Definition 1. It is said that in some experience an event BUT entails followed by the occurrence of an event AT if when the event occurs BUT the event comes AT. Notation of this definition BUT Ì AT. In terms of elementary events, this means that each elementary event included in BUT, is also included in AT.

Definition 2. Events BUT and AT are called equal or equivalent (denoted BUT= AT), if BUT Ì AT and ATÌ A, i.e. BUT and AT 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 AT means that the corresponding subsets BUT and AT do not intersect: BUT∩AT = Æ.

Definition 3. The sum of two events A and AT(denoted FROM= BUT + AT) is called an event FROM, consisting of the onset of at least one of the events BUT or AT(the conjunction "or" for the amount is a keyword), i.e. comes or BUT, or AT, or BUT and AT 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 AT, BUT and FROM, AT and FROM, BUT and AT and FROM, BUT or AT, BUT or FROM, AT or FROM,BUT or AT or FROM.

Definition 4. The product of two events BUT and AT 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 AT 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, AT, FROM is the appearance of the "coat of arms" in the first, second and third trials, respectively, then the event BUT× AT× FROM there is a "coat of arms" drop in all three trials.

Remark 1. For incompatible events BUT and AT 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 AT(denoted BUT– AT) is called an event consisting in the fact that the event BUT will come, and the event AT - no and it is equal BUT– AT= 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).

Rice. 1.1. Operations on events: negation, sum, product and difference

Let us formulate an example as follows: let the experience G consists in shooting 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 AT- according to the events BUT and AT. Then the events A+B(or BUTÈ AT– light area in the figure), A ∙ B(or BUTÇ AT - area in the center) A - B(or BUT\AT - 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 AT there will be an event C = AÇ AT, 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 AT these subsets, but the product of events A ∙ B- intersection BUT∩AT these subsets.

Thus, operations on events can be mapped to operations on sets. This correspondence is given in table. 1.1

Table 1.1

Notation	The Language of Probability Theory	The Language of Set Theory
	Space element. events	Universal set
	elementary event	An element from the universal set
	random event	A subset of elements ω from Ω
	Credible Event	The set of all ω
	Impossible event	Empty set
BUTÌ V	BUT entails AT	BUT- subset AT
A+B(BUTÈ AT)	Sum of events BUT and AT	Union of sets BUT and AT
BUT× V(BUTÇ AT)	Production of events BUT and AT	Intersection of many BUT and AT
A - B(BUT\AT)	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;

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 AT are compatible if the corresponding sets BUT and AT have common elements, and are inconsistent if the corresponding sets BUT and AT 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 AT) – event A leads to event B. In other words, the event AT 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.

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 forelements 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>

1. 
   Space of elementary events. Random event. Reliable event. Impossible event.

Space of elementary events - any set of mutually exclusive outcomes of the experiment, such that each result of interest to us can be uniquely described using the elements of this set. It happens finite and infinite (countable and uncountable)

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.

1. 
   Actions on events: sum, product and difference of events. opposite event. Joint and non-joint events. Complete group of events.

Joint Events - if they can occur simultaneously as a result of the experiment.

^ 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.

1. 
   Classical, statistical and geometric definitions of probability. Basic properties of event probability.

Classic scheme: P(A)=, n is the number of possible outcomes, m is the number of outcomes favoring event A. statistical definition: W(A)=, n is the number of experiments performed, m is the number of experiments performed in which event A appeared. Geometric definition: P(A)= , 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.

1. 
   The theorem of addition of probabilities of incompatible events and consequences from it.

P(A+B) = P(A)+P(B).Consequence 1. P (A 1 + A 2 + ... + A k) \u003d P (A 1) + P (A 2) + ... + P (A k), A 1, A 2, ..., A k - are pairwise incompatible. Consequence 2 . P(A)+P(Ᾱ) = 1. Corollary 3 . The sum of the probabilities of events forming a complete group is 1.

1. 
   Conditional Probability. independent events. Multiplying the probabilities of dependent and independent events.

Conditional Probability - P(B), is calculated on the assumption that event A has already occurred. A and B are independent if the occurrence of one of them does not change the probability of occurrence of the other.

^ 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).

1. 
   ^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

P(A+B) = P(A) + P(B) - P(A∙B)

1. 
   Total Probability Formula. Bayes formulas.

Total Probability Formula

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 (N 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" testing - 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 n trials be carried out, 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

1. 
   ȹ(x) – odd function: ȹ(-x)=- ȹ(x)
2. 
   ȹ(x) – monotonically increasing
3. 
   values ​​ȹ(x) (-0.5;0.5), and lim x →∞ ȹ(x)=0.5; lim x →-∞ ȹ(x)=-0.5

Consequences

1. 
   P n (│m-np│≤Ɛ) ≈ 2 ȹ (Ɛ/(npq) 1/2)
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
3. 
   P n (│(m/n) - p│≈ ∆) ≈ 2 ȹ(∆n 1/2 /(pq)^ 1/2)

m/n relative frequency of occurrence of success in trials

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.

Target: to acquaint students with the rules of addition and multiplication of probabilities, the concept of opposite events on Euler circles.

Probability theory is a mathematical science that studies regularities in random phenomena.

random phenomenon- this is a phenomenon that, with repeated reproduction of the same experience, proceeds each time in a slightly different way.

Here are examples of random events: dice are thrown, a coin is thrown, a target is fired, etc.

All the examples given can be considered from the same point of view: random variations, unequal results of a series of experiments, the basic conditions of which remain unchanged.

It is quite obvious that in nature there is not a single physical phenomenon in which elements of chance would not be present to one degree or another. No matter how precisely and in detail the conditions of the experiment are fixed, it is impossible to ensure that when the experiment is repeated, the results completely and exactly coincide.

Random deviations inevitably accompany any natural phenomenon. Nevertheless, in a number of practical problems, these random elements can be neglected, considering instead of a real phenomenon, its simplified “model” scheme and assuming that, under the given experimental conditions, the phenomenon proceeds in a quite definite way.

However, there are a number of problems where the outcome of the experiment that interests us depends on such a large number of factors that it is practically impossible to register and take into account all these factors.

Random events can be combined with each other in various ways. In this case, new random events are formed.

For a visual representation of events, use Euler diagrams. On each such diagram, a rectangle represents the set of all elementary events (Fig. 1). All other events are depicted inside the rectangle as some part of it, bounded by a closed line. Usually such events depict circles or ovals inside a rectangle.

Consider the most important properties of events using Euler diagrams.

Combining eventsA andB call the event C, consisting of elementary events belonging to the event A or B (sometimes the union is called the sum).

The result of the union can be represented graphically by the Euler diagram (Fig. 2).

Intersection of events A and B call an event C that favors both event A and event B (sometimes the intersections are called the product).

The result of the intersection can be represented graphically by the Euler diagram (Fig. 3).

If events A and B do not have common favorable elementary events, then they cannot occur simultaneously in the course of the same experience. Such events are called incompatible, and their intersection - empty event.

The difference between events A and B call an event C, consisting of elementary events A, which are not elementary events B.

The result of the difference can be represented graphically by the Euler diagram (Fig. 4)

Let the rectangle represent all elementary events. Event A is depicted as a circle inside a rectangle. The rest of the rectangle depicts the opposite of event A, the event (Fig. 5)

Event opposite to event A An event is called an event that is favored by all elementary events that are not favorable to event A.

The event opposite to event A is usually denoted by .

Examples of opposite events.

Combining multiple events is called an event consisting in the occurrence of at least one of these events.

For example, if the experience consists of five shots on a target and the events are given:

A0 - no hits;
A1 - exactly one hit;
A2 - exactly 2 hits;
A3 - exactly 3 hits;
A4 - exactly 4 hits;
A5 - exactly 5 hits.

Find events: no more than two hits and no less than three hits.

Solution: A=A0+A1+A2 - no more than two hits;

B = A3 + A4 + A5 - at least three hits.

Intersection of several events An event consisting in the joint occurrence of all these events is called.

For example, if three shots are fired at a target and the events are considered:

B1 - miss on the first shot,
B2 - miss on the second shot,
VZ - miss on the third shot,

that event is that there will be no hit on the target.

When determining probabilities, it is often necessary to represent complex events as combinations of simpler events, using both union and intersection of events.

For example, let's say three shots are fired at a target, and the following elementary events are considered:

First shot hit
- miss on first shot
- hit on the second shot,
- miss on the second shot,
- hit on the third shot,
- miss on the third shot.

Consider a more complex event B, consisting in the fact that as a result of these three shots there will be exactly one hit on the target. Event B can be represented as the following combination of elementary events:

The event C, consisting in the fact that there will be at least two hits on the target, can be represented as:

Figures 6.1 and 6.2 show the union and intersection of three events.

fig.6

To determine the probabilities of events, not direct direct methods are used, but indirect ones. Allowing the known probabilities of some events to determine the probabilities of other events associated with them. Applying these indirect methods, we always use the basic rules of probability theory in one form or another. There are two of these rules: the rule of adding probabilities and the rule of multiplying probabilities.

The probability addition rule is formulated as follows.

The probability of combining two incompatible events is equal to the sum of the probabilities of these events:

P (A + B) = P (A) + P (B).

The sum of the probabilities of opposite events is equal to one:

P(A) + P() = 1.

In practice, it is often easier to calculate the probability of the opposite event A than the probability of the direct event A. In these cases, calculate P (A) and find

P(A) = 1-P().

Let's look at a few examples of applying the addition rule.

Example 1. There are 1000 tickets in the lottery; of these, a win of 500 rubles falls on one ticket, winnings of 100 rubles on 10 tickets, winnings of 20 rubles on 50 tickets, winnings of 5 rubles on 100 tickets, the rest of the tickets are non-winning. Someone buys one ticket. Find the probability of winning at least 20 rubles.

Solution. Consider the events:

A - win at least 20 rubles,

A1 - win 20 rubles,
A2 - win 100 rubles,
A3 - win 500 rubles.

Obviously, A = A1 + A2 + A3.

According to the rule of addition of probabilities:

P(A) = P(A1) + P(A2) + P(A3) = 0.050 + 0.010 + 0.001 = 0.061.

Example 2. Three ammunition depots are bombed, and one bomb is dropped. The probability of hitting the first warehouse is 0.01; in the second 0.008; in the third 0.025. When one of the warehouses is hit, all three explode. Find the probability that the warehouses will be blown up.

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:

(relation of inclusion of sets: 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. .

(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 did not occur.

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.