Question
Sun April 12, 2015 By: Sumanjali

what is the difference between P(A|B) an P(Awhat is the difference between P(A|B) an P(AB), Question no 89573, MathematicsB)

Expert Reply
Vimala Ramamurthy
Mon April 13, 2015
L e t space A space a n d space B space b e space t w o space e v e n t s space a s s o c i a t e d space w i t h space a space r a n d o m space e x p e r i m e n t comma space t h e n space P open parentheses A intersection B close parentheses space i s space t h e space p r o b a b i l i t y space o f space o c c u r r e n c e space o f space b o t h space A space a n d space B.  C o n s i d e r space t h e space f o l l o w i n g space e x a m p l e. A space b l a c k space a n d space a space r e d space d i c e space a r e space r o l l e d. space L e t space A space b e space t h e space e v e n t space s u c h space t h a t space t h e space r e s u l t i n g space s u m space i s space g r e a t e r space t h a n space 9. L e t space B space b e space t h e space e v e n t space s u c h space t h a t space t h e space b l a c k space d i e space r e s u l t e d space i n space a space 5. T h e n space f i n d space t h e space p r o b a b i l i t y space o f space g e t t i n g space a space s u m space g r e a t e r space t h a n space 9 space a n d space b l a c k space d i e space i s space 5.  S o l u t i o n colon C l e a r l y comma space A equals open curly brackets open parentheses 5 comma 5 close parentheses comma open parentheses 6 comma 4 close parentheses comma open parentheses 4 comma 6 close parentheses comma open parentheses 6 comma 5 close parentheses comma open parentheses 5 comma 6 close parentheses comma open parentheses 6 comma 6 close parentheses close curly brackets B equals open curly brackets open parentheses 5 comma 1 close parentheses comma open parentheses 5 comma 2 close parentheses comma open parentheses 5 comma 3 close parentheses comma open parentheses 5 comma 4 close parentheses comma open parentheses 5 comma 5 close parentheses comma open parentheses 5 comma 6 close parentheses close curly brackets T h u s comma space t h e space p r o b a b i l i t y space o f space g e t t i n g space a space s u m space g r e a t e r space t h a n space 9 space a n d space b l a c k space d i e space i s space 5 space i s space P open parentheses A intersection B close parentheses. A intersection B equals open curly brackets open parentheses 5 comma 5 close parentheses comma open parentheses 5 comma 6 close parentheses close curly brackets P open parentheses A intersection B close parentheses equals 2 over 36 equals 1 over 18  C o n s i d e r space t h e space s a m e space e x a m p l e space o f space h a v i n g space space r e d space a n d space b l a c k space d i e. A space b l a c k space a n d space a space r e d space d i c e space a r e space r o l l e d. space L e t space A space b e space t h e space e v e n t space s u c h space t h a t space t h e space r e s u l t i n g space s u m space i s space g r e a t e r space t h a n space 9. L e t space B space b e space t h e space e v e n t space s u c h space t h a t space t h e space b l a c k space d i e space r e s u l t e d space i n space a space 5. T h e n space f i n d space t h e space p r o b a b i l i t y space o f space g e t t i n g space a space s u m space g r e a t e r space t h a n space 9 comma space g i v e n space t h a t space t h e space b l a c k space d i e space i s space 5.  C l e a r l y comma space A equals open curly brackets open parentheses 5 comma 5 close parentheses comma open parentheses 6 comma 4 close parentheses comma open parentheses 4 comma 6 close parentheses comma open parentheses 6 comma 5 close parentheses comma open parentheses 5 comma 6 close parentheses comma open parentheses 6 comma 6 close parentheses close curly brackets B equals open curly brackets open parentheses 5 comma 1 close parentheses comma open parentheses 5 comma 2 close parentheses comma open parentheses 5 comma 3 close parentheses comma open parentheses 5 comma 4 close parentheses comma open parentheses 5 comma 5 close parentheses comma open parentheses 5 comma 6 close parentheses close curly brackets T h u s comma space t h e space p r o b a b i l i t y space o f space g e t t i n g space a space s u m space g r e a t e r space t h a n space 9 comma space g i v e n space t h a t space t h e space b l a c k space d i e space i s space 5 comma space i s space P open parentheses A vertical line B close parentheses A n d space P open parentheses A vertical line B close parentheses equals fraction numerator P open parentheses A intersection B close parentheses over denominator P open parentheses B close parentheses end fraction A intersection B equals open curly brackets open parentheses 5 comma 5 close parentheses comma open parentheses 5 comma 6 close parentheses close curly brackets P open parentheses A intersection B close parentheses equals 2 over 36 equals 1 over 18 P open parentheses B close parentheses equals 6 over 36 equals 1 over 6 T h u s comma P open parentheses A vertical line B close parentheses equals fraction numerator P open parentheses A intersection B close parentheses over denominator P open parentheses B close parentheses end fraction space space space space space space space space space space space space equals fraction numerator 1 over 18 over denominator 1 over 6 end fraction space space space space space space space space space space space space equals 6 over 18 space space space space space space space space space space space space equals 1 third
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