I have a question I don't understand; I need help how to think about the process. The question is the following:
In the two urns A and B there are three red balls and two green balls. One ball is drawn from the urn containing three balls and it is placed in the other urn.
Define,
Xn= the number of green balls in the urn that after n draws contains two balls, n=1,2,...
X0=2
b) Determine the initial probabilities and the transition matrix P.
So the transition matrix is: P=[ 0 (2/3) (1/3); (1/3) (2/3) 0; (1) 0 0]
But why? I read the matrix such as, lets say I have Xn=2, two green balls in urn B and go to state Xn=0, in other words p20 to p00, how can i start with two green balls in urn B and by taking one ball from urn A end up with 0 green balls in urn B with one draw? That's what I don't understand...
You seem to be assuming that the state index refers to the number of green balls in a fixed urn. But that wouldn’t work well, as the number of balls in each fixed urn varies, so the number of states would vary. Also, the question doesn’t ask about the number of green balls in a particular urn but rather the number of green balls in an urn characterized in a certain way. Note that that characterization ensures that the urn thus characterized always contains the same number of balls. So the states should be defined according to the number of green balls in the urn thus characterized.