Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
21 views

Proof that Polya's Urn is a Martingale

In these Wisconsin Lecture Notes on probability, they sketch that, in Polya's Urn, the fraction of green balls at a step $n$ defines a martingale. I would like to justify these steps and make them ...
0
votes
0answers
13 views

Distribution of color observation in Polya-like urn model

Consider an urn containing balls of $K$ different colors with $a_i$ balls of colour $i$. According to Wikipedia, if we sample $n$ balls in a Polya's fashion, the distribution of the observed colors ...
1
vote
0answers
25 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, ...
2
votes
1answer
33 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 ...
0
votes
1answer
51 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 ...
2
votes
1answer
74 views

Variation of Polya’s urn problem?

I was asked this question at an interview and was stumped by it. The problem is as follows: Alice is playing an online game. She wins the first round and loses the second. The probability she then ...
3
votes
1answer
67 views

Drawing and adding marbles to and from a bag

I have a bag with one black marble and one white marble. Whatever color I draw from the bag, I put it back with another marble of the same color. I'm curious about the composition of the bag as I ...
3
votes
0answers
93 views

Martingale and Super-Martingale

A auto-machine pick up a ball uniformly randomly from a box filled with green balls and red balls in every minute. When the # of green balls or red balls = 0, the auto-machine do not pick. ...
0
votes
1answer
104 views

Pólya urn model and counting 'successes'

Let's consider the Pólya urn model. Inside an urn there are two kinds of balls in two different colours - black and white (b - the amount of black ones, w - the amount of white ones.) We draw one ball,...
2
votes
1answer
115 views

Probability of selecting exactly K colors from an urn of N colors?

Suppose we have an urn with $N$ different colored balls and $b$ balls of each color, for $Nb$ balls total in the urn. From this we randomly draw $r$ balls without replacement. What is the probability ...
0
votes
0answers
212 views

Polya's Urn and Conditional Independence

We use the basic set up of Polya's Urn, with r red balls and b black balls. Let us then denote Xn =1 if the ball picked is red and 0 if ball is black. Let us also denote Sn = (X1 +...+ Xn) to be the ...
2
votes
0answers
66 views

How to prove this conjectured identity related to hypergeometric series?

Question: How to prove the following conjecture? ... Conjecture: For all integers $r\ge 0$ and $b\ge 3$, $$ {\LARGE\sum}_{j=2}^\infty{\LARGE\sum}_{i=j-2}^\infty{(r)_i(b)_j\over (r+b)_{i+j}}\binom{i}...
1
vote
1answer
102 views

Why are urns so common in combinatorics questions?

My research is in combinatorics and graph theory, and I'm used to phrasing basic combinatorics questions in terms of balls and bins. However, many sources talk about balls and urns. I'd never thought ...
1
vote
1answer
67 views

How to calculate expected cost of a lottery game?

I'm trying to guess the design of a very complex lottery mechanism from a game to estimate its costs. Here the mechanism : Each pull costs 1$. Draws are with replacement. Chances to get an item is ...
1
vote
1answer
188 views

Polya urn: Mean number of draws to get a specific sequence of colors?

Polya urn model: At time $0$ an urn initially contains $b$ $\tt{B}$lue balls and $r$ $\tt{R}$ed balls. At time $1$, a ball is drawn uniformly at random (removing it) from the urn, and two balls of the ...
1
vote
0answers
183 views

martingale polya urn

I have this problem which i think I have solved but i am not quite sure if it is correct Consider an urn containing one white and one black ball. At each step a ball is selected uniformly at random ...
3
votes
3answers
110 views

Urns: solving a differential system

I'm working on an urn model where two urns, say A and B, are interacting in that at each step, a ball is drawn from each of them and then replaced adding a ball of the color drawn from the other urn. ...
4
votes
2answers
518 views

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 ...
2
votes
1answer
180 views

sampling without replacement in an urn with unknown parameters

I have an urn containing $N$ balls, of which $n$ ~ Uniform(1, ..., $N$) are white and the rest black. After sampling $i$ balls without replacement (of which $m$ are white), how can I find the ...
1
vote
1answer
109 views

An unusual box?

A box contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at ...
0
votes
1answer
269 views

Expected number of trials (without replacement) - Urn problem

12 white and 8 black balls are placed in an urn. What is the expected time of waiting on taking out the black ball from the urn, if we are taking balls without replacement. It should be solved by ...
1
vote
1answer
294 views

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 ...
2
votes
1answer
105 views

Urn problem, 2 players draw 2 balls

Suppose we have an urn with $x$ white and $y$ black balls, with frequencies $p$ and $q$, respectively. There are 2 players who draw 2 balls each and write down the number $k$ of black balls they got, ...
0
votes
1answer
1k views

Probability (Polya's urn)

An urn contains r red balls and b blue balls. A ball is chosen at random from the urn, its color is noted, and it is returned together with d more balls of the same color. This repeated indefinitely. ...
1
vote
2answers
137 views

Find the probability of both selected balls are the same colour (replacement and adding)

Question: A box initially contains 1 RED ball and 2 WHITE balls all the same size. You select a ball at random and then return it to the box together with another ball of the same colour. The box now ...
3
votes
0answers
87 views

How many total colors in the urn, given that I've seen C colors in M draws?

In this problem I want to estimate the unknown total number of colors of balls in the urn. Say that there are N total balls in the urn, and after m draws (with replacement, if that makes it easier), ...
0
votes
1answer
71 views

Probabilities when picking from two urns

Assume there are two urns with 4 balls each. Each ball has a certain probability to be picked. For example: Urn1: Ball A: 0.7 Ball B: 0.1 Ball C: 0.1 Ball D: 0.1 Urn2: Ball E: 0.3 Ball F: 0.2 Ball ...
0
votes
1answer
70 views

how to get the probability $p=1−(1−1/N)^{Tma_i}$ of extracting at least one ball in the urn

I am reading supplementary information of the paper Activity driven modeling of dynamic networks. It analogys the number of out degree of a activity node by Polya urns problem: it will equal to ...
1
vote
0answers
115 views

Friedman's urn is a supermartingale or a submartingale?

Here is the urn model: At time zero there are $r$ red and $g$ green balls in an urn. At each time-step, we draw out a ball at random and replace it along with $c$ of the same color and $d$ of the ...
1
vote
1answer
73 views

Conditional probabilitiy in balls in urn

A urn has 10 white balls and N black balls. We draw balls without replacement. Given that the second ball drawn from the urn is black, how many black balls must be in the urn such that there is a 1/...
0
votes
1answer
41 views

Does one-to-one correspondence of the set of sample paths of $X$ and $Y$ imply that $X$ and $Y$ are indistinguishable?

I have two discrete stochastic processes (urn models) $X(t)$ and $Y(t)$. There is a function $\varphi(.)$ such that whenever $\{x_0,x_1,...\}$ is a sample path of $X$ then $\{\varphi(x_0),\varphi(x_1),...
0
votes
1answer
34 views

multivariate hyper geometric Vs. Indep. drawing without replacement

I think my problem can be seen as a basic urn setup, with another layer tacked on after the fact. I'm looking at drawing from an urn with 3 biological specimens. Each has genotype AA, Aa, or aa. ...
1
vote
0answers
163 views

Polya urn probabilities [closed]

Consider an urn containing two type of balls (red and blue), initially there are $r_0$ red balls and $b_0$ blue balls. At each stage $n\geq 0$ we choose at random one ball, if the ball is red we put ...
5
votes
2answers
69 views

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 ...
-1
votes
1answer
448 views

What is the distribution of the limit of random variables in a problem involving Polya's Urn?

I know this has been asked elsewhere, but I think the values or random variables are different or something. From Williams' Probability with Martingales: I proved that $M_n$ is a $\sigma(B_1, ..., ...
1
vote
1answer
153 views

References for a Variant of Polya's Urn Scheme

A row of $n_1$ balls, all coloured with colour #1, is laid out before me. I do a set of $k$ replacements (where $k<n_1$) in steps: Step 1: Choose a (uniform) random number $n_2$ on the range $[1,...
0
votes
2answers
445 views

Polya's Urn : proportion Probability

There are two urns with one ball each. Each of subsequent n-2 balls is placed into one of these urns, with probability proportional to the number of balls already in that urn. What is the expected ...
6
votes
2answers
981 views

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 ...
0
votes
1answer
1k views

Consecutive Draws from Urn

An urn contains 4 black balls and 6 white balls. What is the probability of getting 1 black ball and 1 white ball in two consecutive draws from the urn? I know this is supposed to be easy. So i tried ...
1
vote
0answers
100 views

Discrete time Pure-Birth process with population fraction

I would like to solve difference equations for a pure-birth process, where the rate of adding new nodes depends on the fraction of population. $$x_i(t+1)-x_i(t)=\alpha \frac{x_i(t)}{\sum{x_j(t)}}+\...
0
votes
1answer
374 views

Urn problem: with and without replacement the probability of seeing last ball of particular color

suppose you have red and blue balls in a jar and you pick one at a time, if its red you replace it, additionally you do another draw and throw whichever ball comes up. if its blue you throw it away ...
1
vote
1answer
146 views

Counting Operations in the context of an Urn Problem

I was tasked with the following question, regarding the counting of operations in the pseudo code provided that has nested for loops: Let U ={B1,B2,...,Bn} with n >= 3. Interpret the following ...
1
vote
1answer
119 views

Resampling Probability

I've had a student working on some resampling problems. Here's a question I've posed to her: M consecutive draws of size K are taken from an urn with N balls. Balls are replaced between draws. ...
0
votes
2answers
561 views

Formula for urn model with replacement and “color switching”

I tried to figure out following problem, but failed :/ My best guess is that it somehow is a mixture of a binomial and a hypergeometric distribution: I have an urn with N white balls and 0 black ...
3
votes
1answer
4k views

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 ...
4
votes
1answer
1k views

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 ...
3
votes
0answers
211 views

Multicolored Polya Eggenberger Urn distributions

From an urn containing 1 ball each of $n$ different colours, a ball is drawn, its colour noted and then it is returned to the urn along with $k$ balls of the same colour. what is the limit of the ...
3
votes
2answers
101 views

Number of draws to see every object at least once

I woke up this morning with the following question on my mind: What is (the distribution of) the number of draws you need to sample (with replacement) from an urn with $n$ distinct objects until you ...
9
votes
2answers
5k views

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 ...
0
votes
1answer
782 views

Symmetry argument for Pólya's urn scheme

In case of Pólya's urn model, an urn initially contains $r$ red balls and $b$ blue balls. At each stage a ball is randomly selected from the urn and is then returned along with $m$ other balls of the ...