Probability is the mathematical theory we use to describe and quantify uncertainty. Uncertainty can be due to our ignorance, deliberate mixing or shuffling, or due to the essential randomness of Nature. The simplest form of the theory begins with a universe, a finite set of elementary events. We define a weighting function which maps each elementary event into a non-negative number called the weight of the elementary event. We can represent any event by a subset of the universe. The probability of the event is the sum of the weights for all the elementary events in the subset divided by the sum of the weights for the whole universe. Calculation of probabilities can often be determined using CombinaTorics or by applying the definition above directly.
A probability distribution describes a special universe, a set of real numbers (see AnaLysis?) and how probability is distributed among them to determine a random variable. For every random variable, there is a function called the cumulative distribution function which provides the probability that a given value is not exceeded by the random variable.
Statistics is a branch of applied mathematics which includes planning, summarizing, and interpreting uncertain observations. We describe a universe of possible observations and the probabilities we associate with each. This allows us to plan our observations to control their variability, summarize a collection of observations to feature their communality by suppressing details, and reach consensus about what the observations tell us about the world we observe.
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