# 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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### 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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### 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 ...
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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 ...
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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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### 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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### 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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### 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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### 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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### 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, ...
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 ...