# Polya's Urn and Conditional Independence

We use the basic set up of Polya's Urn, with r red balls and b black balls. Let us then denote Xn =1 if the ball picked is red and 0 if ball is black. Let us also denote Sn = (X1 +...+ Xn) to be the number of red balls picked up until time n.

I want to show that Sn+1 and Sn-1 are conditionally independent given Sn for every n.

I realize this pertains to the Markov Property, and I can understand it intuitively, as we simply have to look at how many red balls we have now (at n), and then the value at n+1 is simply one jump away, however I am unsure how you would derive a formal proof.

• Given $S_n$ there are only four possible states for $(S_{n-1},S_{n+1})$. Just write them out and compute all the relevant probabilities by hand. – lulu Jan 16 '18 at 15:29
• Welcome to MSE. Please use MathJax. – José Carlos Santos Jan 16 '18 at 15:29