Probability distribution

A probability distribution describes how the total chance, which always equals 1, is spread among all the possible outcomes of an experiment. To each outcome it assigns the probability of seeing it occur, so that the whole covers every case without forgetting any or counting any twice.

A good image is that of a kilogram of sand spread over the different outcomes. Where the pile is high, the outcome is likely; where it is thin, the outcome is rare. The shape of the pile tells the entire behavior of the chance involved, at a single glance.

The rule for building one is simple: each value is assigned a probability between 0 and 1, and the sum of all these probabilities must be exactly 1. For a fair die, the distribution is flat: each of the six faces receives 1/6, and six times 1/6 indeed gives 1. For the sum of two dice, the profile becomes a bell: a 7 can be obtained in six ways out of thirty-six, that is 6/36, whereas a 2 can be obtained in only one way, that is 1/36.

The distribution should not be confused with a single observed result. A draw gives one value; the distribution, on the other hand, describes the entire landscape of possibilities and their respective weights, before any draw.

We distinguish discrete distributions, which concern separate values such as the faces of a die, from continuous distributions, which spread over an interval, like an instant drawn at random within a span of time. The site's tools rely on well-controlled discrete distributions.

Knowing the distribution of a tool means being able to anticipate what it will produce in the long run: a number generator follows a flat distribution, whereas a draw combining several dice or several wheels naturally gives rise to a bell shape.