Algebraic operations on events define the rules for actions with events and allow one to express one event in terms of another. Operations on events are applicable only to events that represent subsets of the same space of elementary events.
Event actions can be visualized using Venn diagrams. In diagrams, events correspond to different areas on the plane, which conditionally designate subsets of elementary events that make up events. So, in the diagrams of Fig. 1.1, the space of elementary events corresponds to the internal points of the square, the event A _ the internal points of the circle, the event B _ the internal points of the triangle. The fact that events A and B are subsets of the space of elementary events (A, B) is shown in the diagrams in Fig. 1.1a, b.
The sum (union) of events A and B is the event C=A+B (or C=AB), which consists in the fact that at least one of the events A or B will occur. The event C consists of all elementary events belonging to at least one from events A or B, or both events. In the diagram (Fig. 1.2.), the event C corresponds to the shaded area C, representing the union of areas A and B. Similarly, the sum of several events A 1, A 2, ..., A n is the event C, which consists in the fact that at least one of the events will occur And i , i=:
The sum of events unites all elementary events that make up А i , i=. If the events E 1 , E 2 ,…, E n form a complete group, then their sum is equal to a reliable event:
The sum of elementary events is equal to a reliable event
The product (intersection) of events A and B is the event C = AB (or C = AB), which consists in the joint appearance of events A and B. Event C consists of those elementary events that belong to both A and B. Figure 1.3.a event C is represented by the intersection of areas A and B. If A and B are incompatible events, then their product is an impossible event, i.e. AB = (Fig. 1.3.b).
The product of events A 1 , A 2 , ..., A n is an event C, consisting in the simultaneous execution of all events A i , i=:
Products of pairwise incompatible events А 1 , А 2 ,…, А n - impossible events: А i А j =, for any ij. Products of events that make up a complete group are impossible events: Е i Е j =, ij, products of elementary events are also impossible events: ij =, ij.
The difference between events A and B is the event C=A_B (C=AB), which consists in the fact that event A occurs and event B does not occur. Event C consists of those elementary events that belong to A and do not belong to B. Diagram of the difference of events shown in fig. 1.4. The diagram shows that C=A_B=
The opposite event for event A (or its complement) is an event that consists in the fact that event A did not occur. The opposite event completes event A to a complete group and consists of those elementary events that belong to space and do not belong to event A (Fig. 1.5). Thus, is the difference between a certain event and event A: =_A.
Properties of operations on events.
Displacement properties: A + B \u003d B + A, A B \u003d B A.
Associative properties: (A + B) + C \u003d A + (B + C), (AB) C \u003d A (BC).
Distribution property: A(B+C)=AB+AC.
From the definitions of operations on events follow the properties
A+A=A; A+=; A+=A; A·A=A; A·=A; A =
From the definition of the opposite event, it follows that
A+=; A=; =A; =; =; ;
From the diagram in Fig. 1.4, the properties of the difference of joint events are obvious:
If A and B are mutual events, then
The properties of joint events are also obvious.
For opposite events, properties are true that are sometimes called De Morgan's rule or the principle of duality: the operations of union and intersection are reversed when passing to opposite events
The proof of the duality principle can be obtained graphically using Venn diagrams or analytically by applying properties 1-6
It should be noted that actions similar to the actions "reduction of similar terms" and exponentiation in the algebra of numbers are not allowed during operations with events.
For example, for operations with events, the correct actions are:
The erroneous application of actions by analogy with algebraic ones: (A + B) B \u003d A + BB \u003d A leads to an incorrect result (check with Venn diagrams!).
Example 1.11. Prove Identities
a) (A + C) (B + C) \u003d AB + C;
b) AC_B=AC_BC
a) (A + C) (B + C) \u003d AB + CB + AC + CC \u003d AB + C (A + B) + C = \u003d AB + C (A + B) + C \u003d AB + C (A + B+) = AB+C = AB+C;
b) AC_B = AC = CA = C (A_B) = CA_CB = AC_BC
Example 1.12. The prize is drawn between the two finalists of the show program. The draw is made in turn until the first successful attempt, the number of attempts for each participant is limited to three. The first finalist starts first. The following events are considered: A=(the prize was won by the first finalist); B = (the prize was won by the second finalist). 1) Supplement these events to a complete group and compose a reliable event for it. 2) Compose a complete group of elementary events. 3) Express the events of the first complete group in terms of elementary ones. 4) Compose other complete groups of events and record reliable events through them.
1) Events A and B are non-joint, up to the full group they are supplemented by a non-joint event C=(no one won the prize). A certain event = (either the first finalist, or the second, or no one wins the prize) is equal to: = A + B + C.
2) Let's introduce events that describe the outcome of each attempt for each player and do not depend on the competition conditions: А i =(the first finalist successfully completed the i-th attempt), В i =(the second finalist successfully completed the i-th attempt), . These events do not take into account the conditions of the competition, therefore they are not elementary in relation to the fact of winning a prize. But through these events, using operations on events, you can compose a complete group of elementary events that take into account the conditions for winning on the first successful attempt: 1 = (the first finalist won the prize on the first attempt), 2 = (the second finalist won the prize on the first attempt), 3 =(first finalist won prize on second attempt), 4 =(second finalist won prize on second attempt), 5 =(first finalist won prize on third attempt), 6 =(second finalist won prize on third attempt), 7 =( both finalists failed to win the prize in three attempts). According to the terms of the competition
1 \u003d A 1, 2 \u003d, 3 \u003d, 4 \u003d,
5 =, 6 = , 7 = .
Complete group of elementary events: =( 1 ,…, 7 )
3) Events A and B are expressed through elementary events using summation operations, C coincides with an elementary event:
4) Complete event groups also constitute events
The relevant events are:
=(the first finalist will either win the prize or not)=;
=(Second finalist will either win the prize or not)=;
=(prize or not win, or win)=.
Types of random events
Events are called incompatible if the occurrence of one of them excludes the occurrence of other events in the same trial.
Example 1.10. A part is taken at random from a box of parts. The appearance of a standard part excludes the appearance of a non-standard part. Events (a standard part appeared) and (a non-standard part appeared)- incompatible .
Example 1.11. A coin is thrown. The appearance of a "coat of arms" excludes the appearance of a number. Events (a coat of arms appeared) and (a number appeared) - incompatible .
Several events form full group, if at least one of them appears as a result of the test. In other words, the occurrence of at least one of the events of the complete group is reliable event. In particular, if the events that form a complete group are pairwise incompatible, then one and only one of these events will appear as a result of the test. This particular case is of greatest interest to us, since it will be used below.
Example 1.12. Purchased two tickets of the money and clothing lottery. One and only one of the following events will necessarily occur: (winning fell on the first ticket and did not fall on the second), (winning did not fall on the first ticket and fell on the second), (winning fell on both tickets), (no winnings on both tickets) fell out). These events form full group pairwise incompatible events.
Example 1.13. The shooter fired at the target. One of the following two events is sure to occur: a hit or a miss. These two incompatible events form full group .
Events are called equally possible if there is reason to believe that none of them is no more possible than the other.
3. Operations on events: sum (union), product (intersection) and difference of events; vienne diagrams.
Operations on events
Events are denoted by capital letters of the beginning of the Latin alphabet A, B, C, D, ..., supplying them with indices if necessary. The fact that the elemental outcome X contained in the event A, denote .
For understanding, a geometric interpretation using the Vienna diagrams is convenient: we represent the space of elementary events Ω as a square, each point of which corresponds to an elementary event. Random events A and B, consisting of a set of elementary events x i and at j, respectively, are geometrically depicted as some figures lying in the square Ω (Fig. 1-a, 1-b).
Let the experiment consist in the fact that inside the square shown in Figure 1-a, a point is chosen at random. Let us denote by A the event consisting in the fact that (the selected point lies inside the left circle) (Fig. 1-a), through B - the event consisting in the fact that (the selected point lies inside the right circle) (Fig. 1-b ).
A reliable event is favored by any , therefore a reliable event will be denoted by the same symbol Ω.
Two events are identical to each other (A=B) if and only if these events consist of the same elementary events (points).
The sum (or union) of two events A and B is called an event A + B (or ), which occurs if and only if either A or B occurs. The sum of events A and B corresponds to the union of sets A and B (Fig. 1-e).
Example 1.15. The event consisting in the loss of an even number is the sum of the events: 2 fell out, 4 fell out, 6 fell out. That is, (x \u003d even }= {x=2}+{x=4 }+{x=6 }.
The product (or intersection) of two events A and B is called an event AB (or ), which occurs if and only if both A and B occur. The product of events A and B corresponds to the intersection of sets A and B (Fig. 1-e).
Example 1.16. The event consisting of rolling 5 is the intersection of events: odd number rolled and more than 3 rolled, that is, A(x=5)=B(x-odd)∙C(x>3).
Let us note the obvious relations:
The event is called opposite to A if it occurs if and only if A does not occur. Geometrically, this is a set of points of a square that is not included in subset A (Fig. 1-c). An event is defined similarly (Fig. 1-d).
Example 1.14.. Events consisting in the loss of an even and an odd number are opposite events.
Let us note the obvious relations:
The two events are called incompatible if their simultaneous appearance in the experiment is impossible. Therefore, if A and B are incompatible, then their product is an impossible event:
The elementary events introduced earlier are obviously pairwise incompatible, that is,
Example 1.17. Events consisting in the loss of an even and an odd number are incompatible events.
Certain and Impossible Events
credible An event is called an event that will definitely occur if a certain set of conditions is met.
Impossible An event is called an event that certainly will not occur if a certain set of conditions is met.
An event that coincides with the empty set is called impossible event, and an event that coincides with the whole set is called authentic event.
Events are called equally possible if there is no reason to believe that one event is more likely than others.
Probability theory is a science that studies the patterns of random events. One of the main problems in probability theory is the problem of determining a quantitative measure of the possibility of an event occurring.
ALGEBRA OF EVENTS
Operations on events (sum, difference, product)
Each trial is associated with a number of events of interest to us, which, generally speaking, can appear simultaneously. For example, when throwing a dice (i.e., a die with points 1, 2, 3, 4, 5, 6 on its faces), the event is a deuce, and the event is an even number of points. Obviously, these events are not mutually exclusive.
Let all possible results of the test be carried out in a number of the only possible special cases, mutually exclusive of each other. Then:
- each test outcome is represented by one and only one elementary event;
- · any event associated with this test is a set of finite or infinite number of elementary events;
- · an event occurs if and only if one of the elementary events included in this set is realized.
In other words, an arbitrary but fixed space of elementary events is given, which can be represented as a certain area on the plane. In this case, elementary events are points of the plane lying inside. Since an event is identified with a set, all operations that can be performed on sets can be performed on events. That is, by analogy with set theory, one constructs event algebra. In particular, the following operations and relationships between events are defined:
 (set inclusion relation: a set is a subset of a set) - event A entails event B. In other words, event B occurs whenever event A occurs. (set equivalence relation) - an event is identical or equivalent to an event. This is possible if and only if and simultaneously, i.e. each occurs whenever the other occurs.  () - sum of events. This is an event consisting in the fact that at least one of the two events or (not excluding the logical "or") has occurred. In the general case, the sum of several events is understood as an event consisting in the occurrence of at least one of these events. |
 () - product of events. This is an event consisting in the joint implementation of events and (logical "and"). In the general case, the product of several events is understood as an event consisting in the simultaneous implementation of all these events. Thus, events and are incompatible if their product is an impossible event, i.e. . |
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(set of elements belonging but not belonging) - difference of events. This is an event consisting of selections included in but not included in. It lies in the fact that an event occurs, but an event does not occur. |
 The opposite (additional) for an event (denoted) is an event consisting of all outcomes that are not included in.  Two events are said to be opposite if the occurrence of one of them is equivalent to the non-occurrence of the other. An event opposite to an event occurs if and only if the event does not occur. In other words, the occurrence of an event simply means that the event has not occurred. |
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The symmetric difference of two events and (denoted) is called an event consisting of outcomes included in or, but not included in and at the same time. The meaning of the event is that one and only one of the events or occurs. The symmetric difference is denoted: or. |
The sum of all event probabilities in the sample space is 1. For example, if the experiment is a coin toss with Event A = "heads" and Event B = "tails", then A and B represent the entire sample space. Means, P(A) + P(B) = 0.5 + 0.5 = 1.
Example.In the previously proposed example of calculating the probability of extracting a red pen from the pocket of a bathrobe (this is event A), in which there are two blue and one red pen, P(A) = 1/3 ≈ 0.33, the probability of the opposite event - extracting a blue pen - will be
Before moving on to the main theorems, we introduce two more more complex concepts - the sum and the product of events. These concepts are different from the usual concepts of sum and product in arithmetic. Addition and multiplication in probability theory are symbolic operations subject to certain rules and facilitating the logical construction of scientific conclusions.
sum of several events is an event consisting in the occurrence of at least one of them. That is, the sum of two events A and B is called event C, which consists in the appearance of either event A, or event B, or events A and B together.
For example, if a passenger is waiting at a tram stop for one of the two routes, then the event he needs is the appearance of a tram of the first route (event A), or a tram of the second route (event B), or a joint appearance of trams of the first and second routes (event FROM). In the language of probability theory, this means that the event D that the passenger needs is the appearance of either event A, or event B, or event C, which can be symbolically written as:
D=A+B+C
The product of two eventsBUT and AT is an event consisting in the joint occurrence of events BUT and AT. The product of several events the joint occurrence of all these events is called.
In the passenger example above, the event FROM(joint appearance of trams of two routes) is a product of two events BUT and AT, which is symbolically written as follows:
Assume that two physicians are separately examining a patient in order to identify a specific disease. During inspections, the following events may occur:
Detection of diseases by the first physician ( BUT);
Failure to detect the disease by the first doctor ();
Detection of the disease by the second doctor ( AT);
Non-detection of the disease by the second doctor ().
Consider the event that the disease is detected exactly once during the examinations. This event can be implemented in two ways:
The disease is detected by the first doctor ( BUT) and will not find the second ();
Diseases will not be detected by the first doctor () and will be detected by the second ( B).
Let us denote the event under consideration by and write it symbolically:
Consider the event that the disease is discovered in the process of examinations twice (both by the first and the second doctor). Let's denote this event by and write: .
The event, which consists in the fact that neither the first nor the second doctor detects the disease, will be denoted by and we will write: .
Addition rule- if element A can be chosen in n ways, and element B can be chosen in m ways, then A or B can be chosen in n + m ways.
^ multiplication rule - if element A can be chosen in n ways, and for any choice of A, element B can be chosen in m ways, then the pair (A, B) can be chosen in n m ways.
Permutation. A permutation of a set of elements is the arrangement of elements in a certain order. Thus, all the different permutations of a set of three elements are
The number of all permutations of elements is denoted by . Therefore, the number of all different permutations is calculated by the formula
Accommodation. The number of placements of a set of elements by elements is equal to
^ Placement with repetition. If there is a set of n types of elements, and you need to place an element of some type in each of m places (element types can match in different places), then the number of options for this will be n m .
^ Combination. Definition. Combinations from various elements according toelements are called combinations that are made up of data elements by elements and differ by at least one element (in other words,-element subsets of the given set from elements). butback="" onclick="goback(684168)">^ " ALIGN=BOTTOM WIDTH=230 HEIGHT=26 BORDER=0>
Space of elementary events. Random event. Reliable event. Impossible event.
random event - any subset of the space of elementary events.
^ Credible event - is bound to happen as a result of the experiment.
Impossible event - will not occur as a result of the experiment.
Actions on events: sum, product and difference of events. opposite event. Joint and non-joint events. Complete group of events.
^ Incompatible events - if they cannot occur simultaneously as a result of the experiment. It is said that several disjoint events form full group of events, if one of them appears as a result of the experiment.
If the first event consists of all elementary outcomes, except for those included in the second event, then such events are called opposite.
The sum of two events A and B is an event consisting of elementary events belonging to at least one of the events A or B. ^ The product of two events A and B an event consisting of elementary events that belong simultaneously to A and B. The difference between A and B is an event consisting of elements A that do not belong to event B.
Classical, statistical and geometric definitions of probability. Basic properties of event probability.
, g – part of figure G. ^ Basic properties of probability: 1) 0≤P(A)≤1, 2) The probability of a certain event is 1, 3) The probability of an impossible event is 0.
The theorem of addition of probabilities of incompatible events and consequences from it.
Conditional Probability. independent events. Multiplying the probabilities of dependent and independent events.
^ Multiplication of Probabilities: For addicts. Theorem. P (A ∙ B) \u003d P (A) ∙ P A (B). Comment. P(A∙B) = P(A)∙P A (B) = P(B)∙P B (A). Consequence. P (A 1 ∙ ... ∙ A k) \u003d P (A 1) ∙ P A1 (A 2) ∙ ... ∙ P A1-Ak-1 (A k). For independents. P(A∙B) = P(A)∙P(B).
^Ttheorem for adding the probabilities of joint events. Theorem . The probability of the occurrence of at least one of the two joint events is equal to the sum of the probabilities of these events without the probability of their joint occurrence
Total Probability Formula. Bayes formulas.
H 1, H 2 ... H n - form a complete group - hypotheses.
Event A can occur only if H 1, H 2 ... H n appears,
Then P (A) \u003d P (N 1) * P n1 (A) + P (N 2) * P n2 (A) + ... P (N n) * P n n (A)
^ Bayes formula
Let H 1, H 2 ... H n be hypotheses, event A can occur under one of the hypotheses
P (A) \u003d P (N 1) * P n1 (A) + P (N 2) * P n2 (A) + ... P (N n) * P n n (A)
Assume that event A has occurred.
How has the probability of H 1 changed due to the fact that A has occurred? Those. R A (H 1)
R (A * H 1) \u003d R (A) * R A (H 1) \u003d R (H 1) * R n1 (A) => R A (H 1) \u003d (P (H 1) * R n1 ( A))/ P(A)
H 2 , H 3 ... H n are defined similarly
General form:
Р А (Н i)= (Р (Н i)* Р n i (А))/ Р (А) , where i=1,2,3…n.
Formulas allow you to overestimate the probabilities of hypotheses as a result of how the test result becomes known, as a result of which event A appeared.
"Before" the test - a priori probabilities - P (N 1), P (N 2) ... P (N n)
"After" the test - a posteriori probabilities - R A (H 1), R A (H 2) ... R A (H n)
The posterior probabilities, like the prior probabilities, add up to 1.
9. Formulas of Bernoulli and Poisson.
Bernoulli formula
Let there be n trials, in each of which event A may or may not occur. If the probability of the event A in each of these trials is constant, then these trials are independent with respect to A.
Consider n independent trials, in each of which A can occur with probability p. Such a sequence of tests is called the Bernoulli scheme.
Theorem: the probability that in n trials event A will occur exactly m times is equal to: P n (m)=C n m *p m *q n - m
The number m 0 - the occurrence of an event A is called the most probable if the corresponding probability P n (m 0) is not less than other P n (m)
P n (m 0)≥ P n (m), m 0 ≠ m
To find m 0 use:
np-q≤ m 0 ≤np+q
^ Poisson formula
Consider the Bernoulli test:
n is the number of trials, p is the probability of success
Let p be small (p→0) and n large (n→∞)
average number of occurrences of success in n trials
λ=n*p → p= λwe put into the Bernoulli formula:
P n (m)=C n m *p m *(1-q) n-m ; C n m = n!/((m!*(n-m)!) →
→ P n (m)≈ (λ m /m!)*e - λ (Poisson)
If p≤0.1 and λ=n*p≤10, then the formula gives good results.
10. Local and integral theorems of Moivre-Laplace.
Let n be the number of trials, p be the probability of success, n be large and tend to infinity. (n->∞)
^ Local theorem
Р n (m)≈(f(x)/(npg)^ 1/2 , where f(x)= (e - x ^2/2)/(2Pi)^ 1/2
If npq≥ 20 - gives good results, x=(m-np)/(npg)^ 1/2
^ Theorem integral
P n (a≤m≤b)≈ȹ(x 2)-ȹ(x 1),
where ȹ(x)=1/(2Pi)^ 1/2 * 0 ʃ x e (Pi ^2)/2 dt is the Laplace function
x 1 \u003d (a-np) / (npq) ^ 1/2, x 2 \u003d (b-np) / (npq) ^ 1/2
Properties of the Laplace function
ȹ(x) – odd function: ȹ(-x)=- ȹ(x)
ȹ(x) – monotonically increasing
values ȹ(x) (-0.5;0.5), and lim x →∞ ȹ(x)=0.5; lim x →-∞ ȹ(x)=-0.5
P n (│m-np│≤Ɛ) ≈ 2 ȹ (Ɛ/(npq) 1/2)
P n (ɑ≤m/n≤ƥ) ≈ ȹ(z 2)- ȹ(z 1), where z 1=(ɑ-p)/(pq/n)^ 1/2 z 2=(ƥ -p )/(pq/n)^ 1/2
P n (│(m/n) - p│≈ ∆) ≈ 2 ȹ(∆n 1/2 /(pq)^ 1/2)
11. Random value. Types of random variables. Methods for setting a random variable.
SW is a function defined on a set of elementary events.
X,Y,Z is NE, and its value is x,y,z
Random they call a value that, as a result of tests, will take one and only one possible value, not known in advance and depending on random causes that cannot be taken into account in advance.
SW discrete, if the set of its values is finite or counted (they can be numbered). It takes on separate, isolated possible values with certain probabilities. The number of possible values of a discrete CV can be finite or infinite.
SW continuous, if it takes all possible values from some interval (on the whole axis). Its values may differ very little.
^ Discrete SW distribution law m.b. given:
1.table
| X | x 1 | x 2 | … | x n |
| P(X) | p 1 | p 2 | … | p n |
(distribution range)
X \u003d x 1) are incompatible
p 1 + p 2 +… p n =1= ∑p i
2.graphic
Probability distribution polygon
3.analytical
P=P(X)
12. The distribution function of a random variable. Basic properties of the distribution function.
The distribution function of CV X is a function F(X) that determines the probability that CV X will take a value less than x, i.e.
x x = cumulative distribution function
A continuous SW has a continuous, piecewise differentiable function.