Events
A subset of a sample space is called an event. An event that includes a single outcome of an experiment is called a simple event. A compound event is a collection of one or more outcomes for an experiment. The set of possible events, usually denoted by Σ, needs to satisfy the following three postulates:
Note: A set that satisfies the above three requirements is called an algebra. Therefore, Σ is sometimes called the event algebra.
Example: For the experiment of rolling two fair but different dice, the outcome set consists of 36 possible pairs of numbers from [1..6]:
 (1, 1)  (1, 2)  (1, 3)  (1, 4)  (1, 5)  (1, 6)
 (2, 1)  (2, 2)  (2, 3)  (2, 4)  (2, 5)  (2, 6)
 (3, 1)  (3, 2)  (3, 3)  (3, 4)  (3, 5)  (3, 6)
 (4, 1)  (4, 2)  (4, 3)  (4, 4)  (4, 5)  (4, 6)
 (5, 1)  (5, 2)  (5, 3)  (5, 4)  (5, 5)  (5, 6)
 (6, 1)  (6, 2)  (6, 3)  (6, 4)  (6, 5)  (6, 6)
Let the event ω consist of the rolls with a total of 7 points. Then
\[ \omega = \left\{ (1,6), \ (2,5),\ (3,4), \ (4,3), \ (5,2), \ (6,1) \right\} \]
and its cardinal number is |ω| = 5.    ■
A probability or probability measure is a real-valued function Pr: Σ ⟼ [0,1] over some event algebra Σ that satisfies the following properties.
Let us thoroughly analyze every axiom. The first axiom states that probability cannot be negative. The smallest value for Pr(&emega;) is zero and if Pr(ω)=0, then the event ω is called impossible event. When a sample space is finite, then an impossible event is the empty set ∅ because an event is a subset of the sample space, which is the set of all possible outcomes of a probabilistic experiment. So we always have
\[ \Pr \left[ \varnothing \right] = 0 \qquad\mbox{and}\qquad 0 \le \Pr \left[ \omega \right] \le 1 \]
for any event ω. However, when a sample space is infinite, it does not mean that such event will never happen. There could exist a not empty event that may have zero probability. For example, if we weigh a new born baby, his or her weight is random number from a certain finite interval. On the other hand, the probability of the weight of a new born baby is exactly 4 kg has probability zero. The second axiom states that the probability of the whole sample space is equal to one, i.e., 100 percent. The reason for this is that the sample space Ω contains all possible outcomes of our random experiment. Thus, the outcome of each trial always belongs to Ω, i.e., the event Ω always occurs and Pr(Ω)=1. In the example of rolling a die, Ω={1,2,3,4,5,6}, and since the outcome is always among the numbers 1 through 6, Pr[Ω]=1.

The third axiom is probably the most interesting one. The basic idea is that if some events are disjoint (i.e., there is no overlap between them), then the probability of their union must be the summations of their probabilities. Another way to think about this is to imagine the probability of a set as the area of that set in the Venn diagram. If several sets are disjoint such as the ones shown

then the total area of their union is the sum of individual areas. The following example illustrates the idea behind the third axiom.

Example: In a presidential election, there are four candidates. Call them A, B, C, and D. Based on our polling analysis, we estimate that A has a 18.7 percent chance of winning the election, while B has a 36.8 percent chance of winning. What is the probability that A or B win the election?

Notice that the events that {A wins}, {B wins}, {C wins}, and {D wins} are disjoint since more than one of them cannot occur at the same time. For example, if A wins, then B cannot win. From the third axiom of probability, the probability of the union of two disjoint events is the summation of individual probabilities. Therefore,

\begin{align*} \Pr \left[ A \mbox{ wins or } B \mbox{ wins}\right] &= \Pr ({A \mbox{ wins}}∪{B \mbox{ wins}}) \\ &=\Pr ({A \mbox{ wins}})+P({B \mbox{ wins}}) =0.187+0.368 =0.555 . \qquad \blacksquare \end{align*}
 
It is convenient to use the following notations for intersection of events
\[ \Pr \left( A∩B\right) = \Pr \left[ A \mbox{ and } B\right] =\Pr (A\,B) , \]
and for union of events
\[ \Pr (A∪B) =\Pr\left[ A \mbox{ or } B\right] . \]

Borel sets and σ-fields

Later on, we will discuss probabilities for sets from real axis ℝ, and we need to determine what sets from it can have probabilities because not every subset has it. Obviously, we need to have probability for every interval, and also for any set that is a union of finitely many intervals. For technical reasons, not every set can be assigned a probability, but these sets that can be built from intervals should be included.

When one is trying to assign probabilities to every set from ℝ, he or she will get into troubles of two kinds. First, assigning probabilities to the subintervals does not uniquely determine the probabilities of all the other subsets. We would have to describe the assignment in some way for other sets; yet it is impossible even to describe all of these sets, not to mention deciding on their probabilities. Second, even if we could assign probabilities to the rest of the subsets in some way, our assignment would be inconsistent, in that there will be some countable collection of disjoint sets whose union has a probability that is not the sum of the individual sets probabilities. There is a famous Banach--Kuratowski theorem according to which no consistent assignment of probabilities is possible if we require that singletons have probability 0.

It is reasonable to include into the set having probabilities the following subsets as events: intervals, their unions, intersections, and complements. For all such subsets, we know how to define their probabilities. It is important to understand that we use not arbitrary unions or intersections, but only countable collection of events from which we are able to build new events. This will guarantee that probabilities of all events are determined uniquely once we know probabilities of subintervals from which they are constructed. Such subsets are called the Borel subsets named after Émile Borel (1871--1956). A family of sets that is closed under these three set operations---complementation, and countable unions and intersections--is called a σ-field or σ-algebra.

 

  1. Bartoszynski, T. and Halbeisen, L., On a theorem of Banach and Kuratowski and K-Lusin sets, Rocky Mountain Journal of Mathematics, 2003, Vol. 33, Number 4, pp. 1223--1231.
  2. Banach, S. and Kuratowski, K., Sur une generalisation du probleme de la mesure, Fundamenta Mathematicae, 1929, Vol. 14, pp. 127--131.