Assume that removing a ball from bag A or bag B is equally likely, the probability of choosing a blue ball from bag A is 3/10 and the probability of choosing a blue ball from bag B is 4/14 so the probability of choosing a blue ball is 3/10 + 4/14 = 41/70.
Theoretically in 70 trials, 41 of the outcomes would give a blue ball and 21 of these would of been drawn from bag A . Given that the ball drawn was blue the probability that it came from bag A is 21/41.

The only correct solution to date came from Matthew (Smithdon High School)