Urn model

The urn model is the standard thinking tool for reasoning about probabilities. It consists of representing a chance situation as a container filled with balls, usually of different colors, that you draw blindly. This deliberately stripped-down abstraction makes it possible to reduce varied problems to a single simple question: what is the composition of the urn, and how are the balls drawn from it.

The value of the model lies in its clarity. You do not see the balls at the moment of the draw, so no preference can come into play, and each ball present has the same chance of being grabbed as its peers of the same status. Computing the chances then becomes a simple count: the probability of drawing a certain color is the number of balls of that color divided by the total number of balls. An urn containing three red balls and one black thus gives a one in four chance of drawing the black, and three in four of drawing a red.

The model also makes concrete the difference between sampling with and without replacement. If you put the ball back after observing it, the urn regains its initial composition and the chances stay constant. If you keep it out, the urn empties little by little and the probabilities change with each removal. The same image therefore serves to illuminate both modes.

A classic trap is to forget that the chances depend on the actual composition of the urn, and not on what you hope for or on what has just come out. A red ball drawn and then put back does not make the next one "more likely" to be black: the urn ignores the past. Believing otherwise leads to false reasoning.

On the site, all draws come down to this model. Drawing a name, choosing a card, designating a color, or selecting a participant always amounts to reaching into an abstract urn and pulling out one item from a well-defined set.