A box contains nine cards numbered Ithrough $9$ , Box contains five cards numbered Ithrough $5$ . A box chosen at random and a card is drawn, If the number is even, find the probability that the card came from box $A$ ?

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Solution

$\begin{matrix} l e t B_{1}, B_{2} b e t h e f o l l o w i n g : B_{1} : t h e b o x c h o o s e n i s A B_{2} : t h e b o x c h o o s e n i s B C l e a r l y B_{1}, B_{2} a r e m u t u a l l y e x c l u s i v e a n d e x h a u s t i v e e v e n t s a n d h e n c e, t h e y r e p r e s e n t a p o r t i o n o f t h e s a m p l e s p a c e P (B_{1}) = \frac{1}{2} a n d P (B_{2}) = \frac{1}{2} l e t E b e t h e e v e n t t h a t^{'} t h e n u m b e r o n t h e c a r d i s e v e n^{'} ∴ p (\frac{E}{B_{1}}) = o b a b l i t y t h a t t h e n u m b e r i s e v e n g i v e n t h a t i t i s t a k e n o u t f r o m b o x, B_{1} = \frac{4}{9} p (\frac{E}{B_{2}}) = o b a b l i t y t h a t t h e n u m b e r i s e v e n g i v e n t h a t i t i s t a k e n o u t f r o m b o x, B_{2} = \frac{2}{5} H e n c e, b y B a y e^{'} s t h e o r e m p (\frac{B_{1}}{E}) = \frac{P (B_{1}) P (\frac{E}{B_{1}})}{P (B_{1}) P (\frac{E}{B_{1}}) + P (B_{2}) P (\frac{E}{B_{2}})} = \frac{\frac{1}{2} \times \frac{4}{9}}{\frac{1}{2} \times \frac{4}{9} + \frac{1}{2} \times \frac{2}{5}} = \frac{10}{19} \end{matrix}$

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