Question related with Polya urn model The Polya urn model for contagion is as follow.

We start with an urn which contains one white ball and one black ball. At each second we choose a ball at random from the urn and replace it together with one more ball of the same colour.

Calculate the probability that when $n$ balls are in the urn, $i$ of them are white.

How do you prove this probability is equal to $1/(n-1)$?
Are you sure that the probability must be that value?

I'm asking you this, because, if $w(n)$ is the amount of white balls and $b(n)$ the black balls at stage $n$, then the proportion $X(n):=\frac{w(n)}{w(n)+b(n)}$ is a martingale, respect to the natural filtration, then $\mathbb{E}[X(n)]=\mathbb{E}[X(0)]=1/2$, and therefore
$\mathbb{P}[w(n)=n/2]=1\neq \frac{1}{n-1}$.

I think what you are really want to prove is that 

$\mathbb{P}[w(n)=i]=\frac{i}{n-i}$, 

Of course if $i=1$, you get what you put before.
Are you sure that the probability must be that value?

I'm asking you this, because, if $w(n)$ is the amount of white balls and $b(n)$ the black balls at stage $n$, then the proportion $X(n):=\frac{w(n)}{w(n)+b(n)}$ is a martingale, respect to the natural filtration, then $\mathbb{E}[X(n)]=\mathbb{E}[X(0)]=1/2$, and therefore
$\mathbb{P}[w(n)=n/2]=1\neq \frac{1}{n-1}$.

I think what you are really want to prove is that 

$\mathbb{P}[w(n)=i]=\frac{i}{n-i}$, 

Of course if $i=1$, you get what you put before.