Question 430559
The assumptions that 

P(7R3B|bag1) = C(10,3)[(.7)^7(.3)^3] 

P(7R3B|bag2) = C(10,3)[(.3)^7(.7)^3] 
 
are wrong.

The experiment of drawing balls from each bag doesn't follow a binomial distribution, but a hypergeometric distribution.

They should be 
P(7R3B|bag1) = {{{((300C7)*(700C3))/1000C10}}}

P(7R3B|bag2) = {{{((700C7)*(300C3))/1000C10}}} .
 
The use of Bayes' rule is correct.