PURPOSE OF THE LECTURE: to acquaint with elementary information from the theory of sets; formulate the axioms of probability theory, their consequences and the rule for adding probabilities.
Elementary information from set theory
many any collection of objects of arbitrary nature is called, each of which is called set element.
Examples of sets: a lot of students in a lecture; the set of points on a plane that lie inside a circle of radius r; many points on numerical axis, the distance from which to the point b with abscissa a less than d; lots of natural numbers.
Sets are denoted in different ways. Lots of M natural numbers from 1 to 100 can be written as
The set of points on the number axis, the distance from which to the point b with abscissa a less than d, can be written as
where x- the abscissa of the point.
The set of plane points lying inside or on the boundary of a circle of radius r centered at the origin,
where x, y – Cartesian coordinates points.
Another entry of this set
where is one of the polar coordinates of the point.
According to the number of elements, the sets are divided into final and endless. The set is finite and consists of 100 elements. But a set can also consist of one element and even contain no elements at all.
The set of all natural numbers is infinite, just as the set of even numbers is infinite.
Infinite set is called countable if all its elements can be arranged in some sequence and numbered (both sets, and , are countable).
Sets S and C are infinite and uncountable (their elements cannot be numbered).
Two sets A and B match, if they consist of the same elements: and . The coincidence of sets is denoted by an equal sign: A=B. The notation means that the object a is an element of the set BUT or " a belongs BUT". Another entry means that " a do not belong BUT".
A set that does not contain any element is called empty and is denoted by the symbol .
Lots of AT is called a subset (part) of the set BUT if all elements AT are also contained in BUT, and is denoted as or . For example, .
A subset can be equal to the set itself. Graphically, you can depict the relationship between a set and a subset, as shown in Fig. 2.1, where each point of the figure AT belongs to the figure BUT, i.e. .
Union (sum) of sets BUT and AT is called the set consisting of all elements BUT and all elements AT. Thus, a union is a collection of elements that belong to at least one of the combined sets.
For example: .
Geometric interpretation union of two sets BUT and AT shown in fig. 2.2.
The union (sum) of several sets is defined similarly
where the resulting set is the set of all elements included in at least one of the sets: .
Intersection (product) of sets BUT and AT is called a set D, consisting of elements included simultaneously and in BUT, and in :
The geometric interpretation of the intersection is shown in fig. 2.3.
The intersection of several sets is defined similarly
as a set consisting of elements included simultaneously in all sets.
The operations of union (addition) and intersection (multiplication) of sets have a number of properties that are similar to the properties of addition and multiplication of numbers:
1. Displacement property:
2. Associative property:
Adding the empty set and multiplying by the empty set are similar to the corresponding operations on numbers, if you consider zero as the empty set:
Some operations on sets have no analogues in ordinary operations on numbers, in particular
Axioms of probability theory and their consequences.
Probability addition rules
Using elementary information on set theory, one can give a set-theoretic scheme for constructing probability theory and its axiomatics.
In an experiment with a random outcome, there is a set of all possible outcomes of the experiment. Each element of this set is called elementary event, the set itself is elementary event space. Any event BUT in the set-theoretic interpretation there is some subset of the set : . If, in turn, the set BUT splits into several non-intersecting subsets ( at ), then the events are called "variants" of the event BUT. On fig. 2.4 event BUT splits into three options: .
For example, when throwing dice space of elementary events. If event , then event options BUT: ,
A subset of the set itself can also be considered - in this case it will be reliable event. An empty set is added to the entire space of elementary events; this set is also considered as an event, but impossible.
The set-theoretic interpretation of the previously considered properties of events is as follows:
1. Multiple events form full group , if , i.e. their sum (combination) is a reliable event.
2. Two events BUT and AT called incompatible, if the sets corresponding to them do not intersect, i.e. . Several events are called pairwise incompatible, if the appearance of any of them excludes the appearance of each of the others: at .
3. The sum of two events BUT and AT called an event FROM, consisting in the execution of the event BUT or events AT, or both events together. The sum of several events is an event consisting in the execution of at least one of them.
4. The product of two events BUT and AT called an event D, consisting in the joint execution of the event BUT and events AT. A product of several events is an event consisting in the joint execution of all these events.
5. Opposite in relation to the event BUT is called an event consisting in the non-appearance BUT and corresponding complementary event BUT to (see Fig. 2.5).
Based on the above interpretation of events as sets, the axioms of probability theory are formulated.
Every event BUT a certain number is assigned, called the probability of the event. Since any event is a set, the probability of an event is set function.
These event probabilities must satisfy the following axioms:
1. The probability of any event is between zero and one:
2. If BUT and AT are incompatible events, i.e., then
This axiom can be easily generalized with associative property addition for any number of events. If at , then
i.e. the probability of the sum incompatible events is equal to the sum of the probabilities of these events.
This axiom is called addition "theorem"(for the scheme of cases, it can be proven), or rule of addition of probabilities.
3. If available countable set incompatible events ( at ), then
This axiom is not derived from the previous axiom and is therefore formulated as a separate one.
For a scheme of cases (schemes of urns), i.e., for events that have the properties of completeness, incompatibility, and equipotentiality, one can derive the classical formula (1.1) for directly calculating probabilities from the addition rule (2.1).
Let the results of the experiment be presented in the form n incompatible cases. Chance favors event BUT if it represents a subset BUT(), or, in other words, this is a variant of the event BUT. Since they form a complete group, then
According to the rule of addition
where we get
After substituting the obtained expressions into (2.3), we have
Q.E.D.
Formula (2.3) can also be derived for more than two joint events.
For several centuries after the beginning of its systematic study, the basic concepts of probability theory were not yet clearly defined. The fuzziness of the basic definitions often led researchers to conflicting conclusions, and practical probabilistic applications were poorly substantiated. Further development natural science necessitated a systematic study of the basic concepts of probability theory and the determination of the conditions under which it is possible to use its results. Of particular importance was the formal logical substantiation of the theory of probability, which, in particular, in 1900 D. Hilbert was classified as critical issues mathematics.
The formal-logical principle of construction required that the basis of the theory of probability be some axiomatic premises, which are a generalization of the centuries-old human experience. The further development of probabilistic concepts had to be built by means of deduction from axiomatic positions without resorting to fuzzy and intuitive ideas. This point of view was first developed in 1917. Soviet mathematician S.N. Berstein. At the same time, S.N. Bershtein came from qualitative comparison random events according to their greater or lesser probability. A mathematically rigorous construction of the axiomatic probability theory was proposed by A.N. Kolmogorov in 1933, closely linking probability theory with set theory and measure theory. The axiomatic definition of probability as special cases includes both classical and statistical definitions and overcomes the insufficiency of each of them.
The starting point of A.N. Kolmogorov is the set of elementary events ω, in special literature called the phase space and traditionally denoted by Ω. Any observable event whose probability needs to be determined can be represented as some subset of the phase space. Therefore, along with the set Ω, the set Θ of subsets of elementary events is considered, the symbolic designation of which can be arbitrary. A certain event is representable by the entire phase space. A set Θ is called a set algebra if the following requirements are met:
1) Ω ∈ Θ, ∅ ∈ Θ;
2) the fact that A ∈ Ω implies that $\bar A \in \Theta $ also;
3) the fact that A ∈ Θ and B ∈ Θ implies that A ∪ B ∈ Θ and A ∩ B∈ Θ.
If, in addition to the above, the following requirement is met:
4) the fact that A n ∈ Θ (for n = 1,2...) implies that $\mathop \cup \limits_n (A_n) \in \Theta $ and $\mathop \cap \limits_n (A_n ) \in \Theta $, then the set Θ is called σ-algebra. The elements of Θ are called random events.
Operations on random events in the axiomatic theory of probability are understood as operations on the corresponding sets. As a result, it is possible to establish a mutual correspondence between the terms of the language of set theory and the language of probability theory.
As axioms defining probability, A.N. Kolmogorov accepted the following assertions:
Axiom 1. To each random event And aligned non-negative number P (A) , called its probability.
Axiom 2. P(Ω)= 1.
Axiom 3 (axiom of addition). If the events A 1 , A 2 ,...,A n are pairwise incompatible, then
P(A 1 + A 2 +...+ A n) = P(A 1) + P(A 2) +...+ P(A n).
The following statements are consequences of the formulated axioms.
1. The probability of an impossible event is zero: P(∅) = 0.
2. For any event A $P(\bar A) = 1 - P(A)$.
3. Whatever the random event A, 0 ≤ P(A) ≤ 1.
4. If event A entails event B, then P(A) ≤ P(B).
A probability space is usually called a triple of symbols (Ω, Θ, P), where Ω is the set of elementary events ω, Θ – σ is the algebra of subsets of Ω, called random events, and P(A) is the probability defined on σ, the algebra Θ.
Thus, according to the axiomatics of A.N. Kolmogorov, each observed event is assigned a certain non-negative number, called the probability of this event, so that the probability of the entire phase space is equal to 1, and the property sigma additivity. The last property means that in the case of pairwise mutually exclusive events, the probability of occurrence according to at least of one (and due to pairwise incompatibility, exactly one) observed event coincides with the sum of the probabilities of observed events from a given finite or countable set of observed events .
In the case of a definition of probability on a σ - algebra consisting of some subsets of Ω, the first one cannot be extended to other subsets of Ω in such a way that the sigma-additivity property is preserved, unless Ω consists of a finite or countable number of elements. The introduction of sigma-additivity has also led to a number of paradoxes. Therefore, along with sigma-additivity, the property additivity, which is understood as the equivalence of the measure of the union of two incompatible events to the sum of the measures of these events. However, almost immediately it was shown that replacing sigma-additivity with additivity not only does not solve all problems, but also leads to other paradoxical results.
The system of Kolmogorov's axioms is relatively consistent and incomplete, allows you to build probability theory as part of measure theory, and consider the probability as a non-negative normalized additive set function. Although in the theory of probability A.N. Kolmogorov probability is always non-negative, some theorems in probability theory can be generalized to the case when negative numbers act as probabilities, and also obtain other generalizations of probability.
Some fundamental mathematical theories inherit the basic concepts, constructions and terminology of probability theory. Such, in particular, is the possibility theory, which also considers the spaces of possibilities and elementary events, σ - algebra.
Axiomatics of Probability Theory
Suggested above classic definition probabilities along with obvious merits, primarily simplicity and intuitive clarity, has a number of significant drawbacks: it provides only a finite or countable set of elementary events and knowledge of their probabilities is mandatory. All this is by no means always the case, and therefore the introduced definition is not sufficiently general. At present, the axiomatic construction of the theory of probability has become generally accepted.
In mathematics, axioms are propositions that are accepted as true and are not proven within the framework of a given theory. All other provisions of this theory must be derived purely logical way from the accepted axioms. The formulation of the axioms is not initial stage development mathematical science, but is the result of a long accumulation of facts and logical analysis obtained results in order to reveal the really basic primary facts. This is how the axioms of geometry were formed. The theory of probability has passed a similar path, in which the axiomatic construction of its foundations was a matter of the relatively recent past. For the first time, the problem of axiomatic construction of probability theory was solved in 1917 by the Soviet mathematician S.N. Bernstein.
At present, the axiomatics of Academician A.N. Kolmogorov (1933), which connects the theory of probability with the theory of sets and the metric theory of functions.
In the axiomatics of A.N. Kolmogorov, the space (set) of elementary outcomes Ω is primary. What are the elements of this set for logical development probability theory is irrelevant. Next, we consider some system F of subsets of the set Ω; elements of the system F are called random events. Regarding the structure of the system F, the following three requirements are assumed to be satisfied:
1. The subset F contains a certain event Ω as an element.
2. If A and B are two events defined on Ω, are included in the subset F as elements, then the subset F also contains A + B, A ∙ B as elements,
3. If the events А 1 , А 2 , …, defined on Ω, are elements of the subset F, then their sum 
and work 
are also elements of the subset F.
The set F formed in the manner described above called the "σ-algebra of events".
We now turn to the formulation of the axioms that define probability.
Axiom 1.(axiom of existence of probability). Each random event A from the σ-algebra of events F is associated with a non-negative number p(A), called its probability.
Axiom 2.(probability of certain event). The probability of a certain event is equal to 1: Р(Ω)=1. (1.15)
Axiom 3.(axiom of addition). If events A and B are not compatible, then
P(A+B) = P(A)+P(B). (1.16)
Axiom 4.(extended axiom of addition). If the event A is equivalent to the occurrence of at least one of the pairwise incompatible events A 1 , A 2 , …, that is, , then the probability of the event A is equal to
PURPOSE OF THE LECTURE: to acquaint with elementary information from the theory of sets; formulate the axioms of probability theory, their consequences and the rule for adding probabilities.
Elementary information from set theory
many any collection of objects of arbitrary nature is called, each of which is called set element.
Examples of sets: a lot of students in a lecture; the set of points on a plane that lie inside a circle of radius r; set of points on the real axis, the distance from which to the point b with abscissa a less than d; set of natural numbers.
Sets are denoted in different ways. Lots of M natural numbers from 1 to 100 can be written as
The set of points on the number axis, the distance from which to the point b with abscissa a less than d, can be written as
where x- the abscissa of the point.
The set of plane points lying inside or on the boundary of a circle of radius r centered at the origin,
where x, y are the Cartesian coordinates of the point.
Another entry of this set
where is one of the polar coordinates of the point.
According to the number of elements, the sets are divided into final and endless. The set is finite and consists of 100 elements. But a set can also consist of one element and even contain no elements at all.
The set of all natural numbers is infinite, just as the set of even numbers is infinite.
An infinite set is called countable if all its elements can be arranged in some sequence and numbered (both sets, and , are countable).
Sets S and C are infinite and uncountable (their elements cannot be numbered).
Two sets A and B match, if they consist of the same elements: and . The coincidence of sets is denoted by an equal sign: A=B. The notation means that the object a is an element of the set BUT or " a belongs BUT". Another entry means that " a do not belong BUT".
A set that does not contain any element is called empty and is denoted by the symbol .
Lots of AT is called a subset (part) of the set BUT if all elements AT are also contained in BUT, and is denoted as or . For example, .
A subset can be equal to the set itself. Graphically, you can depict the relationship between a set and a subset, as shown in Fig. 2.1, where each point of the figure AT belongs to the figure BUT, i.e. .
Union (sum) of sets BUT and AT is called the set consisting of all elements BUT and all elements AT. Thus, a union is a collection of elements that belong to at least one of the combined sets.
For example: .
Geometric interpretation of the union of two sets BUT and AT shown in fig. 2.2.The union (sum) of several sets is defined similarly
where the resulting set is the set of all elements included in at least one of the sets: .
Intersection (product) of sets BUT and AT is called a set D, consisting of elements included simultaneously and in BUT, and in :
.The geometric interpretation of the intersection is shown in fig. 2.3.
The intersection of several sets is defined similarly
as a set consisting of elements included simultaneously in all sets.
The operations of union (addition) and intersection (multiplication) of sets have a number of properties that are similar to the properties of addition and multiplication of numbers:
1. Displacement property:
2. Associative property:
3. Distribution property:
Adding the empty set and multiplying by the empty set are similar to the corresponding operations on numbers, if you consider zero as the empty set:
Some operations on sets have no analogues in ordinary operations on numbers, in particular
Axioms of probability theory and their consequences.
Probability addition rules
Using elementary information on set theory, one can give a set-theoretic scheme for constructing probability theory and its axiomatics.
In an experiment with a random outcome, there is a set of all possible outcomes of the experiment. Each element of this set is called elementary event, the set itself is elementary event space. Any event BUT in the set-theoretic interpretation there is some subset of the set : . If, in turn, the set BUT splits into several non-intersecting subsets ( at ), then the events are called "variants" of the event BUT. On fig. 2.4 event BUT splits into three options: .
For example, when throwing a dice, the space of elementary events . If event , then event options BUT: ,
A subset of the set itself can also be considered - in this case it will be reliable event. An empty set is added to the entire space of elementary events; this set is also considered as an event, but impossible.The set-theoretic interpretation of the previously considered properties of events is as follows:
1. Multiple events form full group, if , i.e. their sum (combination) is a reliable event.
2. Two events BUT and AT called incompatible, if the sets corresponding to them do not intersect, i.e. . Several events are called pairwise incompatible, if the appearance of any of them excludes the appearance of each of the others: at .
3. The sum of two events BUT and AT called an event FROM, consisting in the execution of the event BUT or events AT, or both events together. The sum of several events is an event consisting in the execution of at least one of them.
4. The product of two events BUT and AT called an event D, consisting in the joint execution of the event BUT and events AT. A product of several events is an event consisting in the joint execution of all these events.
5. Opposite in relation to the event BUT is called an event consisting in the non-appearance BUT and corresponding complementary event BUT to (see Fig. 2.5).
Based on the above interpretation of events as sets, the axioms of probability theory are formulated.Every event BUT a certain number is assigned, called the probability of the event. Since any event is a set, the probability of an event is set function.
These event probabilities must satisfy the following axioms:
1. The probability of any event is between zero and one:
2. If BUT and AT are incompatible events, i.e., then
This axiom can be easily generalized using the associative property of addition to any number of events. If at , then
i.e., the probability of the sum of incompatible events is equal to the sum of the probabilities of these events.
This axiom is called addition "theorem"(for the scheme of cases, it can be proven), or rule of addition of probabilities.
3. If available countable set incompatible events ( at ), then
This axiom is not derived from the previous axiom and is therefore formulated as a separate one.
For a scheme of cases (schemes of urns), i.e., for events that have the properties of completeness, incompatibility, and equipotentiality, one can derive the classical formula (1.1) for directly calculating probabilities from the addition rule (2.1).
Let the results of the experiment be presented in the form n incompatible cases. Chance favors event BUT if it represents a subset BUT(), or, in other words, this is a variant of the event BUT. Since they form a complete group, then
But all cases are incompatible, and the rule of addition of probabilities applies to them
In addition, since all events are equally possible, then
Cases favorable to an event form its variants, and since the probability of each of them is , then by the addition rule we get
But this is the classical formula (1.1).
Consequences of the rule of addition of probabilities
1. The sum of the probabilities of a complete group of incompatible events is equal to one, i.e. if
Proof. Since the events are incompatible, the addition rule applies to them
2. The sum of the probabilities of opposite events is equal to one:
as the events BUT and form a complete group.
The rule is widely used in problems where it is easier to calculate the probability of the opposite event.
3. If events BUT and AT are compatible, i.e., then
Proof. Let's represent as the sum of incompatible (disjoint) options (see Fig. 2.6)According to the rule of addition
where we get
After substituting the obtained expressions into (2.3), we have
Q.E.D.
Formula (2.3) can also be derived for more than two joint events.
Let be the space of elementary events, be the algebra of events (the algebra of subsets of the set). The following five axioms underlie the theory of probability.
1. The algebra of events is - the algebra of events.
The system of events is called - algebra, if for any sequence of events, their union, intersection and additions also belong, i.e. , are also events. Thus, - algebra is a system of events closed under the operations of complement, countable union and countable intersection.
2. On the - algebra of events, for any, a function is defined, called probability and taking numerical values from interval : .
This axiom is the axiom of the existence of probability - as a function of on with values from the interval. The next three axioms define the properties of a function.
3. For any two events such that
Axiom of addition of probabilities.
Hence it follows that for a finite number of incompatible events
4. Let, - pairwise incompatible events: and let. Then
Relation (15.3) is called the axiom of countable additivity of probability or the axiom of continuity of probability. The second is related to the following interpretation of equality (15.3). The event should be understood as the limit of the sequence
In this case, equality (15.3) can be understood as a property of the continuity of the function: or
Which allows the limit operation to be taken out of the function. This is due to the fact that condition (15.5) implies (15.3):
The fifth axiom indicates that the space of elementary events is a certain event. Thus, it contains all the events that can be considered in this problem.
The space of elementary events, - the algebra of events and the probability on, satisfying axioms 1-5, form the so-called probability space, which is usually denoted.
Note that the system of axioms 1-5 is not contradictory, since there exist that satisfy these axioms and is not complete, since the probability can be defined in many ways within the framework of axioms 2-5. The concept of a probability space (or a system of axioms 1-5) contains only the most General requirements submitted to mathematical model random phenomenon, and does not uniquely determine the probability. The latter is possible only if additional conditions given in the formulation of the problem under consideration.
Discrete probability space
A probability space is called discrete if it is finite or countable, - - the algebra of all subsets (including), the probability is defined for each one-point subset of the space of elementary events:
For any event, its probability is determined by the equality
Examples - algebras
17.1. Let be an arbitrary space of elementary events on which no events are specified. To construct an algebra, according to the definition (item 15), it is necessary to consider all additions, unions and intersections set events and include them in - algebra. Because in this case there is a single event, it is possible to construct only its complement. Now there is a system of two events ( ). Further application of the addition, union, intersection operations does not give new events. Thus, in this example- algebra.
17.2. Let be the space of elementary events and be some event that does not coincide with, i.e. . Thus, there is a system of two events. This system can be extended to include new events that are obtained as a result of addition, union, intersection operations on events. It makes sense to continue the procedure of expanding the system of events recurrently until the appearance of new events stops. The limiting system of events is called the algebra generated by the system of events.
Consider the addition operation on system events. Its result is new events not contained in original system, the inclusion of which gives new system events
Obviously, subsequent operations of addition, union, intersection do not give new events that are not contained in (17.1). Thus, the system of events (17.1) is an algebra generated by the system.
17.3. Let's complicate the example. Let be the space of elementary events, be two incompatible events, such that. Thus, there is a system of three events. The union operation on the events of this system results in the appearance of one new event. The resulting system of four events is expanded to eight by including their additions. It is easy to see that the application of the addition, union, intersection operations to these eight events does not generate new events. So the system of eight events
is an algebra generated by a system of events.
17.4. Consider - the space of elementary events and two arbitrary events, fig. 17.1. To construct an algebra generated by a certain system of events, in many cases it is convenient to apply the following method.
We single out all incompatible events, Fig. 17.1. At the same time, etc. - the algebra will contain all events, all unions of events, and also impossible event. Indeed, the operation of intersection of any events from the set generates a single event. The addition operation on events from the set generates an event, which is expressed through the union of events. Consequently, it suffices to consider only the operation of union over events, instead of three operations - addition, intersection, union for the original system of events.
Now, to build - algebra, consider the events, all their combinations and express the resulting events through the original ones. Obviously: , . Pairwise unions give the following events: , ; , ; . Triple unions: , .
Thus, - the algebra contains the events: , ; , ; , as well as and - a total of 16 events.
Note that when defining - algebra, the generating system of events, as a rule, is composed of events observed in the experiment.
We note that the events coincide with the events (8.1), which were considered when deriving the addition formula for frequencies. Indeed, and finally, by the formula (6.1) .
17.5. Consider a generalization of Example 4. Let the original system of events - contain arbitrary events. To construct an algebra, like example 4, we introduce events of the form
where each or, and and. Since each can take two values 0 or 1, the number of all events of the form is equal. These events form a complete group of incompatible events. Thus, events on the - algebra play the role of an orthogonal basis, which makes it possible to represent arbitrary event through incompatible (orthogonal in the sense of the intersection operation) events. In set theory, sets of a kind are called constituents. The constituent apparatus allows us to show that in this example the number of all events - algebra does not exceed (including and), and the number of events reaches maximum value when all are different from (as in example 4). This result makes it possible to judge the high growth rate of the number of events in - algebra depending on - the number of events in the original system. For example 4, the number, therefore, the number of events in - algebra is equal.
