4 Computational Mechanics of Composite Materials P(A)<P(B) (1.10) The definition of probability does not reflect however a natural very practical need of its value determination and that is why the simplified Laplace definition is frequently used for various random events. Definition If n trials forms the random space of elementary events where each experiment has the same probability equal to 1/n,then the probability of the m-element event A is equal to P(A)= (1.11) Next,we will explain the definition,meaning and basic properties of the probability spaces.The probability space(,F,P)is uniquely defined by the space of elementary random events the events field F and probabilistic measure P. The field of events F is the relevant family of subsets of the space of elementary random events This field F is a non-empty,complementary and countable additive set having o-algebra structure. Definition The probabilistic measure P is a function P:F→[0，] (1.12) which is a nonnegative,countable additive and normalized function defined on the fields of random events.The pair (F)is a countable space,while the events are countable subsets of The value P(A)assigned by the probabilistic measure P to event A is called a probability of this event. Definition Two events A and B are independent if they fulfil the following condition: P(AOB)=P(A)P(B) (1.13) while the events A.AA are pair independent,if this condition holds true for any pair from this set. Definition Let us consider the probability space (.F.P)and measurable spaceB. where B is a class of the Borelian sets.Then,the representation4 Computational Mechanics of Composite Materials P() () A ≤ P B (1.10) The definition of probability does not reflect however a natural very practical need of its value determination and that is why the simplified Laplace definition is frequently used for various random events. Definition If n trials forms the random space of elementary events where each experiment has the same probability equal to 1/n, then the probability of the m-element event A is equal to ( ) n m P A = (1.11) Next, we will explain the definition, meaning and basic properties of the probability spaces. The probability space (Ω,F,P) is uniquely defined by the space of elementary random events Ω, the events field F and probabilistic measure P. The field of events F is the relevant family of subsets of the space of elementary random events Ω. This field F is a non-empty, complementary and countable additive set having σ-algebra structure. Definition The probabilistic measure P is a function P : F →[0,1] (1.12) which is a nonnegative, countable additive and normalized function defined on the fields of random events. The pair (Ω,F) is a countable space, while the events are countable subsets of Ω. The value P(A) assigned by the probabilistic measure P to event A is called a probability of this event. Definition Two events A and B are independent if they fulfil the following condition: P( ) A∩ B = P(A)⋅ P(B) (1.13) while the events { } A A An , ,..., 1 2 are pair independent, if this condition holds true for any pair from this set. Definition Let us consider the probability space (Ω,F,P) and measurable space { }n n ℜ , B , where Bn is a class of the Borelian sets. Then, the representation