This problem is perfect for a tree diagram.
The first step is to select a ball from bag 1. P(white)=0.5, P(black)=0.5
The next step is to make the next set of branches -> for each possible event from bag 1, you can select a white ball or a black ball from bag 2.
The third step is to then make the branches for the possible outcomes from bag 3, based on the 4 possible outcomes from bag 1 + bag 2.
The probability of a white ball taken from bag3 is the sum of all the events where a white ball is taken from bag3, in other words, multiply the probabilities for each step of every "branch" that ends with a white ball being selected in the third step.
The answer is P(white ball from bag 3) = 0.5562