# Polya urn probabilities [closed]

Consider an urn containing two type of balls (red and blue), initially there are $r_0$ red balls and $b_0$ blue balls. At each stage $n\geq 0$ we choose at random one ball, if the ball is red we put it back into the urn and we add $r>0$ more balls, if the ball is blue, we return it together with $b>0$ more balls. We denote $r_n:$ amount of red balls at stage $n\geq 0$ and $b_n$ defined analogously.

The idea is that under the conditions of $r_0>b_0$, and $r>b$, compute (or find a non-zero lower bound) the probability of the events $A_n=\{r_n>b_n\}$ and $B_n=\{r_m>b_m\},\forall m\in \{1,...,n\}$.

Thanks in advance

## closed as off-topic by colormegone, Graham Kemp, Stefan Mesken, Claude Leibovici, Daniel W. FarlowMar 18 '16 at 12:02

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• What did you try? Are the $r$ balls all red balls and the $b$ balls all blue balls? – Ritz Mar 18 '16 at 8:40