Show that the number of red balls in Polyas Urn problem is uniformly distributed

Question

I have the classic Polyas Urn problem. Let $t$ be the time steps $t \in \{1, \ldots,N\}$. At $t=1$ there is one red and one green ball in the urn. Let $R_n$ and $G_n$ denote the number of red and green balls before the $n$-th draw. The total number of balls at timestep $n$ are therefore $T_n=R_n+G_n=1+n$.

I want to show that: $$P(R_n=k)=\frac{1}{n}, \, \, \, \, \, \text{for} \, \, \, \, \, n\in\{1,\ldots,N\},k\in\{1,\ldots,n\}$$

I have three questions regarding this problem:

1. Edit I made a mistake here and deleted this question.

2. I derived earlier that for $n \in \{2,\ldots,N\}$ and $2 \le k \le n-1$ the following equalities are true:

$$\begin{aligned} &P(R_n=k \vert R_{n-1}=k)=\frac{n-k}{n} \\[10pt] &P(R_n=k \vert R_{n-1}=k-1)=\frac{k-1}{n} \end{aligned}$$

Can I somehow use these to show that $P(R_n=k)=\frac{1}{n}$?

3. I have looked at lot of the polya's urn questions on this site that seem to deal with uniform distributions. (Intuitive/heuristic explanation of Polya's urn, Question related with Polya urn model, distribution of number of red balls in the first $n$ trials) but they always seem to give some sort of intuitive explanation of why it must be true rather than a derivation. On the other hand some of the derivations I have seen on the web use techniques we haven't covered in class so I am thinking there must be an easier straight forward way to show this but I don't know how.

Answer 1

For your question 2.

The results you have derived can help you demonstrate the property recursively.

The property is evident for $n=1$. considering is is true for $n-1$, you notice that if $R_n=k$ then $R_{n-1}=k$ (the nth draw was not a red ball) or $R_{n-1}=k-1$ (the nth draw was a red ball).

So

$P(R_n=k)=P(R_{n-1}=k).P(R_n=k|R_{n-1}=k)+P(R_{n-1}=k-1).P(R_n=k|R_{n-1}=k-1)$

Then you use $P(R_{n-1}=j)=\frac{1}{n-1}$ (or $0$ when $j\ge n$)

3. You just proved that Polya Urns with this initial conditions provides an equi-probability of the number of red balls in the urn. What is interesting in the Polya Urn experiment, in my opinion, is to think of it as an infinite experiment. You will end up with a frequency of red balls that will stabilize. This frequency is uniformely distributed on the interval $(0,1)$. What is interesting is that when you get the first n draws (for instance, imagine you get 2 red balls for the first draws), then the final frequency you expect is no more uniformely distributed as you intuitively can feel it. It is a distribution in the family of beta distributions. Now considere a slightly different experiment : you are a statistician and you try to estimate the frequency of red balls in a very huge urn. You first suppose that the frequency is uniformely distributed (It is as likely to have 1%, 10%, 50%,... of red balls) then you draw 2 balls that both are red. How should you revise the probability distribution of the frequency of red balls ? Exactly the same as in the Polya Urn experiment when you draw 2 red balls.