Sample spaces and probability spaces

In studying probability, we start by asking what a probability is and how it gets assigned to an event or a statement.  The best way to understand this is by considering the sample space (sometimes called “the universe”) of an experiment, namely, the set of  outcomes such that exactly one outcome occurs when the experiment is performed.  Each of the outcomes is a point in the sample space.  For example, suppose you throw a fair die.  Then, a space consists of the 6 faces, and each face is a point.  However, we might be interested only in even or odd, in which case we might consider a sample space with only two points, even and odd; or, we might be interested only in numbers greater or smaller than 3, in which case we might consider a sample space with only two points, greater then 3 and smaller than 3.  In short, an experiment may have more than one sample space.

A point or a collection of points is an event in the sample space, including the empty set (the set with no points in it) and the whole sample space (the set with all the points in it).  For example, the set {1,2,3} is an event (namely, the event of getting 1 or 2 or 3), and so are the sets {1,2,3,4,5,6} (the whole sample space) and {the even number smaller than 2} (the empty set).  Note that the set of one point, for example {5} constitutes an event.

Let S be a sample space of n points.  Probabilities are numbers satisfying the following 3 requirements:

1. Each point is assigned a non-negative probability (Non-negativity)
2. The sum of all the n probabilities (one for each point) is 1. (Normalization)
3. The probability Pr(A) of an event A is the sum of the probabilities of all the points in A. (Additivity)

A sample space with a probability assignment is a probability space.  Note that because of requirements (1) and (2) a probability is always between 0 and 1, included.

For example, in the case of the die, if we consider the sample space S={1,2,3,4,5,6}, the following 3 are probability assignments:

1. Pr(1)=Pr(2)=Pr(3)=Pr(4)=Pr(5)=Pr(6)=1/6
2. Pr(1)=1/2; Pr(2)=Pr(3)=Pr(4)=0; Pr(5)=1/4; Pr(6)=1/4
3. Pr(1)=Pr(2)=1/8; Pr(3)=Pr(4)=1/4; Pr(5)=0; Pr(6)=1/4

However, the following 2 are not

Pr(1)=Pr(2)=Pr(3)=Pr(4)=Pr(5)=Pr(6)=1/5  (the sum of all the probabilities is greater than 1)

Pr(1)=Pr(2)=Pr(3)=Pr(4)=Pr(5)= 1; Pr(6)= -4 (Pr(6) is negative).

A sure event is one that has probability 1; an impossible event one that has probability 0.

If A is an event, then –A (the complement of A) is the set of points not in A.  For example, consider the probability space S={1,2,3,4,5,6} with probability assignment Pr(1)=Pr(2)=Pr(3)=Pr(4)=Pr(5)=Pr(6)=1/6 (the standard probability space of a fair die).  Then, A={1,3,5,6} is an event, and –A={2,4}. Note that Pr(A)=4/6 and Pr(-A)=2/6.  The point can be generalized, as it follows from requirements (1)-(3) that

Pr(-A)=1-Pr(A).