Conditional Probability, Independence & Bayes · Superchallenge problem

Urn $A$ holds $2$ white and $1$ black ball; urn $B$ holds $1$ white and $2$ black. The first ball is drawn (with replacement) from urn $A$. Thereafter, if the previous ball drawn was white the next is drawn from urn $A$, and if black the next is drawn from urn $B$. Find the probability that the third ball drawn is white.

Express your answer as a common fraction.