Probability

Overview

The axioms and definitions of probability set out the logic and assumptions of a statistical model.

Probabilistic reasoning begins with a set of outcomes over which we assign probabilities. As an example, consider flipping a coin twice. THe different outcomes are, HH, HT ,TH, TT, where H represents a heads, and T represents a tails.

Events

Once we have identified the possible outcomes, we can then identify events, which are just subsets of the set of all outcomes. For example, the event, first coin is a heads is represented by the following set {HH,HT}. That is, there are two outcomes that fall under that event. Note that an event can be a set of one outcome, that is, the event, "first coin is tails, second is heads" is only satisfied by one outcome.

Conditioning

• Conditional Probabilities - are probabilities given a certain set of information. New information will typically change the probability of any given event.

Probability as Measure

The formal development of probability is developed within Measure Theory where a probability space is a measure space where the measure of the universe is set to 1