A bag 'A' contains 2 white and 3 red balls and bag 'B' contains 4 while and 5 red balls. One ball is drawn at random from a randomly chosen bag and is found to be red. If the proability that it was drawn from bag 'B' can be expressed as rational `(p)/(q)` (where p and q are in their lowest form), then find (p+q).

Answer

77

Explanation

Let `E_(1)` be the event that the ball is drawn from bag A, `E_(2)` the event that it is drawn from bag B and E that the ball is red. We have to find `(E_(2)//E)`.
Since both the bags areequally likely to be selected, we have
`p(E_(1))=p(E_(2))=(1)/(2)`
Also `p((E)/(E_(2)))=(3)/(5)` and ` p((E)/(E_(2)))=(5)/(9)`
Hence by Bay's Theorem, we have
`p((E_(2))/(E))=(p(E_(2))p(E//E_(2)))/(p(E_(1))p(E//E_(1))+p(E_(2))p(E//E_(2)))`
`((1)/(2).(5)/(9))/((1)/(2).(3)/(5)+(1)/(2).(5)/(9))=(25)/(52)-=(p)/(q) rArr p+q=77`