AP statistics question A bag contains 2 black and 2 white marbles, and there is a supply of extra marbles of each color. A move consists of randomly drawing 2 marbles from the bag. If the marbles are the same color, they are replaced by two black marbles. If they are different collors they are replaced by two white marbles.

Find the probability that after 4 moves, all the marbles are black in the bag. Give Answer as fraction, reduced to Lowest terms.

Please help. I’m not doing well in statistics and need to answer this problem!
Hint:  There are five states of the bag.  From each state you can go to a maximum of three others.  So there are (at most) 15 transition probabilities to figure-in fact a few less.  For example, from (2B,2W) the chance to go to (4B,0W) is $\frac 16$ as you have to draw the two white balls.  What is the probability distribution after one draw?  Then keep going.
Hint:  There are five states of the bag.  From each state you can go to a maximum of three others.  So there are (at most) 15 transition probabilities to figure-in fact a few less.  For example, from (2B,2W) the chance to go to (4B,0W) is $\frac 16$ as you have to draw the two white balls.  What is the probability distribution after one draw?  Then keep going.