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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 $1/(n-1)$?

N. F. Taussig
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JWen
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    Shorter than what? – zoli Mar 03 '15 at 13:47
  • Actually I'm trying to do a direct proof for this question. I happened to deduct that probability is equal to 1/(n-1), by trying to do n=2,3,4,5. I'm thinking if there is any more complete proof other than deduction. – JWen Mar 05 '15 at 07:53

1 Answers1

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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.

Felipe André Maldonado Caro
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