It is clear that each event has some degree of possibility of its occurrence (of its implementation). In order to quantitatively compare events with each other according to the degree of their possibility, it is obviously necessary to associate a certain number with each event, which is the greater, the more possible the event is. This number is called the probability of the event.

Event Probability- is a numerical measure of the degree of objective possibility of the occurrence of this event.

Consider a stochastic experiment and a random event A observed in this experiment. Let's repeat this experiment n times and let m(A) be the number of experiments in which event A happened.

Relation (1.1)

called relative frequency event A in the series of experiments.

It is easy to verify the validity of the properties:

if A and B are incompatible (AB= ), then ν(A+B) = ν(A) + ν(B) (1.2)

The relative frequency is determined only after a series of experiments and, generally speaking, may vary from series to series. However, experience shows that in many cases, as the number of experiments increases, the relative frequency approaches a certain number. This fact of the stability of the relative frequency has been repeatedly verified and can be considered experimentally established.

Example 1.19.. If you toss one coin, no one can predict which side it will land on. But if you throw two tons of coins, then everyone will say that about one ton will fall up with a coat of arms, that is, the relative frequency of the coat of arms falling is approximately equal to 0.5.

If, as the number of experiments increases, the relative frequency of the event ν(A) tends to some fixed number, then we say that event A is statistically stable, and this number is called the probability of event A.

Probability of an event BUT some fixed number P(A) is called, to which the relative frequency ν(A) of this event tends with an increase in the number of experiments, that is,

This definition is called statistical definition of probability .

Consider some stochastic experiment and let the space of its elementary events consist of a finite or infinite (but countable) set of elementary events ω 1 , ω 2 , …, ω i , … . suppose that each elementary event ω i is assigned a certain number - р i , which characterizes the degree of possibility of the occurrence of this elementary event and satisfies the following properties:

Such a number p i is called elementary event probabilityω i .

Now let A be a random event observed in this experiment, and a certain set corresponds to it

In such a setting event probability BUT is called the sum of the probabilities of elementary events favoring A(included in the corresponding set A):

The probability introduced in this way has the same properties as the relative frequency, namely:

And if AB \u003d (A and B are incompatible),

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

Indeed, according to (1.4)

In the last relation, we have taken advantage of the fact that no elementary event can simultaneously favor two incompatible events.

We especially note that the theory of probability does not indicate how to determine p i , they must be sought from practical considerations or obtained from an appropriate statistical experiment.

As an example, consider the classical scheme of probability theory. To do this, consider a stochastic experiment, the space of elementary events of which consists of a finite (n) number of elements. Let us additionally assume that all these elementary events are equally probable, that is, the probabilities of elementary events are p(ω i)=p i =p. Hence it follows that

Example 1.20. When tossing a symmetrical coin, the coat of arms and tails are equally possible, their probabilities are 0.5.

Example 1.21. When a symmetrical die is thrown, all faces are equally likely, their probabilities are 1/6.

Let now event A be favored by m elementary events, they are usually called outcomes favoring event A. Then

Got classical definition of probability: the probability P(A) of event A is equal to the ratio of the number of outcomes favoring event A to total number outcomes

Example 1.22. An urn contains m white balls and n black ones. What is the probability of drawing a white ball?

Solution. There are m+n elementary events in total. They are all equally incredible. Favorable event A of them m. Consequently, .

The following properties follow from the definition of probability:

Property 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=p, Consequently,

P(A)=m/n=n/n=1.(1.6)

Property 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 t= 0, therefore, P(A)=m/n=0/n=0. (1.7)

Property 3.Probability random event is a positive number between zero and one.

Indeed, only a part of the total number of elementary outcomes of the test favors a random event. That is, 0≤m≤n, which means 0≤m/n≤1, therefore, the probability of any event satisfies the double inequality 0≤ P(A) ≤1. (1.8)

Comparing the definitions of probability (1.5) and relative frequency (1.1), we conclude: the definition of probability does not require testing to be done in reality; the definition of the relative frequency assumes that tests were actually carried out. In other words, the probability is calculated before the experience, and the relative frequency - after the experience.

However, the calculation of probability requires prior information about the number or probabilities of elementary outcomes favoring a given event. In the absence of such preliminary information, empirical data are used to determine the probability, that is, the relative frequency of the event is determined from the results of a stochastic experiment.

Example 1.23. Department of technical control discovered 3 non-standard parts in a batch of 80 randomly selected parts. Relative frequency of occurrence of non-standard parts r (A)= 3/80.

Example 1.24. By purpose.produced 24 shot, and 19 hits were registered. The relative frequency of hitting the target. r (A)=19/24.

Long-term observations have shown that if experiments are carried out under the same conditions, in each of which the number of tests is sufficiently large, then the relative frequency exhibits the property of stability. This property is that in various experiments the relative frequency changes little (the less, the more tests are made), fluctuating around a certain constant number. It turned out that this constant number can be taken as an approximate value of the probability.

The relationship between relative frequency and probability will be described in more detail and more precisely below. Now let us illustrate the stability property with examples.

Example 1.25. According to Swedish statistics, the relative birth rate of girls in 1935 by month is characterized by the following numbers (numbers are arranged in the order of months, starting from January): 0,486; 0,489; 0,490; 0.471; 0,478; 0,482; 0.462; 0,484; 0,485; 0,491; 0,482; 0,473

The relative frequency fluctuates around the number 0.481, which can be taken as approximate value the probability of having girls.

Note that the statistics of different countries give approximately the same value of the relative frequency.

Example 1.26. Repeated experiments were carried out tossing a coin, in which the number of occurrences of the "coat of arms" was counted. The results of several experiments are shown in the table.

1. Statement of the main theorems and probability formulas: addition theorem, conditional probability, multiplication theorem, independence of events, formula full probability.

Goals: creation of favorable conditions for the introduction of the concept of the probability of an event; familiarity with the basic theorems and formulas of probability theory; enter the total probability formula.

Lesson progress:

Random experiment (experiment) is a process in which different outcomes are possible, and it is impossible to predict in advance what the result will be. The possible mutually exclusive outcomes of an experience are called its elementary events . The set of elementary events will be denoted by W.

random event an event is called, about which it is impossible to say in advance whether it will occur as a result of experience or not. Each random event A that occurred as a result of the experiment can be associated with a group of elementary events from W. The elementary events that make up this group are called favorable to the occurrence of event A.

The set W can also be considered as a random event. Since it includes all elementary events, it will necessarily occur as a result of experience. Such an event is called reliable .

If for a given event there are no favorable elementary events from W, then it cannot occur as a result of the experiment. Such an event is called impossible.

Events are called equally possible , if as a result of the test are provided equal opportunity implementation of these events. Two random events are called opposite if, as a result of the experiment, one of them occurs if and only if the other does not occur. The event opposite to event A is denoted by .

Events A and B are called incompatible if the occurrence of one of them excludes the occurrence of the other. Events A 1 , A 2 , ..., A n are called pairwise incompatible, if any two of them are incompatible. Events A 1 , A 2 , ..., An form complete system pairwise incompatible events if, as a result of the test, one and only one of them is sure to occur.

The sum (combination) of events A 1 , A 2 , ..., A n is such an event C, which consists in the fact that at least one of the events A 1 , A 2 , ..., A n has occurred The sum of events is denoted as follows:

C \u003d A 1 + A 2 + ... + A n.

The product (intersection) of events A 1 , A 2 , ..., A n such an event P is called, which consists in the fact that all events A 1 , A 2 , ..., A n occurred simultaneously. The product of events is denoted

The probability P(A) in the theory of probability acts as a numerical characteristic of the degree of possibility of the occurrence of any particular random event A with repeated repetition of tests.

For example, in 1000 throws of a die, the number 4 comes up 160 times. The ratio 160/1000 = 0.16 shows the relative frequency of the number 4 falling out in this series of tests. More generally random event frequency And when conducting a series of experiments, they call the ratio of the number of experiments in which a given event occurred to the total number of experiments:

where P*(A) is the frequency of event A; m is the number of experiments in which event A occurred; n is the total number of experiments.

The probability of a random event A is called a constant number, around which the frequencies of a given event are grouped as the number of experiments increases ( statistical determination of the probability of an event ). The probability of a random event is denoted by P(A).

Naturally, no one will ever be able to do an unlimited number of tests in order to determine the probability. There is no need for this. In practice, the probability can be taken as the frequency of an event at large numbers tests. So, for example, from the statistical patterns of birth established over many years of observation, the probability of the event that the newborn will be a boy is estimated at 0.515.

If during the test there are no reasons due to which one random event would occur more often than others ( equally probable events), we can determine the probability based on theoretical considerations. For example, let's find out in the case of tossing a coin, the frequency of the coat of arms falling out (event A). Various experimenters have shown in several thousand trials that the relative frequency of such an event takes values ​​close to 0.5. given that the appearance of the coat of arms and the opposite side of the coin (event B) are equally likely events if the coin is symmetrical, the judgment P(A)=P(B)=0.5 could be made without determining the frequency of these events. On the basis of the concept of "equal probability" of events, another definition of probability is formulated.

Let the event A under consideration occur in m cases, which are called favorable to A, and do not occur in the remaining n-m, unfavorable to A.

Then the probability of event A is equal to the ratio of the number of elementary events favorable to it to their total number(classical definition of the probability of an event):

where m is the number of elementary events that favor event A; n - The total number of elementary events.

Let's look at a few examples:

Example #1:An urn contains 40 balls: 10 black and 30 white. Find the probability that a randomly chosen ball is black.

The number of favorable cases is equal to the number of black balls in the urn: m = 10. The total number of equally probable events (taking out one ball) is equal to the total number of balls in the urn: n = 40. These events are incompatible, since one and only one ball is taken out. P(A) = 10/40 = 0.25

Example #2:Find the probability of getting an even number when throwing a die.

When throwing a die, six equally possible incompatible events are realized: the appearance of one digit: 1,2,3,4,5 or 6, i.e. n = 6. Favorable cases are the loss of one of the numbers 2,4 or 6: m = 3. The desired probability P(A) = m/N = 3/6 = ½.

As we can see from the definition of the probability of an event, for all events

0 < Р(А) < 1.

Obviously, the probability of a certain event is 1, the probability of an impossible event is 0.

Probability addition theorem: the probability of occurrence of one (no matter what) event from several incompatible events is equal to the sum of their probabilities.

For two incompatible events A and B, the probabilities of these events is equal to the sum of their probabilities:

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

Example #3:Find the probability of getting 1 or 6 when throwing a dice.

Event A (roll 1) and B (roll 6) are equally likely: P(A) = P(B) = 1/6, so P(A or B) = 1/6 + 1/6 = 1/3

The addition of probabilities is valid not only for two, but also for any number of incompatible events.

Example #4:An urn contains 50 balls: 10 white, 20 black, 5 red and 15 blue. Find the probability of a white, or black, or red ball appearing in a single operation of removing a ball from the urn.

The probability of drawing a white ball (event A) is P(A) = 10/50 = 1/5, a black ball (event B) is P(B) = 20/50 = 2/5 and a red ball (event C) is P (C) = 5/50 = 1/10. From here, according to the formula for adding probabilities, we get P (A or B or C) \u003d P (A) + P (B) \u003d P (C) \u003d 1/5 + 2/5 + 1/10 \u003d 7/10

The sum of the probabilities of two opposite events, as follows from the probability addition theorem, is equal to one:

P(A) + P() = 1

In the above example, taking out the white, black and red balls will be the event A 1 , P(A 1) = 7/10. The opposite event of 1 is drawing the blue ball. Since there are 15 blue balls, and the total number of balls is 50, we get P( 1) = 15/50 = 3/10 and P(A) + P() = 7/10 + 3/10 = 1.

If events А 1 , А 2 , ..., А n form a complete system of pairwise incompatible events, then the sum of their probabilities is equal to 1.

In general, the probability of the sum of two events A and B is calculated as

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

Probability multiplication theorem:

Events A and B are called independent If the probability of occurrence of event A does not depend on whether event B occurred or not, and vice versa, the probability of occurrence of event B does not depend on whether event A occurred or not.

The probability of joint occurrence of independent events is equal to the product of their probabilities. For two events P(A and B)=P(A) P(B).

Example: One urn contains 5 black and 10 white balls, the other 3 black and 17 white. Find the probability that the first time balls are drawn from each urn, both balls are black.

Solution: the probability of drawing a black ball from the first urn (event A) - P(A) = 5/15 = 1/3, a black ball from the second urn (event B) - P(B) = 3/20

P (A and B) \u003d P (A) P (B) \u003d (1/3) (3/20) \u003d 3/60 \u003d 1/20.

In practice, the probability of an event B often depends on whether some other event A has occurred or not. In this case, one speaks of conditional probability , i.e. the probability of event B given that event A has occurred. The conditional probability is denoted by P(B/A).

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 expressed in numerical form measure of the possibility of occurrence of some event (A or B) as a result of 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. coin toss - good example: the appearance of tails is automatically the non-appearance of 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 certain 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 an event occurring when, as a result of two attempts with the extraction of balls, only blue balls will be extracted, is 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 of diamonds, if the first card drawn is:

1. Tambourine.
2. 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.

• Probability - degree (relative measure, quantification) the possibility of some event occurring. When the reasons for some possible event to actually occur outweigh the opposite reasons, then this event is called probable, otherwise - unlikely or improbable. The preponderance of positive grounds over negative ones, and vice versa, can be to varying degrees, as a result of which the probability (and improbability) is greater or lesser. Therefore, probability is often assessed at a qualitative level, especially in cases where a more or less accurate quantitative assessment is impossible or extremely difficult. Various gradations of "levels" of probability are possible.

  The study of probability from a mathematical point of view is a special discipline - the theory of probability. In probability theory and mathematical statistics the concept of probability is formalized as a numerical characteristic of an event - a probability measure (or its value) - a measure on a set of events (subsets of a set of elementary events), taking values ​​from

  (\displaystyle 0)

  (\displaystyle 1)

  Meaning

  (\displaystyle 1)

  Corresponds to a valid event. An impossible event has a probability of 0 (the converse is generally not always true). If the probability of an event occurring is

  (\displaystyle p)

  Then the probability of its non-occurrence is equal to

  (\displaystyle 1-p)

  In particular, the probability

  (\displaystyle 1/2)

  Means equal probability of the occurrence and non-occurrence of the event.

  The classical definition of probability is based on the concept of the equiprobability of outcomes. The probability is the ratio of the number of outcomes that favor a given event to the total number of equally likely outcomes. For example, the probability of getting "heads" or "tails" in a random coin toss is 1/2, if it is assumed that only these two possibilities occur and they are equally likely. This classical "definition" of probability can be generalized to the case of an infinite number of possible values ​​- for example, if an event can occur with equal probability at any point (the number of points is infinite) of some limited area of ​​space (plane), then the probability that it will occur in some part of this admissible area is equal to the ratio of the volume (area) of this part to the volume (area) of the area of ​​all possible points.

  The empirical "definition" of probability is related to the frequency of the occurrence of an event, based on the fact that with a sufficiently large number of trials, the frequency should tend to the objective degree of possibility of this event. AT modern presentation probability theory, probability is defined axiomatically as special case abstract theory of measure of a set. However, the link between the abstract measure and the probability, which expresses the degree of possibility of an event, is precisely the frequency of its observation.

  The probabilistic description of certain phenomena has received wide use in modern science, in particular in econometrics, statistical physics macroscopic (thermodynamic) systems, where even in the case of a classical deterministic description of the motion of particles, a deterministic description of the entire system of particles does not seem to be practically possible and appropriate. AT quantum physics the described processes themselves are of a probabilistic nature.

Various definitions of the probability of a random event

Probability theory – mathematical science, which, by the probabilities of some events, allows estimating the probabilities of other events associated with the first ones.

Confirmation that the concept of "probability of an event" has no definition is the fact that in probability theory there are several approaches to explaining this concept:

The classical definition of probability random event .

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

Where

The number of favorable outcomes of experience;

The total number of experience outcomes.

The outcome of experience is called favorable for an event, if an event appeared at this outcome of the experience. For example, if the event is the appearance of a red suit card, then the appearance of an ace of diamonds is an outcome favorable to the event.

Examples.

1) The probability of getting 5 points on the face of the die is equal to, since the die can fall on any of the 6 faces up, and 5 points are only on one face.

2) The probability of a coat of arms falling out when a coin is tossed once is , since a coin can fall with a coat of arms or tails - two outcomes of experience, and the coat of arms is depicted only on one side of the coin.

3) If there are 12 balls in the urn, of which 5 are black, then the probability of taking out a black ball is , since there are 12 outcomes of honey agarics, and 5 of them are favorable

Comment. The classical definition of probability applies under two conditions:

1) all outcomes of the experiment must be equally probable;

2) experience must have a finite number of outcomes.

In practice, it can be difficult to prove that events are equiprobable: for example, when performing an experiment with tossing a coin, the result of the experiment can be influenced by such factors as the asymmetry of the coin, the effect of its shape on the aerodynamic characteristics of the flight, atmospheric conditions, etc., in addition, there are experiments with an infinite number of outcomes.

Example . The child throws the ball and the maximum distance he can throw the ball is 15 meters. Find the probability that the ball will fly beyond the 3m mark.

Solution.The desired probability is proposed to be considered as the ratio of the length of the segment located beyond the mark of 3 m (favorable area) to the length of the entire segment (all possible outcomes):

Example. A point is randomly thrown into a circle of radius 1. What is the probability that the point will fall into a square inscribed in the circle?

Solution.The probability that a point will fall into a square is understood in this case as the ratio of the area of ​​the square (favorable area) to the area of ​​the circle (the total area of ​​the figure where the point is thrown):

The diagonal of a square is 2 and is expressed in terms of its side using the Pythagorean theorem:

Similar reasoning is carried out in space: if a point is randomly selected in the body of volume, then the probability that the point will be in part of the body of volume is calculated as the ratio of the volume of the favorable part to the total volume of the body:

Combining all cases, we can formulate a rule for calculating the geometric probability:

If a point is randomly selected in some area, then the probability that the point will be in part of this area is equal to:

, where

Indicates the measure of the area: in the case of a segment, this is the length, in the case of a flat area, this is the area, in the case of a three-dimensional body, this is the volume, on the surface, the surface area, on the curve, the length of the curve.

An interesting application of the concept of geometric probability is the meeting problem.

A task. (About a meeting)

Two students agreed to meet, for example, at 10 o'clock in the morning on the following conditions: each comes at any time during the hour from 10 to 11 and waits for 10 minutes, after which he leaves. What is the probability of meeting?

Solution.We illustrate the conditions of the problem as follows: on the axis we plot the time for the first of those encountered, and on the axis - the time for the second. Since the experiment lasts one hour, then on both axes we set aside segments of length 1. The moments of time when the meeting arrived at the same time is interpreted by the diagonal of the square.

Let the first one arrive at some point in time . Students will meet if the arrival time of the second student at the meeting point is between

Arguing in this way for any moment of time , we get that the time region interpreting the possibility of a meeting (“intersection of times” of the first and second students being in the right place) is between two straight lines: and . The probability of meeting is determined by the geometric probability formula:

In 1933 Kolmogorov A.M. (1903 - 1987) proposed an axiomatic approach to the construction and presentation of the theory of probability, which has become generally accepted at the present time. When constructing a theory of probability as a formal axiomatic theory, it is required not only to introduce a basic concept - the probability of a random event, but also to describe its properties using axioms (statements that are intuitively true, accepted without proof).

Such statements are statements similar to the properties of the relative frequency of occurrence of an event.

The relative frequency of occurrence of a random event is the ratio of the number of occurrences of an event in trials to the total number of trials performed:

Obviously, for a certain event, for an impossible event, for incompatible events, and the following is true:

Example. Let us illustrate the last statement. Let them take out cards from a deck of 36 cards. Let the event mean the appearance of diamonds, the event means the appearance of hearts, and the event - the appearance of a card of the red suit. Obviously, the events and are incompatible. When a red suit appears, we put a mark near the event, when diamonds appear - near the event, and when worms appear - near the event. It is obvious that the label near the event will be placed if and only if the label is placed near the event or near the event , i.e. .

Let's call the probability of a random event the number associated with the event according to the following rule:

For incompatible events and

So,