Probability

Probability is a number representing an estimate of how likely an event is, ranging from 1.0 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 two basic ways to think about the probability concept:

• Bayesian probability.
• Frequentist ("objective") probability.

The different approaches are largely pedagogical, as some people find one approach or the other much easier.

Bayesian probability

In this approach, probability is taken as a measure of how reasonable a belief is in light of experience or observations. It is based on a rigorous relationship between what are called conditional probabilities and ordinary (non-conditional) probability. It is thus, not simply an intuitive or educated "guess", but something much more specific and precise.

Example of the Bayesian viewpoint

How likely is it that it will rain today? If I'm inside a room with no windows and cannot look outside and see whether there are any clouds in the sky or wshether the wind is blowing, then I do not have this information available, and cannot use it to inform my estimate of how likely it is that it going to rain. But of the last 100 cloudy days I've experienced, I've noticed that it rained on 20 of them, and that of days without a cloud in the sky it has only rained on five (because a storm blew in later in the day), I will conclude that rain is more likely on cloudy days. In fact, this can be made precise using a formula known as Bayes' theorem which expresses the probability of rain given that it is cloudy in terms of the probability that it will rain on a cloudy day (something I can estimate by direct observation), the probability that it will rain on a given day, and the probability that it will be cloudy on a given day (both of which I can estimate by direct observation).

Objective probability

In this approach one views probabilities as "propensities" of the actual system under study - f.i. a fair coin will have a "propensity" to show heads 50% of the time. This approach is more restrictive than the Bayesian interpretation: F.i. there is no way to assign a probability as to whether or not there exists life in the Andromeda galaxy this way, since no "propensities" have been measured.

Example of the objective viewpoint

We are given a die, and a list of it's measured (or theoretical) "propensities", i.e. probabilities for each possible outcome. We then use this information to calculate the "propensities"/probabilities of certain outcomes. If our (empirical) results seem improbable, we may decide to do experiments to re-measure the "propensities".

More technical information

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

Links

• statistics
• subjective probability
• objective probability

External links

• Intros
  • http://www.mathgoodies.com/lessons/vol6/intro_probability.html
  • http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html

• More advanced
  • http://bayes.wustl.edu/
  • http://plato.stanford.edu/entries/probability-interpret/