1.1 Probability systems

Probability measures

In the module "Likelihood" we introduce the notion of a basic space $\Omega$ of all possible outcomes of a trial or experiment, events as subsets of the basic space determined by appropriate characteristics of the outcomes, and logical or Boolean combinationsof the events (unions, intersections, and complements) corresponding to logical combinations of the defining characteristics.

Occurrence or nonoccurrence of an event is determined by characteristics or attributes of the outcome observed on a trial. Performing the trial is visualized as selecting an outcome from the basic set.An event occurs whenever the selected outcome is a member of the subset representing the event. As described so far, the selection process could be quite deliberate, with a prescribed outcome,or it could involve the uncertainties associated with “chance.” Probability enters the picture only in the latter situation. Before the trial is performed, there is uncertainty about which of these latent possibilities will be realized. Probability traditionally is a number assigned to an event indicating the likelihood of the occurrence of that event on any trial.

We begin by looking at the classical model which first successfully formulated probability ideas in mathematical form. We use modern terminology and notation todescribe it.

Classical probability

1. The basic space Ω consists of a finite number N of possible outcomes.
   • There are thirty six possible outcomes of throwing two dice.
   • There are ${\displaystyle C(52,5)=\frac{52!}{5!47!}=2598960}$ different hands of five cards(order not important).
   • There are $2^5=32$ results (sequences of heads or tails) of flipping five coins.
2. Each possible outcome is assigned a probability $1/N$
3. If event (subset) A has N _A elements, then the probability assigned event A is
   $$P\left(A\right)=N_A/N\text{(i.e.,}\text{the}\text{fraction}\text{favorable}\text{to}A\text{)}$$

With this definition of probability, each event A is assigned a unique probability, which may be determined by counting N _A , the number of elements in A (in the classical language, the number of outcomes "favorable" to the event) and N the total number of possible outcomes in the sure event Ω .