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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141 views

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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Polya's Urn: Prove that: $N_n = \frac{(n+1)!}{B_n!(n-B_n)!} p^{B_n}(1-p)^{n-B_n}$ is a Martingale.

So basically Polya's Urn is where at time 0 you have one black ball and one white ball in an urn. At each integer time, a ball is chosen at random from the urn and replaced together with a new ball of ...
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73 views

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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42 views

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 ...
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22 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. ...
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87 views

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 ...
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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 ...
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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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1answer
75 views

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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Probability distribution defining sequence of Bernoulli trials where probability of success depends on previous success?

I have a process in mind that I'm trying to derive an expected value for: Step 1: A group of size n-1 exists. Each individual in the group observes a guaranteed event with probability r (binomially ...
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107 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 ...
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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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87 views

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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43 views

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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148 views

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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38 views

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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73 views

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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39 views

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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161 views

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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95 views

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{...
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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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1answer
50 views

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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1answer
44 views

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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142 views

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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184 views

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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384 views

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....
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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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Polya's urn scheme from Feller Volume 1

(Feller Volume 1, (5.8.20)) Prove by induction: for any $m < n$ the probabilities that the $m$th and the $n$th drawings produce (black, black) or (black, red) are $\frac{b(b+c)}{(b+r)(b+r+c)}$, $\...
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Urn model for birth statistic

I have an exercise which reads like this: A hospital has a statistic about how many boys and girls are born. 40% of all newborns are female, the others are male. Calculate the probability for the ...
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53 views

A nonlinear polya urn model

We have $b$ black balls and $w$ white balls at time $t$. At the time $t+1$, we generate two uniformly distributed random variables $X,Y$ in $[0,1]$. If $bX \ge wY$, we add a black ball; otherwise, we ...
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333 views

Poly Urn - is it a Markov chain?

THE PROCESS: Consider an urn which initially has $w$ white balls and $b$ black balls. Make a sequence of draws from the urn as follows. Draw one of the balls in the urn at random, then put this ball ...
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Meaning of 'independency'

Given a sequence of random variables $X_n$, what does it mean that they are not an "independency"? I have never met such terminology before. Is it just bad English to say that they are not mutually ...
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79 views

Bounding ratio of joint and marginal pmfs for urn model

Setup Suppose I have $N=mn$ balls in an urn, with $N_1$ red and $N_0$ black ($N=N_1+N_0$). Now suppose that I perform $m$ draws of $n$ balls ($n\ll N$), each time without replacement. Let $Y_1,\dots,...
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Polya's Urn process

I am aware of the mathematical definitions and its general results of the polyps urn process. For example if you draw a green ball, you add (replacement) the same colored ball back in to the urn (in ...
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151 views

probability of the $n$-th ball drawn is white is same as probability of first ball drawn

Consider an urn which initially has w white balls and b black balls. Draw one of the balls in the urn at random, then put this ball back into the urn and add another ball to the urn of the same color ...
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1answer
115 views

Probability for $n$-th draw is same as the first one

Consider an urn, which initially has $w$ white balls and $b$ black balls. Make a sequence of draws from the urn as follows. Draw one of the balls in the urn at random, then put this ball back into the ...
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How to calculate expectation precisely

I have a next model. Let two balls be colored in different colors: $\color{red}{\text{red}}$ and $\color{blue}{\text{blue}}$. Colors have intensities, from $0$ upward. Suppose, the maximum intensity ...
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451 views

Polya urn - Computation of an expectation

Suppose that at time 0 I have a urn with a red ball and a white ball. At time $n$ I take a random ball with uniform probability and return this ball with one extra ball of the same colour to the urn. ...
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131 views

Number of ways to pick out pairs of balls from an urn

We have an urn with $2N$ balls, of four colors $(a,b,c,d)$. I want to know how many ways there are to pick pairs of balls until the urn is empty. Note that $(x,y)$ pair is equivalent to $(y,x)$. Also ...
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The cross entropy of Pólya Gamma distribution

Is there analytical solution for computing the cross entropy of Pólya Gamma distribution? For example, $$\int_{\omega} P_{PG}(\omega|1,c)*\log P_{PG}(\omega|1,0)d\omega$$
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1 Black and 1 White Ball in an Urn Black Balls Added After Each Black Draw

As in title I am looking for the answer to the following problem: We have an urn with 1 black and 1 white ball in it. If you pick a black ball, then you put in another black ball each time until you ...
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1answer
116 views

Calculate the probability that the $i$-th draw returns a red ball.

$R$, $B$, and $k$ are positive integers. An urn initially contains $R$ red and $B$ black balls. A ball is drawn at random and its colour is noted. The ball is put back in the urn along with k balls of ...
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1answer
79 views

Polya's scheme for two urns

I don't know why I can't solve this problem using Polya's method .$$$$ There are two boxes,one of which contains 5 black and 7 white marbles and the other have 2 black and 2 white marbles .randomly ...
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How many draws from an urn given c colors have been drawn?

I have an urn with N balls each with a different color and I am sampling them with replacement. I keep track of which colors I have seen. Given the number of unique colors observed I would like to ...
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76 views

What is the sample space in conditional probability?

In conditional probabilities, the sample space at every step is dependent on previous outcomes. How do we then define a sample space? In particular, why is it possible to define the sample space of, ...
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1answer
200 views

Write down the sample space of this experiment and give the distribution.

An urn contains $4$ red balls and $6$ black balls. You draw with replacement $2$ balls. a) Write down the sample space of this experiment and list $3$ events $A$, $B$ and $C$, whose intersection is ...
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286 views

Urn problem: probability of m drawing balls for n simulation

Problem Definition : Inputs: j - the number of ball colours k - quantity of each ball colour i - total number of balls available (j*k) m - number of ball to randomly pick n - number of times to run ...