# 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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### Considering that the two selected balls are white, what is the probability that urn III was chosen?

Consider three numbered urns. Urn I contains 6 white balls and 2 red balls, urn II contains 2 white balls and 6 red balls, and urn III contains 4 white balls and 4 red balls. To choose an urn, roll a ...
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### Polya's Urn solution to question

A Polya urn has two balls, one red, and one blue. One of these is chosen uniformly at random. It is put back, with another of the same color. Again, a ball is chosen uniformly at random, and put back, ...
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### Interesting problem on Polya's Urn [closed]

Suppose we have b blue balls and r red balls in an urn. n times, we pick a ball out of the urn, put it back, then add a new ball of the opposite color to the urn. We end with a total of r + b + n ...
1 vote
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### Distribution and Martingale in Polya's urn

This is exercise 10.1 from Probability with Martingale by David Williams: At time $0,$ an urn contains $1$ black ball and $1$ white ball. At each time $1,2,3,...,$ a ball is chosen at random from the ...
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### Polya Urn Scheme marginal distribution of draws

I'm going through the really well written notes https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.212.2959&rep=rep1&type=pdf. I'm at the point where they discuss about the Polya Urn ...
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### Polya urn with probability of ball replacement

Suppose we have an urn that initially contains $a$ red and $b$ green balls, where $a$ and $b$ are positive integers. At each discrete time, we select a ball from the urn and then return the ...
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### Is there a name for distinguishing these two types of polya urns? (Add another ball of same colour vs Only add for white balls)

I am wondering if there is a name for distinguishing these two kinds of Polya urns: When you draw a ball of any colour, replace it, and add one ball of the same colour. When you draw a white ball, ...
1 vote
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### Pólya's urn question

There are w white balls and b blue balls in a urn. We take one ball, see its color, return it to the urn and add another ball of the same color. Let's call $W_n ≡$ we get a white ball at the n-th ...
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### Exchangable series and replacement matrix of a Polya urn model

I'm given Polya Urn model with following parameters: initial state $w,b = (3, 1)$ and replacement matrix = $\bigg( \begin{matrix} a & 3 \\ 3 & 3 \end{matrix} \bigg)$ $X_i = 1$ if $w$ is drawn ...
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1 vote
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### Polya's Urn - probability of taking $k$ white balls in the first $n$ extractions

An urn contains $p$ black balls and $b$ white balls. A ball is taken from the urn and then the same ball is added back to the urn together with another ball of same color. This process is repeated ...
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### Closed Form for Number of Samples Required when Picking Without Replacement

So maybe i missed something simple, but I could not find material related to the following problem: Assume we have an urn with $N$ balls of which $k$ are colored black. Assume that $k$ is some fixed ...
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1 vote
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### What is the probability of drawing a red ball in the fourth trial? Find a simpler solution.

A bag has $r$ red balls and $b$ black balls. All balls are identical except for their colours. In a trial, a ball is randomly drawn from the bag, its colour is noted and the ball is placed back into ...
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### Batched Urn/coupon problem

The problem setting is as follows. We have an initial collection (urn?) of $N$ different coloured balls - say $C$ distinct colours. The distribution of different colors is non-uniform and unknown. In ...
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### Polya's urn as a counterexample for the Kolmogorov 0-1 law

Consider a simple formulation for the Polya urn model. An urn contains two balls at time 0, one is white and the other is black. At time $n\in\mathbb{N}$, one of the balls is chosen uniformly at ...
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### n black balls k white balls in M bins, what is the probability of selecting a black ball from any bin.

There are $n$ black balls and $k$ white, with $M$ bins. The process for filling the bins is as follows: Consider each of the $N=n+k$ balls, one at a time. For each ball, select 1 of the $M$ bins ...
1 vote
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### How can I solve this Martingale Problem with fractions

I am working on an exercise from previous years in my university that states that : In a village there are living N people and they decide with an interesting way about some actions. Specifically, if ...
1 vote
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### Probability of exactly one same-colored pair when drawing six balls from an urn consisting of five same-colored pairs.

Imagine there is an urn with $10$ colored balls. There are five colors present in the urn, and two balls of each color. What is the probability that there is exactly one pair of the same color when ...
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### A proof for Polya's urn model

Question An urn initially contains $r$ red and $b$ blue balls. At each stage, a ball is randomly selected and returned along with $m$ other balls of the same colour. Let $X_k$ be the number of red ...
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1 vote
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### Show that the number of red balls in Polyas Urn problem is uniformly distributed

I have the classic Polyas Urn problem. Let $t$ be the time steps $t \in \{1, \ldots,N\}$. At $t=1$ there is one red and one green ball in the urn. Let $R_n$ and $G_n$ denote the number of red and ...
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### Pólya's urn - Probability of drawing two same-colour balls in succession

Pólya's urn is a statistical model which uses the following scheme: there is an urn containing n black balls and m white balls. When a ball is drawn, it is then returned with an additional ball of the ...
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### Drawing from an urn - probability of drawing maximum 2 of each color.

I have an urn with 100 balls; 10 colors with each 10 balls. Now I draw 10 balls from the urn without replacement - I want to calculate the probability of getting 3 or more of any color. How do I ...
1 vote
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### Urn model: closed formula for: at least a given number of blue balls

I was discussing the following question with a friend but we are stuck, this is probably obvious for someone with a mathematical background. Say we have an urn with $N$ balls of which a $M$ are blue. ...
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### Polya's Urn for Three Colors Instead of Two

Is it possible to extend Polya's Urn problem to balls with $3$ different colors instead of just $2$? ie. An urn contains $1$ red, $1$ blue, and $1$ green ball. At each turn you draw one ball and put ... 30 views

### Negative binomial (and multinomial) extension to the minimum(failures,successes)

The negative binomial distribution defines the probability of N draws to obtain K successes. The negative multinomial gives the joint over k1...km given k0 successes. I need to find the PMF over n ...
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1 vote
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### Marginal Dirichlet Negative Binomial Distribution and the Multinomial Inverse Polya Urn

I have the following 'URN-like' problem - assume an urn the contains balls with m different colors. As in the standard Polya scheme, every time a ball is sampled, it is returned the urn in addition ...
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1 vote
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### There are red and blue balls in the urn, only put red ball back. What is the expected value?

There are b blue balls and r red balls in the urn. All balls currently in the urn have an equal probability of being selected each. When drawing a red ball, put it back in the urn, but a blue ball ...
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### Struggling with intuition about this probability question. Symmetry argument of two balls drawn from an urn.

So the question is as follows: An urn contains m red balls and n blue balls. Two balls are drawn uniformly at random from the urn, without replacement. (a) What is the probability that the first ... 1 vote
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### Estimating a parameter in a changing attachment probability of an urn model given the draw sequence

I want to find a way of measuring a parameter in a changing attachment probability of an urn model given the draw sequence. I hope this is the right place to ask these kind of questions, I apologize ...
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1 vote
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### Computing Beta Function for Polya's Urn Scheme

Suppose we have an urn that contains $r$ red and $g$ green balls. At each time we draw a ball, we look at the color, replace it, and add $c$ more balls of that color. Let $X_n$ be the ratio of green ...
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### Polya's Urn: Need Help Calculating Expected Value of Concentration of Red Balls at Stage n

An urn U contains $r_0$ red balls and $g_0$ green balls. At each stage a ball is selected at random from the urn, we observe its color, we return it to the urn and then we add another ball of the same ...
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### Probability of first to draw red ball from urn without replacement

There are R red and G green balls in a box. Alice and Bob pick one ball from the box in turns without replacement. First one to draw a red ball wins. What is the probability that Alice wins if she ...
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### Laplace urn scheme

(Laplace example). We have $r + 1$ urn. In each urn, there is a $r$ of balls, with Urn number i contains both white and $r-i$ and black balls. The drawing takes place in the following way. We choose ...
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### Find the probability of the event that we run out of white balls first

I have $a$ white balls, $b$ black balls, $c$ red balls in a basket. We take balls one by one and do not return. I have to find the probability of the event that we run out of white balls first. If I ...
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### Urn contains $i$ white balls and $(3-i)$ black balls. Let $T$ be the time there are only white balls left in the urn. Find $E[T]$

An urn contains $i$ white balls and $(3-i)$ black balls. Let $T$ be the time there are only white balls left in the urn. Find $E[T]$. At each time step we choose a ball at random and replace with a ...
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### Markov Chain and Polya's Urn Problem

An Urn contains a red and black ball. At each time $n \ge 1$, a ball is taken randomly, its color noted, and both this ball and another ball of the same color are put back into the urn (ie. the number ...
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### Urn problem with two ways of expectation computation

Let $X$ denote the number of white balls selected when $k$ balls are chosen randomly from an urn containing $n$ white balls and $m$ blacks. For $i＝1....k; j＝1....n$, \begin{align*} X_i&＝\begin{...
1 vote
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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. ...
1 vote
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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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1 vote
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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 ...
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### Combinatorics Formula Reference

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 ...
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### Beginner urn probability question

I am having issues getting my head around an urn problem. I have seven urns. First three urns contain 4 balls (1 red, 3 blue) and the last 4 urns contain 3 balls (1 red, 2 blue). If picking totally ...
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### Pólya’s Urn, but No Induction

Recall Pólya’s Urn: Initially there is 1 red and 1 blue ball in an urn. In each step, we select a ball from the urn uniformly at random, and then put it back together with a new ball of the same color.... 746 views

### Conditional Probability - Urn Problem

Given are two urns, urn $U_1$ contains $3$ black and $2$ white balls, and urn $U_2$ contains $2$ black and $3$ white balls. A fair coin is flipped to decide which urn we should draw from. We draw $2$ ...
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