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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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Question on the expected number of same color balls left in a urn

I'm working on a problem where I am given an urn with $a$ white balls and $b$ black balls. One ball at a time is selected randomly until there is only balls of the same color. I am asked to find the ...
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Polya urn scheme by induction task

I've been trying to solve the task given in "An Introduction to Probability Theory and Its Applications" by William Feller. The task is to show by induction that the probability of a black ...
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Asymptotic behaviour of the difference process of the Polya urn

Assume we have a Polya urn with $(a,b)$ initial (time t=0) red and green balls, respectively. At each point in time we draw a ball at random and add it back together with further $S$ balls. This means ...
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We draw a ball, look at it, then return it to the urn and add another ball of the same color. Pr: "after $n$ draws the urn contains $j$ white balls"?

The urn contains $1$ white ball and $1$ black ball. We make a sequence of $n$ draws according to the following scheme: we draw a ball, look at it, then return it to the urn and add another ball of the ...
thefool's user avatar
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How to get this formula for expectation of continuous-time urn process

We define the continuous-time, multi-type branching process $(X(t))_{t\ge0}$ as follows: $(X(0))=\alpha\in\mathbb{R}^d$, where $\alpha=(\alpha_1,\dots,\alpha_d)$ is the urn initial composition, ...
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A multidimensional Central Limit Theorem for Pólya Urns

I know that if we fix $\alpha_1,\dots,\alpha_d\ge1$ and $S\ge1$ integers, and we define the $d$-dimensional process $(Z(n))_{n\ge0}$ as follows: let $V$ be a Dirichlet random variable of parameter $\...
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Probability of $k$ successes in $n$ trials where chance of success is $\frac1x$

I'm looking for a formula to find the probability of exactly $k$ successes in $n$ trials where the initial probability of success is $\frac1x$, with $x\ge1$, and $x$ is incremented on each trial. The ...
alec's user avatar
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Convergence of Pólya urns: what does it mean "expanding real polynomials in a basis"?

I'm reading these notes, and in the proof of this Proposition 2, which states: Let $d\ge 2$ and $S\ge 1$ be integers. Let also $(\alpha_1,\dots,\alpha_d)\in\mathbb{N}^d \setminus\{ 0 \}$. Let $(P_n)_{...
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On the proof of convergence of Pòlya urns

I'm reading the proof of Proposition $2$ of the Appendix here. The proposition states Let $d\ge 2$ and $S\ge 1$ be integers. Let also $(\alpha_1,\dots,\alpha_d)\in\mathbb{N}^d \setminus\{ 0 \}$. Let $...
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Urns and marbles problem with shuffling

An urn "A" contains 2 white marbles. Another urn "B" contains 1 white marble and 1 black marble. One white marble is drawn, not being known which urn it was drawn from. Then, the ...
conierju's user avatar
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Polya's urn scheme - Proof by induction [duplicate]

I created this proof (with the help of this post) because I was struggling with this type of problem, and I wanted to share it here in case someone else needs it. Exercise 17 Let's describe the ...
Gabriel Ribeiro's user avatar
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2 answers
326 views

Pólya's urn - Probability first ball is blue given subsequent draws

I refer to the solution contained in this post here. I know it must be simple but I cannot deduce why $$ \begin{align*} P(B_1|B_2\cap \dots \cap B_{n+1})&=P(B_{n+1}|B_1\cap\dots\cap B_n) \end{...
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balls in bins with finite capacity and overflow: mean ball overflow

Say you have $M$ bins and $N$ balls, and each bin can only be filled with $C$ balls. At step zero, all bins are empty. Then at each step, one of the $M$ bins is chosen with equal probability, and a ...
user3433489's user avatar
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1 answer
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Polya's urn, should I use martingales or LLN

I am trying to prove the following question, but I am finding it a bit tricky to determine the distribution of $X_i$ (the number of red balls drawn in the $i$-th round) and thus I do not know which ...
idlatva's user avatar
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Limit of a probability for Polya’s Urn (from Durrett’s textbook)

My question comes from 1st expression in Page 227 of Durrett’s Probability textbook (5th edition). It says: by taking $n \to \infty$ and $m/n \to x,$ we have $$\frac{n!}{m!(n-m)!} \frac{(m+1)!(n-m)!}{(...
Yung-Hsiang Huang's user avatar
2 votes
1 answer
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Establishing bounds on Pólya's Urn for three colors and two draws after $N$ draws

I encountered the following Pólya's Urn setup: Suppose originally we have $1$ red ball, $1$ blue ball, and $k$ green balls. Each turn we draw a ball from the urn uniformly at random and replace it ...
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Polya's Urn Application, Tossing Paper into a Bin

My question is regarding the problem in this video: https://www.youtube.com/watch?v=BcIDeZ2nLf0&list=PLADLdjl79_mhdmhdxA1Gk8sklGEWJkEKq You are tossing 300 pieces of crumpled paper into a bin. You ...
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Variation of the Polya Urn Model

An urn contains ${b \ge 1}$ blue balls and one red ball. At each stage a ball is randomly chosen from the urn and then replaced along with a new ball of the same color. Let $T$ denote the first step ...
KoKo Chogiwa's user avatar
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Arranging triplets from an urn and partitioning them.

Say we have an urn containing 3N orbs of three different colors: red, green and blue. N of them of each color. The orbs are Arranged necessarily in sets of three, we'll call them troikas. there are 10 ...
Antonio Bernardo's user avatar
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What is the probability distribution (Moran Process where the other colour is always removed)

The Problem Suppose there are $H$ balls in a jar, with $X_i(0)$ white balls and $H-X_i(0)$ blacks balls. At each step, we randomly choose a ball from the jar. If a white ball is chosen, we replace it ...
kiston's user avatar
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Probability of getting two balls of different colours.

Question: An urn contain 9 balls three red four blue and two are green. Three balls are drawn at random without replacement from the urn. What is the probability that all three balls different colour. ...
Pratham Sinha's user avatar
4 votes
1 answer
207 views

Azuma's inequality for a simple case of Polya's urn

Suppose that an urn contains one red ball and one blue ball. A ball is drawn from the urn uniformly at random. After that, the ball is put back into the urn and another ball of the same colour is ...
3nondatur's user avatar
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2 votes
1 answer
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What is the probability that a marble from the urn has been picked up by exactly $n$ people?

An urn starts with $m$ marbles and is then approached by $p$ people, each of which picks up $k$ marbles, discarding one and returning the rest to the urn. The urn now has $m - p$ marbles remaining. ...
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3 votes
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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 ...
GWA 's user avatar
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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, ...
rudytheduck's user avatar
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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 ...
John He's user avatar
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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 ...
Jonathan1234's user avatar
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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 ...
Marcos RF's user avatar
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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 ...
David's user avatar
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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 ...
Bruno Reis's user avatar
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2 votes
1 answer
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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 ...
Hajaku's user avatar
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2 answers
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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 ...
Ramasamy Kandasamy's user avatar
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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 ...
tdc's user avatar
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3 votes
1 answer
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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 ...
Irene's user avatar
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1 answer
287 views

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 ...
AshleyPerry's user avatar
1 vote
1 answer
77 views

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 ...
Homer Jay Simpson's user avatar
1 vote
3 answers
227 views

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 ...
user6322383's user avatar
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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 ...
Ethan Mark's user avatar
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1 vote
1 answer
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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 ...
Nullspace's user avatar
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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 ...
Michaela's user avatar
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3 answers
99 views

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 ...
LasagneAlForno's user avatar
1 vote
1 answer
41 views

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. ...
Beginner's user avatar
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2 answers
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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 ...
user avatar
2 votes
0 answers
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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 ...
Isaac's user avatar
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1 answer
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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 ...
Isaac's user avatar
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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 ...
Momo N's user avatar
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4 votes
3 answers
500 views

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 ...
user avatar
1 vote
0 answers
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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 ...
Nik's user avatar
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1 answer
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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 ...
user389056's user avatar
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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 ...
momo's user avatar
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