Definition 1. It is said that in some experience an event BUT entails followed by the occurrence of an event IN if when the event occurs BUT the event comes IN. Notation of this definition BUT Ì IN. In terms of elementary events, this means that each elementary event included in BUT, is also included in IN.
Definition 2. Events BUT And IN are called equal or equivalent (denoted BUT= IN), if BUT Ì IN And INÌ A, i.e. BUT And IN consist of the same elementary events.
Credible Event is represented by an enclosing set Ω, and an impossible event is an empty subset of Æ in it. Inconsistency of events BUT And IN means that the corresponding subsets BUT And IN do not intersect: BUT∩IN = Æ.
Definition 3. The sum of two events A And IN(denoted FROM= BUT + IN) is called an event FROM, consisting of the onset of at least one of the events BUT or IN(the conjunction "or" for the amount is a keyword), i.e. comes or BUT, or IN, or BUT And IN together.
Example. Let two shooters shoot at the target at the same time, and the event BUT consists in the fact that the 1st shooter hits the target, and the event B- that the 2nd shooter hits the target. Event A+ B means that the target is hit, or, in other words, that at least one of the shooters (1st shooter or 2nd shooter, or both shooters) hit the target.
Similarly, the sum of a finite number of events BUT 1 , BUT 2 , …, BUT n (denoted BUT= BUT 1 + BUT 2 + … + BUT n) the event is called BUT, consisting of the occurrence of at least one from events BUT i ( i = 1, … , n), or an arbitrary set BUT i ( i = 1, 2, … , n).
Example. The sum of events A, B, C is an event consisting of the occurrence of one of the following events: BUT, B, C, BUT And IN, BUT And FROM, IN And FROM, BUT And IN And FROM, BUT or IN, BUT or FROM, IN or FROM,BUT or IN or FROM.
Definition 4. The product of two events BUT And IN called an event FROM(denoted FROM = A ∙ B), consisting in the fact that as a result of the test, an event also occurred BUT, and event IN simultaneously. (The conjunction "and" for producing events is the key word.)
Similarly to the product of a finite number of events BUT 1 , BUT 2 , …, BUT n (denoted BUT = BUT 1 ∙BUT 2 ∙…∙ BUT n) the event is called BUT, consisting in the fact that as a result of the test all the specified events occurred.
Example. If events BUT, IN, FROM is the appearance of a "coat of arms" in the first, second and third trials, respectively, then the event BUT× IN× FROM there is a "coat of arms" drop in all three trials.
Remark 1. For incompatible events BUT And IN fair equality A ∙ B= Æ, where Æ is an impossible event.
Remark 2. Events BUT 1 , BUT 2, … , BUT n form a complete group of pairwise incompatible events if .
Definition 5. opposite events two uniquely possible incompatible events that form a complete group are called. Event opposite to event BUT, is indicated. Event opposite to event BUT, is an addition to the event BUT to the set Ω.
For opposite events, two conditions are simultaneously satisfied A ∙= Æ and A+= Ω.
Definition 6. difference events BUT And IN(denoted BUT– IN) is called an event consisting in the fact that the event BUT will come, and the event IN - no and it is equal BUT– IN= BUT× .
Note that the events A + B, A ∙ B, , A - B it is convenient to interpret graphically using the Euler-Venn diagrams (Fig. 1.1).
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Rice. 1.1. Operations on events: negation, sum, product and difference |
Let us formulate an example as follows: let the experience G consists in firing at random over the region Ω, the points of which are elementary events ω. Let hitting the region Ω be a certain event Ω, and hitting the region BUT And IN- according to the events BUT And IN. Then the events A+B(or BUTÈ IN– light area in the figure), A ∙ B(or BUTÇ IN - area in the center) A - B(or BUT\IN - light subdomains) will correspond to the four images in Fig. 1.1. Under the conditions of the previous example with two shooters shooting at a target, the product of events BUT And IN there will be an event C = AÇ IN, consisting in hitting the target with both arrows.
Remark 3. If operations on events are represented as operations on sets, and events are represented as subsets of some set Ω, then the sum of events A+B match union BUTÈ IN these subsets, but the product of events A ∙ B- intersection BUT∩IN these subsets.
Thus, operations on events can be mapped to operations on sets. This correspondence is given in table. 1.1
Table 1.1
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Notation |
The Language of Probability Theory |
The Language of Set Theory |
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Space element. events |
Universal set |
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|
elementary event |
An element from the universal set |
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|
random event |
A subset of elements ω from Ω |
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|
Credible Event |
The set of all ω |
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|
Impossible event |
Empty set |
|
|
BUTÌ V |
BUT entails IN |
BUT- subset IN |
|
A+B(BUTÈ IN) |
Sum of events BUT And IN |
Union of sets BUT And IN |
|
BUT× V(BUTÇ IN) |
Production of events BUT And IN |
Intersection of many BUT And IN |
|
A - B(BUT\IN) |
Event Difference |
Set difference |
Actions on events have the following properties:
A + B = B + A, A ∙ B = B ∙ A(displacement);
(A+B) ∙ C = A× C + B× C, A ∙ B + C =(A + C) × ( B + C) (distributive);
(A+B) + FROM = BUT + (B + C), (A ∙ B) ∙ FROM= BUT ∙ (B ∙ C) (associative);
A + A = A, A ∙ A = A;
BUT + Ω = Ω, BUT∙ Ω = BUT;
Developments
Event. elemental event.
Space of elementary events.
Reliable event. Impossible event.
identical events.
Sum, product, difference of events.
opposite events. incompatible events.
Equivalent events.
Under event in probability theory is understood any fact that may or may not occur as a result of experience withrandom outcome. The simplest result of such an experiment (for example, the appearance of "heads" or "tails" when tossing a coin, hitting the target when shooting, the appearance of an ace when removing a card from the deck, randomly dropping a number when throwing a dieetc.) is calledelementary event .
The set of all elementary events E called element space tare events . Yes, at throwing a dice, this space consists of sixelementary events, and when a card is removed from the deck - from 52. An event can consist of one or more elementary events, for example, the appearance of two aces in a row when removing a card from the deck, or the loss of the same number when throwing a die three times. Then one can define event as an arbitrary subset of the space of elementary events.
a certain event the whole space of elementary events is called. Thus, a certain event is an event that must necessarily occur as a result of a given experience. When a dice is thrown, such an event is its fall on one of the faces.
Impossible event () is called an empty subset of the space of elementary events. That is, an impossible event cannot occur as a result of this experience. So, when throwing a dice, an impossible event is its fall on the edge.
Developments BUT And IN calledidentical (BUT= IN) if the event BUToccurs when and only when an event occursIN .
They say that the event BUT triggers an event IN ( BUT IN), if from the condition"event A happened" should "Event B happened".
Event FROM called sum of events BUT And IN (FROM = BUT IN) if the event FROM occurs if and only if either BUT, or IN.
Event FROM called product of events BUT And IN (FROM = BUT IN) if the event FROM happens when and only when it happens andBUT, And IN.
Event FROM called difference of events BUT And IN (FROM = BUT – IN) if the event FROM happens then and Only then, when it happens event BUT, and the event does not occur IN.
Event BUT"called opposite eventBUTif the event didn't happen BUT. So, a miss and a hit when shooting are opposite events.
Developments BUT And IN calledincompatible (BUT IN = ) , if their simultaneous occurrence is impossible. For example, dropping and "tails", and"eagle" when tossing a coin.
If during the experiment several events can occur and each of them, according to objective conditions, is no more possible than the other, then such events are calledequally possible . Examples of equally likely events: the appearance of a deuce, an ace and a jack when a card is removed from the deck, loss of any of the numbers from 1 to 6 when throwing a dice, etc.
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.
Joint and non-joint events.
The two events are called joint in a given experiment, if the appearance of one of them does not exclude the appearance of the other. Examples : Hitting an indestructible target with two different arrows, rolling the same number on two dice.
The two events are called incompatible(incompatible) in a given trial if they cannot occur together in the same trial. Several events are said to be incompatible if they are pairwise incompatible. Examples of incompatible events: a) hit and miss with one shot; b) a part is randomly extracted from a box with parts - the events “standard part removed” and “non-standard part removed”; c) the ruin of the company and its profit.
In other words, events BUT And IN are compatible if the corresponding sets BUT And IN have common elements, and are inconsistent if the corresponding sets BUT And IN have no common elements.
When determining the probabilities of events, the concept is often used equally possible events. Several events in a given experiment are called equally probable if, according to the symmetry conditions, there is reason to believe that none of them is objectively more possible than the others (the loss of a coat of arms and tails, the appearance of a card of any suit, the selection of a ball from an urn, etc.)
Associated with each trial is a series of events that, generally speaking, can occur simultaneously. For example, when throwing a die, an event is a deuce, and an 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
ü each test outcome is represented by one and only one elementary event;
ü any event associated with this test is a set of finite or infinite number of elementary events;
ü an event occurs if and only if one of the elementary events included in this set is realized.
An arbitrary but fixed space of elementary events can be represented as some 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. By analogy with set theory, one constructs event algebra. In this case, the following operations and relationships between events can be defined:
AÌ B(set inclusion relation: set BUT is a subset of the set IN) – event A leads to event B. In other words, the event IN occurs whenever an event occurs A. Example - Dropping a deuce entails dropping an even number of points.
(set equivalence relation) – event identically or equivalent to event . This is possible if and only if and simultaneously , i.e. each occurs whenever the other occurs. Example - event A - failure of the device, event B - failure of at least one of the blocks (parts) of the device.
() – sum of events. This is an event consisting in the fact that at least one of the two events or (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. Example - the target is hit by the first gun, the second or both at the same time.
() – 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. . Example - event A - taking out a card of a diamond suit from the deck, event B - taking out an ace, then - the appearance of a diamond ace has not occurred.
A geometric interpretation of operations on events is often useful. The graphical illustration of operations is called Venn diagrams.
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 necessary for the passenger consists in the appearance of either event A, or event B, or event C, which is symbolically written as:
D=A+B+C
The product of two eventsBUT And IN is an event consisting in the joint occurrence of events BUT And IN. 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 IN, 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 ( IN);
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: .
Basic theorems of probability theory
The probability of the sum of two incompatible events is equal to the sum of the probabilities of these events.
Let's write the addition theorem symbolically:
P(A + B) = P(A) + P(B),
where R- the probability of the corresponding event (the event is indicated in brackets).
Example . The patient has stomach bleeding. This symptom is recorded in ulcerative vessel erosion (event A), rupture of varicose veins of the esophagus (event B), stomach cancer (event C), gastric polyp (event D), hemorrhagic diathesis (event F), obstructive jaundice (event E) and end gastritis (eventG).
The doctor, based on the analysis of statistical data, assigns a probability value to each event:
In total, the doctor had 80 patients with gastric bleeding (n= 80), of which 12 had ulcerative vessel erosion (), at6 - rupture of varicose veins of the esophagus (), 36 had stomach cancer () etc.
To prescribe an examination, the doctor wants to determine the likelihood that stomach bleeding is associated with stomach disease (event I):
The likelihood that gastric bleeding is associated with stomach disease is quite high, and the doctor can determine the tactics of examination based on the assumption of stomach disease, justified at a quantitative level using probability theory.
If joint events are considered, the probability of the sum of two events is equal to the sum of the probabilities of these events without the probability of their joint occurrence.
Symbolically, this is written as follows:
If we imagine that the event BUT consists in hitting a target shaded with horizontal stripes while shooting, and the event IN- in hitting a target shaded with vertical stripes, then in the case of incompatible events, according to the addition theorem, the probability of the sum is equal to the sum of the probabilities of individual events. If these events are joint, then there is some probability corresponding to the joint occurrence of events BUT And IN. If you do not introduce a correction for the deductible P(AB), i.e. on the probability of the joint occurrence of events, then this probability will be taken into account twice, since the area shaded by both horizontal and vertical lines is an integral part of both targets and will be taken into account both in the first and in the second summand.
On fig. 1 a geometric interpretation is given that clearly illustrates this circumstance. In the upper part of the figure there are non-intersecting targets, which are an analogue of incompatible events, in the lower part - intersecting targets, which are an analogue of joint events (one shot can hit both target A and target B at once).
Before moving on to the multiplication theorem, it is necessary to consider the concepts of independent and dependent events and conditional and unconditional probabilities.
Independent an event B is an event A whose probability of occurrence does not depend on the occurrence or non-occurrence of event B.
addicted An event B is an event A whose probability of occurrence depends on the occurrence or non-occurrence of event B.
Example . An urn contains 3 balls, 2 white and 1 black. When choosing a ball at random, the probability of choosing a white ball (event A) is: P(A) = 2/3, and black (event B) P(B) = 1/3. We are dealing with a scheme of cases, and the probabilities of events are calculated strictly according to the formula. When the experiment is repeated, the probabilities of occurrence of events A and B remain unchanged if after each choice the ball is returned to the urn. In this case, events A and B are independent. If the ball chosen in the first experiment is not returned to the urn, then the probability of the event (A) in the second experiment depends on the occurrence or non-occurrence of the event (B) in the first experiment. So, if event B appeared in the first experiment (a black ball was chosen), then the second experiment is carried out if there are 2 white balls in the urn and the probability of the occurrence of event A in the second experiment is: P(A) = 2/2= 1.
If the event B did not appear in the first experiment (a white ball is chosen), then the second experiment is carried out if there are one white and one black balls in the urn and the probability of the occurrence of event A in the second experiment is equal to: P(A) = 1/2. Obviously, in this case, events A and B are closely related and the probabilities of their occurrence are dependent.
Conditional Probability event A is the probability of its occurrence, provided that event B has appeared. The conditional probability is symbolically denoted P(A/B).
If the probability of an event occurring BUT does not depend on the occurrence of the event IN, then the conditional probability of the event BUT is equal to the unconditional probability:
If the probability of occurrence of event A depends on the occurrence of event B, then the conditional probability can never be equal to the unconditional probability:
Revealing the dependence of various events among themselves is of great importance in solving practical problems. So, for example, an erroneous assumption about the independence of the appearance of certain symptoms in the diagnosis of heart defects using a probabilistic method developed at the Institute of Cardiovascular Surgery. A. N. Bakuleva, caused about 50% of erroneous diagnoses.
