09-28 Conditional Probability and Independence (3)

# 09-28 Conditional Probability and Independence (3) - B T R...

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Unformatted text preview: B T R Y 4 8 / S T S C I 4 8 F a l l 2 1 1 4 5 T h e B a y e s F o r m u l a F o r e v e n t s E ⊂ S a n d F ⊂ S s u c h t h a t P ( E ) > , P ( F | E ) = P ( E | F ) P ( F ) P ( E | F ) P ( F ) + P ( E | F c ) P ( F c ) P r o o f : U s i n g t h e d e fi n i t i o n o f c o n d i t i o n a l p r o b a b i l i t y : P ( F | E ) = P ( E F ) P ( E ) = P ( E | F ) P ( F ) P ( E ) . B T R Y 4 8 / S T S C I 4 8 F a l l 2 1 1 4 6 N o w , w r i t e S = F ∪ F c . T h e n , n o t i c e t h a t P ( E ) = P ( E S ) = P ( E F ) + P ( E F c ) = P ( E | F ) P ( F ) + P ( E | F c ) P ( F c ) H e n c e , P ( F | E ) = P ( E | F ) P ( F ) P ( E ) = P ( E | F ) P ( F ) P ( E | F ) P ( F ) + P ( E | F c ) P ( F c ) . square B T R Y 4 8 / S T S C I 4 8 F a l l 2 1 1 4 7 E x a m p l e ( 3 . 3 . 3 a ) : A n a c c i d e n t- p r o n e p e r s o n w i l l h a v e a n a c c i d e n t w i t h i n a fi x e d o n e- y e a r p e r i o d w i t h p r o b a b i l i t y . 4 . T h i s p r o b a b i l i t y d e c r e a s e s t o . 2 f o r a n o n- a c c i d e n t- p r o n e p e r s o n . S u p p o s e t h a t t h e p r o b a b i l i t y o f a r a n d o m l y s e l e c t e d p e r s o n b e i n g a c c i d e n t p r o n e i s . 3 ( i . e . , i n t h e p o p u l a t i o n , 3 % a r e a c c i d e n t- p r o n e ) . L e t A = { a c c i d e n t p r o n e } a n d B = { h a s a c c i d e n t } . P r o b l e m s p e c i fi e s t h a t P ( B | A ) = . 4 , P ( B | A c ) = . 2 , P ( A ) = . 3 . Q 1 : W h a t i s t h e p r o b a b i l i t y t h a t a r a n d o m l y s e l e c t e d p e r s o n w i l l h a v e a n a c c i d e n t w i t h i n a fi x e d o n e- y e a r p e r i o d ? T h a t i s : P ( B ) ? Q 2 : S u p p o s e a p e r s o n h a s a n a c c i d e n t w i t h i n a y e a r . W h a t i s t h e p r o b a b i l i t y t h a t h e / s h e i s a c c i d e n t p r o n e ? T h a t i s : P ( A | B ) ? B T R Y 4 8 / S T S C I 4 8 F a l l 2 1 1 4 8 E x a m p l e ( 3 . 3 . 3 c ) : C o n s i d e r a m u l t i p l e c h o i c e t e s t q u e s t i o n . L e t p b e t h e p r o b a b i l i t y t h a t t h e s t u d e n t k n o w s t h e a n s w e r a n d 1 − p t h e p r o b a b i l i t y t h a t t h e s t u d e n t g u e s s e s . A s s u m e t h a t a s t u d e n t w h o g u e s s e s w i l l b e c o r r e c t w i t h p r o b a b i l i t y 1 m , w h e r e m i s t h e t o t a l n u m b e r o f m u l t i p l e- c h o i c e a l t e r n a t i v e s . W h a t i s t h e c o n d i t i o n a l p r o b a b i l i t y t h a t a s t u d e n t k n o w s t h e a n s w e r , g i v e n t h a t h e / s h e a n s w e r e d i t c o r r e c t l y ? B T R Y 4 8 / S T S C I 4 8 F a l l 2 1 1 4 9 S o l u t i o n : L e t C = e v e n t t h a t t h e s t u d e n t a n s w e r s t h e q u e s t i o n c o r r e c t l y ....
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## This note was uploaded on 10/06/2010 for the course BTRY 4080 taught by Professor Schwager during the Fall '06 term at Cornell.

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09-28 Conditional Probability and Independence (3) - B T R...

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