1. ## Markov chain question

A single-celled organism contains N particles, some of which are of type A, the others of
type B . The cell is said to be in state i , where $\displaystyle$0\leq  i \leq N, if it contains exactly i particles
of type A. Daughter cells are formed by cell division, but rst each particle replicates itself;
the daughter cell inherits N particles chosen at random from the 2i particles of type A
and 2N-2i of type B in the parent cell.

Find the absorption probabilities and expected times to absorption for the case N = 3.

2. 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 $\displaystyle \binom{6}{3}$ 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).