The absorbing states are where i = 0 and i = 3. The cell doubles in size to contain either (where i = 0) 6 particles of type B or (where i = 3) 6 particles of type A. Then it splits in two, and is replaced by two daughters which are each in the same state as was the parent. And likewise the daughters' daughters etc.

Where i = 1, the cell doubles to contain A A B B B B. Make a list to find what proportion of the ways that that double cell could split in two would find both daughters in state 1, i.e. A B B, and what proportion would produce one daughter in state 2 and the other in state 0. Then you can show that the probability of going (in a line of descent from state 1) to state 1 is 3/5, to state 2 is 1/5, and to state 0 is also 1/5.

Do similarly for state 2.

Then make and use a transition matrix as described here in order to find whether you are most likely to slide into state 0 or 3 from either of states 1 or 2 (the absorption probabilities) and the most likely number of generations till that would happen (expected times).