I seem to fail to understand: The chances clearly change depending on what happened. E.g., with initially g=1 and r=1, after the first draw we have w.l.o.g. g=2 and r=1 and chances are certainly not fifty-fifty anymore! In other words, in your formulas "r/(r+g) or g/(r+g)" the g and r do change (as far as I understand). Best regards, jj * Dan Asimov <dasimov@earthlink.net> [Sep 27. 2015 08:50]:
I just skimmed an online paper that appears to describe the Polya urn model, as follows:
Start with an urn containing r red balls and g green balls.
At the nth stage, n = 1,2,3,..., pick a ball from the urn at random.
Then put the ball back in the urn and *add* to the urn one more ball of the same color.
It's surprising but easy to check that at each stage, the probability of picking a red or green ball is r/(r+g) or g/(r+g), respectively — the same as on the first stage.
Hence, the probabilities for the outcome (ball color) at the (n+1)st stage are independent of anything that occurred at previous stages. This is the defining property of a "martingale", and so the Polya urn model is a martingale.
—Dan
On Sep 20, 2015, at 9:13 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Would anyone please be so kind as to say what the Polya urn model is?
(Not Jim, because he obviously doesn't want to bother.)
—Dan
On Sep 20, 2015, at 4:35 AM, James Propp <jamespropp@gmail.com> wrote:
Does anyone have a pointer to the cute way of seeing that the Polya urn model gives rise to the uniform distribution at the nth level, by contriving a uniformly random permutation of 1,2,...,n whose first (or last) element corresponds to the number of white-vs.-black balls at the end?
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