Hello guys. I have the following question.

We have a Markov chain with this transition matrix P : [1/3, 1/3, 1/3; 0 2/3 1/3; 1/2, 1/2, 0]

So 1/3. 1/3, 1/3 is my first row, 0, 2/3, 1/3 the second row and 1/2, 1/2, 0 the third row.

And I have to find the mean time to hit state 3 starting from state 1, say b(1 to 3)

I said b(1 to 3)= 1+ 1/3*(b(1 to 3)) + 1/3* b(2 to 3)

Now b(2 to 3) = 1+2/3 * b(2 to 3) so b(2 to 3)=3

So I find b(1 to 3)=3

I have a question. What I did is based on our lectures. I understand all the steps except why do we add 1 to find b(1 to 3) and b(2 to 3) i.e why not be b(1 to 3)= 1/3 * (b(3 to 3))+ 1/3*(b(1 to 3)) + 1/3* b(2 to 3) AND b(2 to 3) = 1/3 * b(3 to 3) +2/3 * b(2 to 3) so b(2 to 3)=3

The lecturer said to us is like a forced step but for example the probability to go from 1 to 3 is 1/3 and not 1 and the probability to go from 2 to 3 is 1/3 again, not 1!

I hope you can understand my notation!

So can someone explain me this and also tell me if what I did is correct? Thank you very much!!!