# Questions tagged [polya-urn-model]

In the basic Pólya urn model, the urn contains x white and y black balls; one ball is drawn randomly from the urn and its color observed; it is then replaced in the urn, and an additional ball of the same color is added to the urn.

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### probability with two variables - Taking balls out of the basket

There's 3 balls in the basket - White, red and black. Three people chose one ball after the other with return. $X$ is the various colors that got chosen. $Y$ is the number of people that chose white. ...
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### Find the probability that both the balls drawn from urn $U_2$ are red.

Urn $U_i \;\;(i = 1, 2)$ contains $N_i$ balls out of which $r_i$ are red and $N_i − r_i$ are black. A sample of $n \;(1 ≤ n ≤ N_1)$ balls is chosen at random (without replacement) from urn $U_1$ and ...
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### Solve Polya urn with generating function?

The (simple) Polya urn contains $a \in \mathbf N$ black and $b \in \mathbf N$ white balls at the initial time $t=0$, and, at each time $t \in \mathbf Z_{+}$, a ball is picked uniformly at random in ...
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### Expected time to get only green balls

A jar contains $n$ balls. Initially, all of the balls are red. Every minute, a ball is drawn at random from the jar, and then replaced with a green ball. Let $T$ be the number of minutes until the jar ...
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### Pólya's Urn Long-Term Probability

I am tasked with the following problem: Let $M_n$ be the fraction of white balls in Pólya's urn after $n$ draws, where you start with one white and one black ball, and after every draw, you add ...
I'm trying to organize the formulas for permutations and combinations. Let's say we sample $r$ objects from an urn which contains $n$ objects, $m$ of which are duplicates. So if the objects in the ...