2.2 Probability space
2.2.1 Classical probability spaces
=
n
P
i=1
P(Ei). This is what we call the additivity of probability function. If n = 2, then for two mutually exclusive events E1 and E2 the above condition yields P(E1+E2) =P(E1) +P(E2).
If the sample space is finite, then the sequence of events in condition (c) is the impossible event with a finite number of exceptions. Therefore, in a finite sample space sigma additivity is identical to the additivity of function P.
In the definition of probability the requirements (a), (b) and (c) are called Kol-mogorov’s axioms of probability. All statements valid for probability can be derived from these axioms.
2.2 Probability space
The triplet of a given sample space Ω, the event algebra A consisting of certain subsets of Ω and the probability function P defined on the event algebraA is calledKolmogorov probability space (or probability space), and is denoted as (Ω,A, P).
In the case of an experiment, the sample space Ω consists of the possible outcomes of the experiment. The choice of the event algebra is our decision. It is worthwhile to choose one which consists of observable events and which allows us to formulate all arising problems, but it should not be too large, which would cause mathematical difficulties in the definition of the probability function. The probabilities of the events should be defined in accordance with our experience.
There are different possibilities to assume a probability function for a given sample space and event algebra. For example, in the case of rolling a die there can be different probabilities when the die is fair and when it has an inhomogeneous mass distribution.
In the following, we present some frequently encountered types of probability spaces.
2.2.1 Classical probability spaces
Several experiments have a finite number of outcomes, and in many cases we can notice that all outcomes have the same chance of occurring. This is the case when we flip a coin or roll a die, provided that the coin has the shape of a straight cylinder and a homogeneous mass distribution, and the die has the shape of a cube and also has a
homogeneous mass distribution. Such a coin or cube is called fair. The so-called classical probability space is created with the aim of describing experiments of this kind.
The triplet (Ω,A, P) is called classical probability space if the sample space Ω is a finite set, with each subset being an event, i.e., A is the power set of Ω, moreover, the probability function is
P(E) := the number of elementary events in E
the number of elementary events in Ω, E ⊂Ω. (2.1) Remark: This formula can be derived from the facts that the sample space is finite and all the elementary events have the same chance of occurring. Determining the numerator and the denominator is a combinatorial problem.
Example 2.3. (Famous classical probability spaces)
(a) In the example of the coin flip (see also Example 1.1(c)) let the sample space be Ω := {h, t}, let the event algebra be defined as the power set of Ω, i.e., A :=
{∅,{h},{t},{h, t}}, and the probabilities of the events as P(∅) := 0, P({h}) := 1
2, P({t}) := 1
2, P({h, t}) := 1.
(b) In the example of rolling a die (see also Example 1.1(a)), by denoting the outcomes by the numbers of dots on the face-up side, the sample space isΩ := {1,2,3,4,5,6}, let the event algebra be the power set of Ω, which has 26 = 64 elements, and let the probability of an event E be P(E) := k6 if E occurs in the case of k elementary events, i.e., if the number of elementary events of E is k.
In a classical probability space pairwise mutually exclusive events cannot form an infinite sequence, therefore in the classical case instead of sigma additivity it is enough to require the additivity of probability function.
Choosing an element randomly from a given set means that all the elements have the same chance of being chosen. If the sample space is a finite set, then the experiment is described by a classical probability space. If the sample space is infinite, then a random choice can only be supposed to have zero probability (see the Remark below formula (2.1) and Example4.5).
Example 2.4. An urn containsN balls,K of them are red, and N −K are green. We independently draw n balls at random, one after the other, in each case replacing the ball before the next independent draw. This procedure is called sampling with replacement.
What is the probability that k red balls are drawn following this procedure?
Solution: The experiment is drawing n balls one after the other with replacement.
The outcomes are variations with repetition of size n of the N balls in the urn, and the number of these variations is Nn. If we draw k red balls and n−k green balls, then the order of colours is again a permutation with repetition, the number of these orders being k! (n−k)!n! = nk
. For each order of colours the order of the red balls as well the order of the green balls is a variation with repetition, so the number of possible orders of the red balls is Kk, and that of the green balls is (N −K)n−k. Any order of drawing of the red balls can be combined with any order of drawing of the green balls, therefore we can draw k red balls in nk
Kk(N −K)n−k ways.
By the assumption that the experiment is described by a classical probability space, the probability that k red balls are drawn (Ek) is
P(Ek) =
The product of probability spaces
For describing the independent execution of experiments we will need to introduce the notion of the product of probability spaces. First we define the product of classical probability spaces, and then the product of arbitrary ones.
Let (Ω1,A1, P1) and (Ω2,A2, P2) be classical probability spaces. We define theproduct of these probability spaces as the probability space (Ω,A, P) if its sample space is the set Ω := Ω1×Ω2, its event algebra Aconsists of all subsets of the product sample space Ω, and its probability function is
P: A →R, P(E) := |E|
|Ω|, E ∈ A, where |E| denotes the number of elementary events in E.
Here the Cartesian product Ω = Ω1×Ω2 consists of all those ordered pairs (ω1, ω2) for
We remark that an event of the product space cannot always be given as the Cartesian product of two events.
If two probability spaces are not classical, their product can nevertheless be defined.
Let the sample space of the product (Ω,A, P) of probability spaces (Ω1,A1, P1) and
(Ω2,A2, P2) be Ω := Ω1 ×Ω2. Let A be the smallest event algebra containing the set {E1 ×E2 :E1 ∈ A1, E2 ∈ A2} (one can show that it exists). It can be proven that the function
P0(E1×E2) := P1(E1)·P2(E2), E1 ∈ A1, E2 ∈ A2
has exactly one extension to the event algebra A, this will be the probability functionP of the product.