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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.

In statistics, a Pólya urn model (also known as a Pólya urn scheme or simply as Pólya's urn), named after George Pólya, is a type of statistical model used as an idealized mental exercise framework, unifying many treatments.

In an urn model, objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. 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, and the selection process is repeated. Questions of interest are the evolution of the urn population and the sequence of colors of the balls drawn out (Wikipedia).

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Optimal strategy for 1-player "snowball" game

This particular game, which we can call "snowball", has one player and an operator who runs it. It begins with an urn with one blue ball and one red ball inside. The first round then starts, with each round working as follows: The operator picks a…
badroit
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a problem on Polya's urn scheme

In an urn with $b$ blue and $r$ red balls, each time (call it a trial) a ball is chosen at random and then put again in the urn along with $c$ extra balls of the same color. Now probability of getting a blue ball in the 1st trial = $\frac{b}{b+r}$.…
user21982
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Intuitive/heuristic explanation of Polya's urn

Suppose we have an urn with one red ball and one blue ball. At each step, we take out a single ball from the urn and note its color; we then put that ball back into the urn, along with an additional ball of the same color. This is Polya's urn, and…
Michael G
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Polya's urn model - limit distribution

Let an urn contain $w$ white and $b$ black balls. Draw a ball randomly from the urn and return it together with another ball of the same color. Let $b_n$ be the number of black balls and $w_n$ the number of white balls after the $n$-th…
Max93
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Show rigorously that Pólya urn describes a martingale

We work with the famous Pólya urn problem. At the beginning one has $r$ red balls and $b$ blue ball in the urn. After each draw we add $t$ balls of the same color in the urn. $(X_n)_{n \in \mathbb N}$ is the share of red balls in the urn after…
zesy
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Understanding a generalisation of Pòlya's urn (k balls drawn each turn instead of 1)

In interested in a variation on Pòlya's urn. My aim is mostly just to understand its behaviour, but I also want to know whether there's anything special about it, i.e. whether it is in some sense a natural generalisation of the original. In Pòlya's…
N. Virgo
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Negative Hypergeometric Distribution expectation

I am reading Introduction to Probability by Blitzstein and Hwang - Expectation. The book states : An urn contains $w$ white balls and $b$ black balls, which are randomly drawn one by one without replacement. The number of black balls drawn before…
Quasar
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Probability of non-intersecting Polya Urn sequences

Suppose you have a standard Polya Urn process: as the initial step you have $1$ red ball and $1$ blue ball in an urn; at further each step you draw a ball and then replace it together with another ball of the same colour, so at step $n+1$ (i.e.…
Henry
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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 balls drawn in the first $k$ draws. Conjecture the…
Ethan Mark
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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 the urn and put back in the urn together with a…
Olivier
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What is the number of binary strings of length N with exactly R runs of ones, with C total ones?

I'm concerned with the total number of ones, and the total number of runs, but not with the size of any of the runs. For example, $N=8$, $R=3$, $C=5$ includes 11101010, 01101011 among the 24 total possible strings. I can compute these for small $N$…
monguin
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Probability of every ball occurring in multiple independent random samples

An urn contains 5 distinct numbered balls. You choose 2 without replacement. You then reset the urn and choose another 2 without replacement. Do this one more time. Now you have three random samples of size 2. What is the probability that all of the…
Paul Loach
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Expected number of red balls removed from an urn before the first black ball

Question: An urn contains n+m balls of which n are red and m are black. They are withdrawn from the urn one at a time and without replacement. Let $X$ be the number of red balls removed before the first black ball is chosen. We are interested in…
user211962
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How long will it take for Polya's Urn to have the same Ratio as it had in the Beginning?

Recently I learned that in Polya's Urn starting with $m$ red balls and $n$ blue balls ... as time goes to infinity, the ratio of red to blue balls converges to its initial ratio (Polya's urn model - limit distribution). Intuitively, I interpret this…
konofoso
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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, meaning that, at time $0$, there are $\alpha_i$…
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