Urn problem

In probability theory and statistics, an urn problem is an idealized thought experiment in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties. A number of important variations are described below.

An urn model is either a set of probabilities that describe events within an urn problem, or it is a probability distribution, or a family of such distributions, of random variables associated with urn problems.

Basic urn model
In this basic urn model in probability theory, the urn contains x white and y black balls, well-mixed together. One ball is drawn randomly from the urn and its color observed; it is then placed back in the urn (or not), and the selection process is repeated.

Possible questions that can be answered in this model are:
 * Can I infer the proportion of white and black balls from n observations? With what degree of confidence?
 * Knowing x and y, what is the probability of drawing a specific sequence (e.g. one white followed by one black)?
 * If I only observe n balls, how sure can I be that there are no black balls? (A variation on the first question)

Example
One example of an urn problem is the binomial distribution: the distribution of the number of successful draws (trials), i. e. extraction of white balls, given n draws. This is of great importance for players because basically all in-game random processes incolving random containers like Pro Kit Boxes or Card Packs are realizations of the urn model: Instead of balls, cards are drawn from the container. Obtaining a desired card (e.g. a blueprint or an engine card) is a success, not obtaining it is a failure.