈❤❡❝❦✐♥❣ ■♥❞❡♣❡♥❞❡♥❝❡
▲❡t E ❛♥❞ F ❜❡ ❡✈❡♥ts ✐♥ ❛ s❛♠♣❧❡ s♣❛❝❡ Ω✳ ❚♦ ❝❤❡❝❦ ✐❢ E ❛♥❞ F ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts✱ ✶✳ ❈♦♠♣✉t❡ P(E)✱ P(F)✱ ❛♥❞ P(F|E)✳ ✷✳ ❈♦♠♣❛r❡ P(F|E) ❛♥❞ P(F)✳ ❆r❡ t❤❡② ❡q✉❛❧ ♦r ♥♦t❄ ◮ ■❢ P(F|E) ❛♥❞ P(F) ❛r❡ ❡q✉❛❧✱ t❤❡♥ E ❛♥❞ F ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts✳ ◮ ■❢ P(F|E) ❛♥❞ P(F) ❛r❡ ❞✐✛❡r❡♥t ♥✉♠❜❡rs✱ t❤❡♥ E ❛♥❞ F ❛r❡ ♥♦t ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts✳
SLIDE 28
■♥❞❡♣❡♥❞❡♥❝❡ ✇✐t❤ ❛ ❋✉❧❧ ❉❡❝❦
❆ ❞❡❝❦ ♦❢ ✶✵ ❝❛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❛✇♥✳✑ ❲❡ s❛✇ t❤❛t P(F|E) = ✶ ✺, ❛♥❞ P(F) = ✶ ✺ ✭❜♦t❤ ❡q✉❛❧ ✵✳✷✮✳ ❍❡♥❝❡ E ❛♥❞ F ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts✳
SLIDE 29
❉r❛✇✐♥❣ ❛ ❈❛r❞ ✷
❆ ❞❡❝❦ ♦❢ ✽ ❝❛r❞s ❤❛s ✷ s✉✐ts ❛♥❞ ✺ r❛♥❦s✱ ❜✉t ❝❛r❞s ❇✹ ❛♥❞ ❇✺ ❛r❡ ♥♦t ✐♥❝❧✉❞❡❞✿ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸
SLIDE 30
❄✭✻✳✶✮✿ ❉r❛✇✐♥❣ ❛ ❈❛r❞ ✷
❆ ❞❡❝❦ ♦❢ ✽ ❝❛r❞s ❤❛s ✷ s✉✐ts ❛♥❞ ✺ r❛♥❦s✱ ❜✉t ❝❛r❞s ❇✹ ❛♥❞ ❇✺ ❛r❡ ♥♦t ✐♥❝❧✉❞❡❞✿ ❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ◮ ▲❡t E ❜❡ t❤❡ ❡✈❡♥t ✏❆ ❝❛r❞ ✇✐t❤ s✉✐t ❇ ✐s ❞r❛✇♥✳✑ ❋✐♥❞ P(E)✳ A) ✶/✷ B) ✸/✶✵ C) ✸/✺ D) ✸/✽ E) ✶/✸ ◮ ▲❡t F ❜❡ t❤❡ ❡✈❡♥t ✏❆ ❝❛r❞ ✇✐t❤ r❛♥❦ ✷ ✐s ❞r❛✇♥✳✑ ❋✐♥❞ P(F)✳ A) ✶/✺ B) ✶/✽ C) ✷/✺ D) ✶/✷ E) ✷/✽ ◮ ❚❤❡♥ E F ✐s t❤❡ ❡✈❡♥t ✏❈❛r❞ ❇✷ ✐s ❞r❛✇♥✳✑ ❋✐♥❞ P(E F)✳ A) ✶/✷ B) ✶/✶✵ C) ✶/✽ D) ✶/✺ E) ✻/✻✹
SLIDE 31
❉r❛✇✐♥❣ ❛ ❈❛r❞ ✇✐t❤ ❛ ❉❡s✐r❡❞ ❙✉✐t ✷
❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞r❛✇✐♥❣ ❛ ❝❛r❞ ✇✐t❤ s✉✐t ❇ ✐s✿ P(E) = ✸ s✉❝❤ ❝❛r❞s ✽ ❝❛r❞s t♦t❛❧ = ✸ ✽.
SLIDE 32
❉r❛✇✐♥❣ ❛ ❈❛r❞ ✇✐t❤ ❛ ❉❡s✐r❡❞ ❘❛♥❦ ✷
❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞r❛✇✐♥❣ ❛ ❝❛r❞ ✇✐t❤ r❛♥❦ ✷ ✐s✿ P(F) = ✷ s✉❝❤ ❝❛r❞s ✽ ❝❛r❞s t♦t❛❧ = ✷ ✽ = ✶ ✹.
SLIDE 33
❉r❛✇✐♥❣ ❛ ❙♣❡❝✐✜❝ ❈❛r❞ ✷
❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❞r❛✇✐♥❣ ❝❛r❞ ❇✷ ✭s✉✐t ❇ ❛♥❞ r❛♥❦ ✷✮ ✐s✿ P(E
- F) = ✶ s✉❝❤ ❝❛r❞ ❇✷
✽ ❝❛r❞s t♦t❛❧ = ✶ ✽.
SLIDE 34
❉✐✛❡r❡♥t ❖❜s❡r✈❛t✐♦♥
◆♦t✐❝❡ t❤❛t P(E) = ✸ ✽, P(F) = ✶ ✹, P(E
- F) = ✶
✽. ❚❤✐s t✐♠❡✱ P(E)·P(F) = ✸ ✽ · ✶ ✹ = ✸ ✸✷ = ✶ ✽. ▼✉❧t✐♣❧②✐♥❣ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ t✇♦ ❡✈❡♥ts ❞✐❞ ♥♦t ❡q✉❛❧ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥✳
SLIDE 35
❉✐✛❡r❡♥t ❖❜s❡r✈❛t✐♦♥
◆♦t✐❝❡ t❤❛t P(E) = ✸ ✽, P(F) = ✶ ✹, P(E
- F) = ✶
✽. ❚❤✐s t✐♠❡✱ P(E)·P(F) = ✸ ✽ · ✶ ✹ = ✸ ✸✷ = ✶ ✽. ▼✉❧t✐♣❧②✐♥❣ t❤❡ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ t✇♦ ❡✈❡♥ts ❞✐❞ ♥♦t ❡q✉❛❧ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡✐r ✐♥t❡rs❡❝t✐♦♥✳
SLIDE 36
❄✭✻✳✷✮✿ ❉r❛✇✐♥❣ ❛ ❈❛r❞✿ ❈♦♥❞✐t✐♦♥❛❧ Pr♦❜❛❜✐❧✐t②
❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ◆♦✇ ❝♦♠♣✉t❡✿ ◮ P(F|E)✿ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s r❛♥❦ ✷✱ ❣✐✈❡♥ t❤❛t ✐t ❤❛s s✉✐t ❇✳ A) ✸/✽ B) ✶/✽ C) ✶/✸ D) ✷/✽ E) ✷/✸ ◮ P(E|F)✿ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s s✉✐t ❇✱ ❣✐✈❡♥ t❤❛t ✐t ❤❛s r❛♥❦ ✷✳ A) ✷/✽ B) ✶/✷ C) ✸/✷ D) ✶/✽ E) ✸/✽ ❲❡ ✇✐❧❧ ❝♦♠♣❛r❡ t❤❡s❡ t♦ ❡❛❝❤ ♦t❤❡r✱ ❛♥❞ t♦ P(E) ❛♥❞ P(F)✳
SLIDE 37
❉r❛✇✐♥❣ ❛ ❈❛r❞✿ P(F|E) ✭●✐✈❡♥ ❙✉✐t✮ ✷
❙✉✐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❤❡ t❤r❡❡ ❝❛r❞s ✇✐t❤ s✉✐t ❇✳
SLIDE 38
❉r❛✇✐♥❣ ❛ ❈❛r❞✿ P(F|E) ✭●✐✈❡♥ ❙✉✐t✮ ✷
❙✉✐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❤❡ t❤r❡❡ ❝❛r❞s ✇✐t❤ s✉✐t ❇✳
SLIDE 39
❉r❛✇✐♥❣ ❛ ❈❛r❞✿ P(F|E) ✭●✐✈❡♥ ❙✉✐t✮ ✷
❙✉✐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❤❡ t❤r❡❡ ❝❛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 ❇ ❛♥❞ r❛♥❦ ✷ ✸ ❝❛r❞s ✇✐t❤ s✉✐t ❇ = ✶ ✸.
SLIDE 40
❉r❛✇✐♥❣ ❛ ❈❛r❞✿ P(E|F) ✭●✐✈❡♥ ❘❛♥❦✮ ✷
❙✉✐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❛♥❦ ✷✳
SLIDE 41
❉r❛✇✐♥❣ ❛ ❈❛r❞✿ P(E|F) ✭●✐✈❡♥ ❘❛♥❦✮ ✷
❙✉✐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❛♥❦ ✷✳
SLIDE 42
❉r❛✇✐♥❣ ❛ ❈❛r❞✿ P(E|F) ✭●✐✈❡♥ ❘❛♥❦✮ ✷
❙✉✐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❛♥❦ ✷✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s s✉✐t ❇ ❣✐✈❡♥ t❤❛t ✐t ❤❛s r❛♥❦ ✷ ✐s P(E|F) = ✶ ❝❛r❞ ✇✐t❤ r❛♥❦ ✷ ❛♥❞ s✉✐t ❇ ✷ ❝❛r❞s ✇✐t❤ r❛♥❦ ✷ = ✶ ✷.
SLIDE 43
❄✭✻✳✸✮ ■♥❞❡♣❡♥❞❡♥t❄
❖✉r ❝❛❧❝✉❧❛t✐♦♥s s♦ ❢❛r✿ ◮ P(E) = ✸/✽ ◮ P(F) = ✷/✽ ◮ P(F|E) = ✶/✸ ◮ P(E|F) = ✶/✷ ❆r❡ t❤❡ ❡✈❡♥ts E ❛♥❞ F ✐♥❞❡♣❡♥❞❡♥t❄ ❨❡s ♦r ♥♦❄
SLIDE 44
❖❜s❡r✈❛t✐♦♥✿ P(E) ✈s✳ P(E|F)
◮ ◆♦t✐❝❡ t❤❛t P(E) = ✸ ✽, ❜✉t P(E|F) = ✶ ✷. ◆♦r♠❛❧❧②✱ ✇❡ ❤❛✈❡ ❛ ✸ ✐♥ ✽ ❝❤❛♥❝❡ ✭✵✳✸✼✺✮ ♦❢ ❞r❛✇✐♥❣ ❛ ❝❛r❞ ✇✐t❤ s✉✐t ❇✱ ❜✉t ✐❢ ✇❡ ❛r❡ ❣✐✈❡♥ t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s r❛♥❦ ✷✱ t❤❛t r❛✐s❡s t❤❡ ❝❤❛♥❝❡s ♦❢ ❞r❛✇✐♥❣ ❛ ❝❛r❞ ✇✐t❤ s✉✐t ❇ t♦ ✶✴✷ ✭✵✳✺✮✳
SLIDE 45
❖❜s❡r✈❛t✐♦♥✿ P(F) ✈s✳ P(F|E)
◮ ◆♦t✐❝❡ t❤❛t P(F) = ✶ ✹, ❜✉t P(F|E) = ✶ ✸. ◆♦r♠❛❧❧②✱ ✇❡ ❤❛✈❡ ❛ ✶ ✐♥ ✹ ❝❤❛♥❝❡ ✭✵✳✷✺✮ ♦❢ ❞r❛✇✐♥❣ ❛ ❝❛r❞ ✇✐t❤ r❛♥❦ ✷✱ ❜✉t ✐❢ ✇❡ ❛r❡ ❣✐✈❡♥ t❤❛t t❤❡ ❞r❛✇♥ ❝❛r❞ ❤❛s s✉✐t ❇✱ t❤❛t r❛✐s❡s t❤❡ ❝❤❛♥❝❡s ♦❢ ❞r❛✇✐♥❣ ❛ ❝❛r❞ ✇✐t❤ r❛♥❦ ✷ t♦ ✶✴✸ ✭❛❜♦✉t ✵✳✸✸✸✸✮✳
SLIDE 46
■♥❞❡♣❡♥❞❡♥❝❡ ✇✐t❤ ▼✐ss✐♥❣ ❈❛r❞s
❙✉✐t ❭ ❘❛♥❦ ✶ ✷ ✸ ✹ ✺ ❆ ❆✶ ❆✷ ❆✸ ❆✹ ❆✺ ❇ ❇✶ ❇✷ ❇✸ ❙✐♥❝❡ P(F) = ✶ ✹, ❜✉t P(F|E) = ✶ ✸, t❤❡ ❡✈❡♥ts E ❛♥❞ F ❛r❡ ♥♦t ✐♥❞❡♣❡♥❞❡♥t✳
SLIDE 47