Elementary outcomes are the classic definition of probability. Typical mistakes in solving problems for the classical definition of probability

Fundamentals of Probability Theory

Plan:

1. Random events

2. Classical definition of probability

3. Calculation of event probabilities and combinatorics

4. Geometric probability

Theoretical information

Random events.

random phenomenon- a phenomenon, the outcome of which is unambiguously determined. This concept can be interpreted in a fairly broad sense. Namely: everything in nature is quite accidental, the appearance and birth of any individual is a random phenomenon, the choice of goods in a store is also a random phenomenon, getting a mark on an exam is a random phenomenon, illness and recovery are random phenomena, etc.

Examples of random phenomena:

~ Shooting is carried out from a gun set at a given angle to the horizon. Hitting it on the target is accidental, but hitting a projectile in a certain "fork" is a pattern. You can specify the distance closer than and beyond which the projectile will not fly. Get some "fork dispersion of shells"

~ The same body is weighed several times. Strictly speaking, different results will be obtained each time, albeit differing by a negligibly small amount, but different.

~ An aircraft flying along the same route has a certain flight corridor within which the aircraft can maneuver, but it will never have exactly the same route

~ An athlete will never be able to run the same distance with the same time. His results will also be within a certain numerical range.

Experience, experiment, observation are tests

Trial- observation or fulfillment of a certain set of conditions that are performed repeatedly, and regularly repeated in this and the same sequence, duration, while observing other identical parameters.

Let's consider performance by the sportsman of a shot on a target. In order for it to be produced, it is necessary to fulfill such conditions as the preparation of the athlete, loading the weapon, aiming, etc. "Hit" and "miss" are events as a result of a shot.

Event– qualitative test result.

An event may or may not occur Events are indicated by capital Latin letters. For example: D ="The shooter hit the target". S="White ball drawn". K="Random lottery ticket without winning.".

Tossing a coin is a test. The fall of her "coat of arms" is one event, the fall of her "number" is the second event.

Any test involves the occurrence of several events. Some of them may be needed at a given time by the researcher, while others may not be needed.

The event is called random, if under the implementation of a certain set of conditions S it can either happen or not happen. In what follows, instead of saying "the set of conditions S is fulfilled," we will say briefly: "the test was carried out." Thus, the event will be considered as the result of the test.

~ The shooter shoots at a target divided into four areas. The shot is a test. Hitting a certain area of the target is an event.

~ There are colored balls in the urn. One ball is drawn at random from the urn. Removing a ball from an urn is a test. The appearance of a ball of a certain color is an event.

Types of random events

1. Events are said to be incompatible if the occurrence of one of them excludes the occurrence of other events in the same trial.

~ A part was taken at random from a box with parts. The appearance of a standard part excludes the appearance of a non-standard part. Events € a standard part appeared" and with a non-standard part appeared" - incompatible.

~ A coin is thrown. The appearance of the "coat of arms" excludes the appearance of the inscription. The events "a coat of arms appeared" and "an inscription appeared" are 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 a certain 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 special case is of greatest interest to us, since it is used below.

~ Two tickets of the money and clothing lottery were purchased. One and only one of the following events must occur:

1. "the winnings fell on the first ticket and did not fall on the second",

2. "the winnings did not fall on the first ticket and fell on the second",

3. "the winnings fell on both tickets",

4. "both tickets did not win."

These events form a complete group of pairwise incompatible events,

~ The shooter fired a shot at the target. One of the following two events is sure to occur: hit, miss. These two disjoint events also form a complete group.

2. Events are called equally possible if there is reason to believe that neither is more possible than the other.

~ The appearance of a "coat of arms" and the appearance of an inscription when a coin is tossed are equally possible events. Indeed, it is assumed that the coin is made of a homogeneous material, has a regular cylindrical shape, and the presence of a coinage does not affect the loss of one or another side of the coin.

~ The appearance of one or another number of points on a thrown dice is an equally probable event. Indeed, it is assumed that the die is made of a homogeneous material, has the shape of a regular polyhedron, and the presence of points does not affect the loss of any face.

3. The event is called authentic, if it cannot happen

4. The event is called not reliable if it can't happen.

5. The event is called opposite to some event if it consists of the non-occurrence of the given event. Opposite events are not compatible, but one of them must necessarily occur. Opposite events are commonly referred to as negations, i.e. a dash is written above the letter. The events are opposite: A and Ā; U and Ū, etc. .

The classical definition of probability

Probability is one of the basic concepts of probability theory.

There are several definitions of this concept. Let us give a definition that is called classical. Next, we point out the weaknesses of this definition and give other definitions that make it possible to overcome the shortcomings of the classical definition.

Consider the situation: A box contains 6 identical balls, 2 being red, 3 being blue and 1 being white. Obviously, the possibility of drawing a colored (i.e., red or blue) ball at random from an urn is greater than the possibility of drawing a white ball. This possibility can be characterized by a number, which is called the probability of an event (the appearance of a colored ball).

Probability- a number characterizing the degree of possibility of occurrence of the event.

In the situation under consideration, we denote:

Event A = "Pulling out a colored ball".

Each of the possible outcomes of the test (the test consists in extracting a ball from the urn) is called elementary (possible) outcome and event. Elementary outcomes can be denoted by letters with indexes below, for example: k 1 , k 2 .

In our example, there are 6 balls, so there are 6 possible outcomes: a white ball appeared; a red ball appeared; a blue ball appeared, and so on. It is easy to see that these outcomes form a complete group of pairwise incompatible events (only one ball will necessarily appear) and they are equally likely (the ball is taken out at random, the balls are the same and thoroughly mixed).

Elementary outcomes, in which the event of interest to us occurs, we will call favorable outcomes this event. In our example, the event is favored BUT(the appearance of a colored ball) the following 5 outcomes:

Thus the event BUT observed if one occurs in the test, no matter which, of the elementary outcomes that favor BUT. This is the appearance of any colored ball, of which there are 5 pieces in the box

In the considered example of elementary outcomes 6; of which 5 favor the event BUT. Hence, P(A)= 5/6. This number gives that quantification of the degree of possibility of the appearance of a colored ball.

Probability definition:

Probability of event A is the ratio of the number of outcomes favorable to this event to the total number of all equally possible incompatible elementary outcomes that form a complete group.

P(A)=m/n or P(A)=m: n, where:

m is the number of elementary outcomes that favor BUT;

P- the number of all possible elementary outcomes of the test.

It is assumed here that the elementary outcomes are incompatible, equally probable and form a complete group.

The following properties follow from the definition of probability:

1. The probability of a certain event is equal to one.

Indeed, if the event is reliable, then each elementary outcome of the test favors the event. In this case m = n hence p=1

2. The probability of an impossible event is zero.

Indeed, if the event is impossible, then none of the elementary outcomes of the trial favors the event. In this case m=0, hence p=0.

3.The probability of a random event is a positive number between zero and one. 0t< n.

In subsequent topics, theorems will be given that allow, from the known probabilities of some events, to find the probabilities of other events.

Measurement. There are 6 girls and 4 boys in the group of students. What is the probability that a randomly selected student will be a girl? will it be a young man?

p dev = 6 / 10 = 0.6 p jun = 4 / 10 = 0.4

The concept of "probability" in modern rigorous courses of probability theory is built on a set-theoretic basis. Let's take a look at some of this approach.

Suppose that as a result of the test one and only one of the following events occurs: w i(i=1, 2, .... n). Events w i, is called elementary events (elementary outcomes). O it follows that the elementary events are pairwise incompatible. The set of all elementary events that can appear in a trial is called elementary event spaceΩ (Greek letter omega capital), and the elementary events themselves - points in this space..

Event BUT identified with a subset (of the space Ω) whose elements are elementary outcomes favoring BUT; event AT is a subset Ω whose elements are outcomes that favor AT, etc. Thus, the set of all events that can occur in the test is the set of all subsets of Ω. Ω itself occurs with any outcome of the test, therefore Ω is a certain event; an empty subset of the space Ω is an -impossible event (it does not occur for any outcome of the test).

Elementary events are distinguished from among all events by topics, "each of them contains only one element Ω

To every elementary outcome w i match a positive number p i is the probability of this outcome, and the sum of all p i equal to 1 or with the sign of the sum, this fact will be written as an expression:

By definition, the probability P(A) events BUT is equal to the sum of the probabilities of elementary outcomes favoring BUT. Therefore, the probability of a certain event is equal to one, impossible - to zero, arbitrary - is between zero and one.

Let us consider an important particular case, when all outcomes are equally probable. The number of outcomes is equal to n, the sum of the probabilities of all outcomes is equal to one; hence the probability of each outcome is 1/n. Let the event BUT favors m outcomes.

Event Probability BUT is equal to the sum of the probabilities of outcomes favoring BUT:

P(A)=1/n + 1/n+…+1/n = n 1/n=1

The classical definition of probability is obtained.

There is still axiomatic approach to the concept of "probability". In the system of axioms proposed. Kolmogorov A.N., undefined concepts are elementary event and probability. The construction of a logically complete probability theory is based on the axiomatic definition of a random event and its probability.

Here are the axioms that define the probability:

1. Every event BUT assigned a non-negative real number P(A). This number is called the probability of the event. BUT.

2. The probability of a certain event is equal to one:

3. The probability of occurrence of at least one of the pairwise incompatible events is equal to the sum of the probabilities of these events.

Based on these axioms, the properties of probabilities for the relationship between them are derived as theorems.

Probability event is the ratio of the number of elementary outcomes that favor a given event to the number of all equally possible outcomes of experience in which this event may occur. The probability of an event A is denoted by P(A) (here P is the first letter of the French word probabilite - probability). According to the definition
(1.2.1)
where is the number of elementary outcomes favoring event A; - the number of all equally possible elementary outcomes of experience, forming a complete group of events.
This definition of probability is called classical. It arose at the initial stage of the development of probability theory.

The probability of an event has the following properties:
1. The probability of a certain event is equal to one. Let's designate a certain event by the letter . For a certain event, therefore
(1.2.2)
2. The probability of an impossible event is zero. We denote the impossible event by the letter . For an impossible event, therefore
(1.2.3)
3. The probability of a random event is expressed as a positive number less than one. Since the inequalities , or are satisfied for a random event, then
(1.2.4)
4. The probability of any event satisfies the inequalities
(1.2.5)
This follows from relations (1.2.2) -(1.2.4).

Example 1 An urn contains 10 balls of the same size and weight, of which 4 are red and 6 are blue. One ball is drawn from the urn. What is the probability that the drawn ball is blue?

Decision. The event "the drawn ball turned out to be blue" will be denoted by the letter A. This test has 10 equally possible elementary outcomes, of which 6 favor the event A. In accordance with formula (1.2.1), we obtain

Example 2 All natural numbers from 1 to 30 are written on identical cards and placed in an urn. After thoroughly mixing the cards, one card is removed from the urn. What is the probability that the number on the card drawn is a multiple of 5?

Decision. Denote by A the event "the number on the taken card is a multiple of 5". In this trial, there are 30 equally possible elementary outcomes, of which 6 outcomes favor event A (numbers 5, 10, 15, 20, 25, 30). Hence,

Example 3 Two dice are thrown, the sum of points on the upper faces is calculated. Find the probability of the event B, consisting in the fact that the top faces of the cubes will have a total of 9 points.

Decision. There are 6 2 = 36 equally possible elementary outcomes in this trial. Event B is favored by 4 outcomes: (3;6), (4;5), (5;4), (6;3), so

Example 4. A natural number not exceeding 10 is chosen at random. What is the probability that this number is prime?

Decision. Denote by the letter C the event "the chosen number is prime". In this case, n = 10, m = 4 (primes 2, 3, 5, 7). Therefore, the desired probability

Example 5 Two symmetrical coins are tossed. What is the probability that both coins have digits on the top sides?

Decision. Let's denote by the letter D the event "there was a number on the top side of each coin". There are 4 equally possible elementary outcomes in this test: (G, G), (G, C), (C, G), (C, C). (The notation (G, C) means that on the first coin there is a coat of arms, on the second - a number). Event D is favored by one elementary outcome (C, C). Since m = 1, n = 4, then

Example 6 What is the probability that the digits in a randomly chosen two-digit number are the same?

Decision. Two-digit numbers are numbers from 10 to 99; there are 90 such numbers in total. 9 numbers have the same digits (these are the numbers 11, 22, 33, 44, 55, 66, 77, 88, 99). Since in this case m = 9, n = 90, then
,
where A is the "number with the same digits" event.

Example 7 From the letters of the word differential one letter is chosen at random. What is the probability that this letter will be: a) a vowel b) a consonant c) a letter h?

Decision. There are 12 letters in the word differential, of which 5 are vowels and 7 are consonants. Letters h this word does not. Let's denote the events: A - "vowel", B - "consonant", C - "letter h". The number of favorable elementary outcomes: - for event A, - for event B, - for event C. Since n \u003d 12, then
, and .

Example 8 Two dice are tossed, the number of points on the top face of each dice is noted. Find the probability that both dice have the same number of points.

Decision. Let us denote this event by the letter A. Event A is favored by 6 elementary outcomes: (1;]), (2;2), (3;3), (4;4), (5;5), (6;6). In total there are equally possible elementary outcomes that form a complete group of events, in this case n=6 2 =36. So the desired probability

Example 9 The book has 300 pages. What is the probability that a randomly opened page will have a sequence number that is a multiple of 5?

Decision. It follows from the conditions of the problem that there will be n = 300 of all equally possible elementary outcomes that form a complete group of events. Of these, m = 60 favor the occurrence of the specified event. Indeed, a number that is a multiple of 5 has the form 5k, where k is a natural number, and , whence . Hence,
, where A - the "page" event has a sequence number that is a multiple of 5".

Example 10. Two dice are thrown, the sum of points on the upper faces is calculated. What is more likely to get a total of 7 or 8?

Decision. Let's designate the events: A - "7 points fell out", B - "8 points fell out". Event A is favored by 6 elementary outcomes: (1; 6), (2; 5), (3; 4), (4; 3), (5; 2), (6; 1), and event B - by 5 outcomes: (2; 6), (3; 5), (4; 4), (5; 3), (6; 2). There are n = 6 2 = 36 of all equally possible elementary outcomes. Hence, and .

So, P(A)>P(B), that is, getting a total of 7 points is a more likely event than getting a total of 8 points.

Tasks

1. A natural number not exceeding 30 is chosen at random. What is the probability that this number is a multiple of 3?
2. In the urn a red and b blue balls of the same size and weight. What is the probability that a randomly drawn ball from this urn is blue?
3. A number not exceeding 30 is chosen at random. What is the probability that this number is a divisor of zo?
4. In the urn a blue and b red balls of the same size and weight. One ball is drawn from this urn and set aside. This ball is red. Then another ball is drawn from the urn. Find the probability that the second ball is also red.
5. A natural number not exceeding 50 is chosen at random. What is the probability that this number is prime?
6. Three dice are thrown, the sum of points on the upper faces is calculated. What is more likely - to get a total of 9 or 10 points?
7. Three dice are tossed, the sum of the dropped points is calculated. What is more likely to get a total of 11 (event A) or 12 points (event B)?

Answers

1. 1/3. 2 . b/(a+b). 3 . 0,2. 4 . (b-1)/(a+b-1). 5 .0,3.6 . p 1 \u003d 25/216 - the probability of getting 9 points in total; p 2 \u003d 27/216 - the probability of getting 10 points in total; p2 > p1 7 . P(A) = 27/216, P(B) = 25/216, P(A) > P(B).

Questions

1. What is called the probability of an event?
2. What is the probability of a certain event?
3. What is the probability of an impossible event?
4. What are the limits of the probability of a random event?
5. What are the limits of the probability of any event?
6. What definition of probability is called classical?

In the economy, as well as in other areas of human activity or in nature, we constantly have to deal with events that cannot be accurately predicted. Thus, the volume of sales of goods depends on demand, which can vary significantly, and on a number of other factors that are almost impossible to take into account. Therefore, when organizing production and sales, one has to predict the outcome of such activities on the basis of either one's own previous experience, or similar experience of other people, or intuition, which is also largely based on experimental data.

In order to somehow evaluate the event under consideration, it is necessary to take into account or specially organize the conditions in which this event is recorded.

The implementation of certain conditions or actions to identify the event in question is called experience or experiment.

The event is called random if, as a result of the experiment, it may or may not occur.

The event is called authentic, if it necessarily appears as a result of this experience, and impossible if it cannot appear in this experience.

For example, snowfall in Moscow on November 30th is a random event. The daily sunrise can be considered a certain event. Snowfall at the equator can be seen as an impossible event.

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

Events are called incompatible if they cannot be observed together in the same experience. Thus, the presence of two and three cars in one store for sale at the same time are two incompatible events.

sum events is an event consisting in the occurrence of at least one of these events

An example of a sum of events is the presence of at least one of two products in a store.

work events is called an event consisting in the simultaneous occurrence of all these events

An event consisting in the appearance of two goods at the same time in the store is a product of events: - the appearance of one product, - the appearance of another product.

Events form a complete group of events if at least one of them necessarily occurs in the experience.

Example. The port has two berths for ships. Three events can be considered: - the absence of vessels at the berths, - the presence of one vessel at one of the berths, - the presence of two vessels at two berths. These three events form a complete group of events.

Opposite two unique possible events that form a complete group are called.

If one of the events that are opposite is denoted by , then the opposite event is usually denoted by .

Classical and statistical definitions of the probability of an event

Each of the equally possible test results (experiments) is called an elementary outcome. They are usually denoted by letters . For example, a dice is thrown. There can be six elementary outcomes according to the number of points on the sides.

From elementary outcomes, you can compose a more complex event. So, the event of an even number of points is determined by three outcomes: 2, 4, 6.

A quantitative measure of the possibility of occurrence of the event under consideration is the probability.

Two definitions of the probability of an event are most widely used: classic and statistical.

The classical definition of probability is related to the notion of a favorable outcome.

Exodus is called favorable this event, if its occurrence entails the occurrence of this event.

In the given example, the event under consideration is an even number of points on the dropped edge, has three favorable outcomes. In this case, the general
the number of possible outcomes. So, here you can use the classical definition of the probability of an event.

Classic definition equals the ratio of the number of favorable outcomes to the total number of possible outcomes

where is the probability of the event , is the number of favorable outcomes for the event, is the total number of possible outcomes.

In the considered example

The statistical definition of probability is associated with the concept of the relative frequency of occurrence of an event in experiments.

The relative frequency of occurrence of an event is calculated by the formula

where is the number of occurrence of an event in a series of experiments (tests).

Statistical definition. The probability of an event is the number relative to which the relative frequency is stabilized (established) with an unlimited increase in the number of experiments.

In practical problems, the relative frequency for a sufficiently large number of trials is taken as the probability of an event.

From these definitions of the probability of an event, it can be seen that the inequality always holds

To determine the probability of an event based on formula (1.1), combinatorics formulas are often used to find the number of favorable outcomes and the total number of possible outcomes.

Brief theory

For a quantitative comparison of events according to the degree of possibility of their occurrence, a numerical measure is introduced, which is called the probability of an event. The probability of a random event a number is called, which is an expression of a measure of the objective possibility of the occurrence of an event.

The values that determine how significant are the objective grounds for counting on the occurrence of an event are characterized by the probability of the event. It must be emphasized that probability is an objective quantity that exists independently of the cognizer and is conditioned by the totality of conditions that contribute to the occurrence of an event.

The explanations that we have given to the concept of probability are not a mathematical definition, since they do not define this concept quantitatively. There are several definitions of the probability of a random event, which are widely used in solving specific problems (classical, axiomatic, statistical, etc.).

The classical definition of the probability of an event reduces this concept to a more elementary concept of equally probable events, which is no longer subject to definition and is assumed to be intuitively clear. For example, if a dice is a homogeneous cube, then the fallout of any of the faces of this cube will be equally probable events.

Let a certain event be divided into equally probable cases, the sum of which gives the event. That is, the cases from , into which it breaks up, are called favorable for the event, since the appearance of one of them ensures the offensive.

The probability of an event will be denoted by the symbol .

The probability of an event is equal to the ratio of the number of cases favorable to it, out of the total number of unique, equally possible and incompatible cases, to the number, i.e.

This is the classical definition of probability. Thus, to find the probability of an event, it is necessary, after considering the various outcomes of the test, to find a set of the only possible, equally possible and incompatible cases, calculate their total number n, the number of cases m that favor this event, and then perform the calculation according to the above formula.

The probability of an event equal to the ratio of the number of outcomes of experience favorable to the event to the total number of outcomes of experience is called classical probability random event.

The following properties of probability follow from the definition:

Property 1. The probability of a certain event is equal to one.

Property 2. The probability of an impossible event is zero.

Property 3. The probability of a random event is a positive number between zero and one.

Property 4. The probability of the occurrence of events that form a complete group is equal to one.

Property 5. The probability of the occurrence of the opposite event is defined in the same way as the probability of the occurrence of event A.

The number of occurrences that favor the occurrence of the opposite event. Hence, the probability of the opposite event occurring is equal to the difference between unity and the probability of the event A occurring:

An important advantage of the classical definition of the probability of an event is that with its help the probability of an event can be determined without resorting to experience, but on the basis of logical reasoning.

When a set of conditions is met, a certain event will definitely happen, and the impossible will definitely not happen. Among the events that, when a complex of conditions is created, may or may not occur, the appearance of some can be counted on with more reason, on the appearance of others with less reason. If, for example, there are more white balls in the urn than black ones, then there are more reasons to hope for the appearance of a white ball when taken out of the urn at random than for the appearance of a black ball.

Problem solution example

Example 1

A box contains 8 white, 4 black and 7 red balls. 3 balls are drawn at random. Find the probabilities of the following events: - at least 1 red ball is drawn, - there are at least 2 balls of the same color, - there are at least 1 red and 1 white ball.

The solution of the problem

We find the total number of test outcomes as the number of combinations of 19 (8 + 4 + 7) elements of 3 each:

Find the probability of an event– drawn at least 1 red ball (1,2 or 3 red balls)

Required probability:

Let the event- there are at least 2 balls of the same color (2 or 3 white balls, 2 or 3 black balls and 2 or 3 red balls)

Number of outcomes favoring the event:

Required probability:

Let the event– there is at least one red and one white ball

(1 red, 1 white, 1 black or 1 red, 2 white or 2 red, 1 white)

Number of outcomes favoring the event:

Required probability:

Answer: P(A)=0.773;P(C)=0.7688; P(D)=0.6068

Example 2

Two dice are thrown. Find the probability that the sum of the points is at least 5.

Decision

Let the event be the sum of points not less than 5

Let's use the classical definition of probability:

Total number of possible trial outcomes

The number of trials that favor the event of interest to us

On the dropped face of the first dice, one point, two points ..., six points can appear. similarly, six outcomes are possible on the second die roll. Each of the outcomes of the first die can be combined with each of the outcomes of the second. Thus, the total number of possible elementary outcomes of the test is equal to the number of placements with repetitions (selection with placements of 2 elements from a set of volume 6):

Find the probability of the opposite event - the sum of points is less than 5

The following combinations of dropped points will favor the event:

№

1st bone

2nd bone

The geometric definition of probability is presented and the solution of the well-known meeting problem is given.

Initially, being just a collection of information and empirical observations of the game of dice, the theory of probability has become a solid science. Fermat and Pascal were the first to give it a mathematical framework.

From reflections on the eternal to the theory of probability

Two individuals to whom the theory of probability owes many fundamental formulas, Blaise Pascal and Thomas Bayes, are known as deeply religious people, the latter was a Presbyterian minister. Apparently, the desire of these two scientists to prove the fallacy of the opinion about a certain Fortune, bestowing good luck on her favorites, gave impetus to research in this area. After all, in fact, any game of chance, with its wins and losses, is just a symphony of mathematical principles.

Thanks to the excitement of the Chevalier de Mere, who was equally a gambler and a person who was not indifferent to science, Pascal was forced to find a way to calculate the probability. De Mere was interested in this question: "How many times do you need to throw two dice in pairs so that the probability of getting 12 points exceeds 50%?". The second question that interested the gentleman extremely: "How to divide the bet between the participants in the unfinished game?" Of course, Pascal successfully answered both questions of de Mere, who became the unwitting initiator of the development of the theory of probability. It is interesting that the person of de Mere remained known in this area, and not in literature.

Previously, no mathematician has yet made an attempt to calculate the probabilities of events, since it was believed that this was only a guesswork solution. Blaise Pascal gave the first definition of the probability of an event and showed that this is a specific figure that can be justified mathematically. Probability theory has become the basis for statistics and is widely used in modern science.

What is randomness

If we consider a test that can be repeated an infinite number of times, then we can define a random event. This is one of the possible outcomes of the experience.

Experience is the implementation of specific actions in constant conditions.

In order to be able to work with the results of experience, events are usually denoted by the letters A, B, C, D, E ...

Probability of a random event

To be able to proceed to the mathematical part of probability, it is necessary to define all its components.

The probability of an event is a numerical measure of the possibility of the occurrence of some event (A or B) as a result of an experience. The probability is denoted as P(A) or P(B).

Probability theory is:

reliable the event is guaranteed to occur as a result of the experiment Р(Ω) = 1;
impossible the event can never happen Р(Ø) = 0;
random the event lies between certain and impossible, that is, the probability of its occurrence is possible, but not guaranteed (the probability of a random event is always within 0≤P(A)≤1).

Relationships between events

Both one and the sum of events A + B are considered when the event is counted in the implementation of at least one of the components, A or B, or both - A and B.

In relation to each other, events can be:

Equally possible.
compatible.
Incompatible.
Opposite (mutually exclusive).
Dependent.

If two events can happen with equal probability, then they equally possible.

If the occurrence of event A does not nullify the probability of occurrence of event B, then they compatible.

If events A and B never occur at the same time in the same experiment, then they are called incompatible. Tossing a coin is a good example: coming up tails is automatically not coming up heads.

The probability for the sum of such incompatible events consists of the sum of the probabilities of each of the events:

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

If the occurrence of one event makes the occurrence of another impossible, then they are called opposite. Then one of them is designated as A, and the other - Ā (read as "not A"). The occurrence of event A means that Ā did not occur. These two events form a complete group with a sum of probabilities equal to 1.

Dependent events have mutual influence, decreasing or increasing each other's probability.

Relationships between events. Examples

It is much easier to understand the principles of probability theory and the combination of events using examples.

The experiment that will be carried out is to pull the balls out of the box, and the result of each experiment is an elementary outcome.

An event is one of the possible outcomes of an experience - a red ball, a blue ball, a ball with the number six, etc.

Test number 1. There are 6 balls, three of which are blue with odd numbers, and the other three are red with even numbers.

Test number 2. There are 6 blue balls with numbers from one to six.

Based on this example, we can name combinations:

Reliable event. In Spanish No. 2, the event "get the blue ball" is reliable, since the probability of its occurrence is 1, since all the balls are blue and there can be no miss. Whereas the event "get the ball with the number 1" is random.
Impossible event. In Spanish No. 1 with blue and red balls, the event "get the purple ball" is impossible, since the probability of its occurrence is 0.
Equivalent events. In Spanish No. 1, the events “get the ball with the number 2” and “get the ball with the number 3” are equally likely, and the events “get the ball with an even number” and “get the ball with the number 2” have different probabilities.
Compatible events. Getting a six in the process of throwing a die twice in a row are compatible events.
Incompatible events. In the same Spanish No. 1 events "get the red ball" and "get the ball with an odd number" cannot be combined in the same experience.
opposite events. The most striking example of this is coin tossing, where drawing heads is the same as not drawing tails, and the sum of their probabilities is always 1 (full group).
Dependent events. So, in Spanish No. 1, you can set yourself the goal of extracting a red ball twice in a row. Extracting it or not extracting it the first time affects the probability of extracting it the second time.

It can be seen that the first event significantly affects the probability of the second (40% and 60%).

Event Probability Formula

The transition from fortune-telling to exact data occurs by transferring the topic to the mathematical plane. That is, judgments about a random event like "high probability" or "minimum probability" can be translated to specific numerical data. It is already permissible to evaluate, compare and introduce such material into more complex calculations.

From the point of view of calculation, the definition of the probability of an event is the ratio of the number of elementary positive outcomes to the number of all possible outcomes of experience with respect to a particular event. Probability is denoted by P (A), where P means the word "probability", which is translated from French as "probability".

So, the formula for the probability of an event is:

Where m is the number of favorable outcomes for event A, n is the sum of all possible outcomes for this experience. The probability of an event is always between 0 and 1:

0 ≤ P(A) ≤ 1.

Calculation of the probability of an event. Example

Let's take Spanish. No. 1 with balls, which is described earlier: 3 blue balls with numbers 1/3/5 and 3 red balls with numbers 2/4/6.

Based on this test, several different tasks can be considered:

A - red ball drop. There are 3 red balls, and there are 6 variants in total. This is the simplest example, in which the probability of an event is P(A)=3/6=0.5.
B - dropping an even number. There are 3 (2,4,6) even numbers in total, and the total number of possible numerical options is 6. The probability of this event is P(B)=3/6=0.5.
C - loss of a number greater than 2. There are 4 such options (3,4,5,6) out of the total number of possible outcomes 6. The probability of the event C is P(C)=4/6=0.67.

As can be seen from the calculations, event C has a higher probability, since the number of possible positive outcomes is higher than in A and B.

Incompatible events

Such events cannot appear simultaneously in the same experience. As in Spanish No. 1, it is impossible to get a blue and a red ball at the same time. That is, you can get either a blue or a red ball. In the same way, an even and an odd number cannot appear in a die at the same time.

The probability of two events is considered as the probability of their sum or product. The sum of such events A + B is considered to be an event that consists in the appearance of an event A or B, and the product of their AB - in the appearance of both. For example, the appearance of two sixes at once on the faces of two dice in one throw.

The sum of several events is an event that implies the occurrence of at least one of them. The product of several events is the joint occurrence of them all.

In probability theory, as a rule, the use of the union "and" denotes the sum, the union "or" - multiplication. Formulas with examples will help you understand the logic of addition and multiplication in probability theory.

Probability of the sum of incompatible events

If the probability of incompatible events is considered, then the probability of the sum of events is equal to the sum of their probabilities:

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

For example: we calculate the probability that in Spanish. No. 1 with blue and red balls will drop a number between 1 and 4. We will calculate not in one action, but by the sum of the probabilities of the elementary components. So, in such an experiment there are only 6 balls or 6 of all possible outcomes. The numbers that satisfy the condition are 2 and 3. The probability of getting the number 2 is 1/6, the probability of the number 3 is also 1/6. The probability of getting a number between 1 and 4 is:

The probability of the sum of incompatible events of a complete group is 1.

So, if in the experiment with a cube we add up the probabilities of getting all the numbers, then as a result we get one.

This is also true for opposite events, for example, in the experiment with a coin, where one of its sides is the event A, and the other is the opposite event Ā, as is known,

Р(А) + Р(Ā) = 1

Probability of producing incompatible events

Multiplication of probabilities is used when considering the occurrence of two or more incompatible events in one observation. The probability that events A and B will appear in it at the same time is equal to the product of their probabilities, or:

P(A*B)=P(A)*P(B)

For example, the probability that in No. 1 as a result of two attempts, a blue ball will appear twice, equal to

That is, the probability of the occurrence of an event when, as a result of two attempts with the extraction of balls, only blue balls will be extracted is equal to 25%. It is very easy to do practical experiments on this problem and see if this is actually the case.

Joint Events

Events are considered joint when the appearance of one of them can coincide with the appearance of the other. Despite the fact that they are joint, the probability of independent events is considered. For example, throwing two dice can give a result when the number 6 falls on both of them. Although the events coincided and appeared simultaneously, they are independent of each other - only one six could fall out, the second die has no influence on it.

The probability of joint events is considered as the probability of their sum.

The probability of the sum of joint events. Example

The probability of the sum of events A and B, which are joint in relation to each other, is equal to the sum of the probabilities of the event minus the probability of their product (that is, their joint implementation):

R joint. (A + B) \u003d P (A) + P (B) - P (AB)

Assume that the probability of hitting the target with one shot is 0.4. Then event A - hitting the target in the first attempt, B - in the second. These events are joint, since it is possible that it is possible to hit the target both from the first and from the second shot. But the events are not dependent. What is the probability of the event of hitting the target with two shots (at least one)? According to the formula:

0,4+0,4-0,4*0,4=0,64

The answer to the question is: "The probability of hitting the target with two shots is 64%."

This formula for the probability of an event can also be applied to incompatible events, where the probability of the joint occurrence of an event P(AB) = 0. This means that the probability of the sum of incompatible events can be considered a special case of the proposed formula.

Probability geometry for clarity

Interestingly, the probability of the sum of joint events can be represented as two areas A and B that intersect with each other. As you can see from the picture, the area of their union is equal to the total area minus the area of their intersection. This geometric explanation makes the seemingly illogical formula more understandable. Note that geometric solutions are not uncommon in probability theory.

The definition of the probability of the sum of a set (more than two) of joint events is rather cumbersome. To calculate it, you need to use the formulas that are provided for these cases.

Dependent events

Dependent events are called if the occurrence of one (A) of them affects the probability of the occurrence of the other (B). Moreover, the influence of both the occurrence of event A and its non-occurrence is taken into account. Although events are called dependent by definition, only one of them is dependent (B). The usual probability was denoted as P(B) or the probability of independent events. In the case of dependents, a new concept is introduced - the conditional probability P A (B), which is the probability of the dependent event B under the condition that the event A (hypothesis) has occurred, on which it depends.

But event A is also random, so it also has a probability that must and can be taken into account in the calculations. The following example will show how to work with dependent events and a hypothesis.

Example of calculating the probability of dependent events

A good example for calculating dependent events is a standard deck of cards.

On the example of a deck of 36 cards, consider dependent events. It is necessary to determine the probability that the second card drawn from the deck will be a diamond suit, if the first card drawn is:

Tambourine.
Another suit.

Obviously, the probability of the second event B depends on the first A. So, if the first option is true, which is 1 card (35) and 1 diamond (8) less in the deck, the probability of event B:

P A (B) \u003d 8 / 35 \u003d 0.23

If the second option is true, then there are 35 cards in the deck, and the total number of tambourines (9) is still preserved, then the probability of the following event is B:

P A (B) \u003d 9/35 \u003d 0.26.

It can be seen that if event A is conditional on the fact that the first card is a diamond, then the probability of event B decreases, and vice versa.

Multiplication of dependent events

Based on the previous chapter, we accept the first event (A) as a fact, but in essence, it has a random character. The probability of this event, namely the extraction of a tambourine from a deck of cards, is equal to:

P(A) = 9/36=1/4

Since the theory does not exist by itself, but is called upon to serve practical purposes, it is fair to note that most often the probability of producing dependent events is needed.

According to the theorem on the product of the probabilities of dependent events, the probability of occurrence of jointly dependent events A and B is equal to the probability of one event A multiplied by the conditional probability of event B (depending on A):

P (AB) \u003d P (A) * P A (B)

Then in the example with a deck, the probability of drawing two cards with a suit of diamonds is:

9/36*8/35=0.0571 or 5.7%

And the probability of extracting not diamonds at first, and then diamonds, is equal to:

27/36*9/35=0.19 or 19%

It can be seen that the probability of occurrence of event B is greater, provided that a card of a suit other than a diamond is drawn first. This result is quite logical and understandable.

Total probability of an event

When a problem with conditional probabilities becomes multifaceted, it cannot be calculated by conventional methods. When there are more than two hypotheses, namely A1, A2, ..., A n , .. forms a complete group of events under the condition:

P(A i)>0, i=1,2,…
A i ∩ A j =Ø,i≠j.
Σ k A k =Ω.

So, the formula for the total probability for event B with a complete group of random events A1, A2, ..., A n is:

A look into the future

The probability of a random event is essential in many areas of science: econometrics, statistics, physics, etc. Since some processes cannot be described deterministically, since they themselves are probabilistic, special methods of work are needed. The probability of an event theory can be used in any technological field as a way to determine the possibility of an error or malfunction.

It can be said that, by recognizing the probability, we somehow take a theoretical step into the future, looking at it through the prism of formulas.