Conditional Probability, Independence & Bayes · Challenge problem

Box $A$ contains $2$ white and $3$ black balls; box $B$ contains $4$ white balls and $1$ black ball. One ball is chosen at random from box $A$ and moved, unseen, into box $B$. Then one ball is drawn at random from box $B$, and it is white. What is the probability that the ball moved from $A$ was white? Express your answer as a common fraction.