A Brief Introduction to Probability Theory

If a coin is tossed, then there are two possible outcomes: heads and tails. Statements such as ‘the chance that the outcome will be heads is 50%’ are formalized in probability theory, giving a better way to reason about the odds of a certain outcome of an experiment. The possible outcomes of an experiment are called the sample space. For example, the sample space of the coin tossing experiment is Ω = {head, tail}. Certain subsets of the sample space are called events, and the probability of these events is determined by a probability measure. For instance, if we roll a dice, then one of its six face values is the outcome of the experiment, so the sample space is Ω = {1, 2, 3, 4, 5, 6}. An event is a subset of the sample space Ω. The event {1, 2} occurs when the dice shows a face value less than three. The probability measures describes the odds that a certain event occurs, for instance Pr[{1, 2}] = 1/3 means that the event {1, 2} will occur with probability 1/3. A probability measure is not necessarily defined on all subsets of the sample space Ω, but just on all subsets of Ω that are considered events. Nevertheless, we want to have a uniform way to reason about the probability of events. This is accomplished by requiring that the collection of events form a σ-algebra. A σ-algebra F is a collection of subsets of the sample space Ω such that the following requirements are satisfied: S1 The empty set is contained in F . S2 If a set E is contained in F , then its complement Ec is contained in F . S3 The countable union of sets in F is contained in F .