I guess you should only need to show P[Xn+1 = xn+1 | X1 = x1, X2 = x2, ..., Xn = xn] = P[Xn+1 = xn+1 | Xn = xn]. (Markov property).
In your case it holds, since G is only a function of Xn; and Yi are independent random variables. P[Xn+1 = xn+1 | Xn = xn] = P[G(xn, y)=xn+1] where y is the...
If I understood correctly, you have a 3-state continuous-time Markov chain. You know the transition rates (birth/death rates) so you know the infinitesimal generator matrix (Q), which gives you the probabilities of a transition in a short time interval. I guess you are trying to find the...
Let Xn denote the color of the ball selected from the nth urn.
The probability that you are looking for is P[Xn=w]. We have P[Xn=w]=P[Xn=w|Xn-1=w]P[Xn-1=w] + P[Xn=w|Xn-1=b]P[Xn-1=b].
We know P[Xn-1=b]=1-P[Xn-1=w]. So we have P[Xn=w]=P[Xn=w|Xn-1=w]P[Xn-1=w] + P[Xn=w|Xn-1=b](1-P[Xn-1=w])...