Probability

A probability is a number representing the likelihood of a random event or an uncertain proposition occurring, ranging from 1 representing certainty down to 0 for impossibility.

Probability is the topic of probability theory, a branch of mathematics concerned with analysis of random phenomena. Like algebra, geometry and other parts of mathematics, probability theory has its origins in the natural world. Humans routinely deal with incomplete and/or uncertain information in daily life: in decisions such as crossing the road ("Will this approaching car respect the red light?"), eating food ("Am I certain this food is not contaminated?"), and so on. Probability theory is a mathematical tool intended to formalize this ubiquitous mental process. The probability concept is a part of this theory, and is intended to formalize uncertainty.

There are three basic ways to think about the probability concept:


 * Bayesian probability.
 * Frequentist probability.
 * Axiomatic probability.

Bayesian probability
According to Bayes Theorem, probability is taken as a measure of how reasonable a belief is in light of prior observations and theoretical considerations.

Example of the Bayesian viewpoint
A Bayesian may assign a probability of 1/2 to the proposition that there was life on Mars a billion years ago. A frequentist would not do that, since one cannot say that the event that there was life on Mars a billion years ago happens in half of all cases; there are no such cases.

Frequentist probability
In this approach one views probabilities as reflecting the frequencies of the various outcomes, if an infinite number of tests is carried out.

Example of the frequentist viewpoint
The probability of "heads" when flipping a fair coin is 50%, because when we flip it an infinite number of times, that's what the frequency will be.

The axiomatic approach
Neither the Bayesian nor the frequentist approach really give us a satisfactory formal structure for development of a rigid theory.

The axiomatic approach takes a different tack. Instead of focusing on the question "What is probability?" we step back and ask "How does probability work?"

We then focus on abstract mathematical concepts such as sets, measure, sigma algebras and a set of rules we expect probability to follow, known as Kolmogorov's axioms.

But the ultimate justification for this approach rests on experience, too. Fortunately, the results derived by applying these axioms accord very nicely with experience. We can also derive Bayes' theorem as a consequence, and we can show that in a large number of trials, the frequencies will give us a good estimate of the probabilities; or at least it will do so on average.

Example of the axiomatic approach
Given a standard deck of card. We want to draw a card at random.

This experiment can be modeled by a set of 52 possible outcomes. Any set with 52 elements has exactly $$2^{52}$$ subsets. To each of those subsets we may assign a certain number (a "probability"), so that certain axioms are satisfied. We choose to assign probability .25 to the subset consisting of all 13 hearts. We also assign .25 the the subset consisting of all spades.

According to the axioms, we must then assign probability 0.5 to the subset consisting of every spade and heart in the deck.

In non-axiomatic informal terms, we would describe this result thus:

If the probability that a card (drawn at random from a standard deck of cards) is a heart is .25, and the probability that it is a spade is also .25, and if I know that the being a heart and being a spade are mutually exclusive possibilities (i.e., a card cannot be both), then the probability that it is a heart or a spade is 0.5 = 0.25 + 0.25.

More technical information

 * Bayes' theorem
 * principle of maximum entropy
 * Probability distributions
 * Kolmogorov's axioms
 * probability theory