
Dependent urns problem
This problem has been suggested by my professor.
Maybe it is trivial but I am not able to solve it. Thank you in advance for your help!
At time t=0, there are 4 urns: A, B, C, D. Every urn contains a positive number of red and black balls.
Urn A contains b_A0>0 black balls and r_A0>0 red balls. Urn B b_B0>0 and r_B0>0, etc.
Let n_A0=b_A0+r_A0, n_B0=b_B0+r_B0, etc...
The four urn are connected as follows
ABCD
We focus our attention on urns B and C
Urn B has two neighbors: A and C
Urn C has two neighbors: B and D
At time t=1, the two urns are sampled according to the following rule:
Urn B
 a ball is sampled at random from the urn and the probability that this ball is black is not b_B0/n_B0, but (b_B0+b_A0+b_C0)/(n_A0+n_B0+n_C0). In other words, the probability of picking a black ball from urn B depends on the total number of black balls in its neighborhood. It is like A, B and C form one big urn, containing all their balls.
An analogous reasoning is valid for red balls.
 once we have discovered the color of the sampled ball, we replace it into urn B and we add one extra ball of the same color (as in a Polya urn) to B.
 if X_t is the color of the sampled ball at time t and we assume that X_t=1 means "black ball sampled", we have that X_t is a Bernoulli variable with parameter (b_Bt1+b_At1+b_Ct1)/(n_At1+n_Bt1+n_Ct1).
We have that b_Bt=b_Bt1+1if X_t=1 and b_Bt=b_Bt1 if X_t=0.
Urn C
Always at time t=1, we also sample urn C (according to the same mechanism with reinforcement). In this case we consider the variable Y_t, which indicates the sampled ball from urn C at time t.
Even Y_t is a Bernoulli variable with parameter (b_Bt1+b_Ct1+b_Dt1)/(n_Bt1+n_Ct1+n_Dt1).
My problem is the following: it is obvious that the two urns are not independent. In fact the evolution of black (or red) balls in urn A also depends on the composition of urn B and viceversa.
What is the joint probability of X_t and Y_t, i.e. P(X_t=x;Y_t=y) where x,y=0,1?
Notice that for t=0, P(X_0=x;Y_0=y)=P(X_0=x)P(Y_0=y), i.e. the product of two Bernoulli variables.
But this is not true for t>=1.
Thanks!!!