The classical concept of probability. Statistical definition of probability

The probability of an event is understood as some numerical characteristic of the possibility of the occurrence of this event. There are several approaches to determining probability.

Probability of an 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. So the probability of an event A is determined by the formula

Where m is the number of elementary outcomes favoring A, n- the number of all possible elementary outcomes of the test.

Example 3.1. In the experiment with throwing a dice, the number of all outcomes n is 6 and they are all equally possible. Let the event A means the appearance of an even number. Then for this event, favorable outcomes will be the appearance of numbers 2, 4, 6. Their number is 3. Therefore, the probability of the event A is equal to

Example 3.2. What is the probability that the digits in a randomly chosen two-digit number are the same?

Two-digit numbers are numbers from 10 to 99, there are 90 such numbers in total. 9 numbers have the same numbers (these are the numbers 11, 22, ..., 99). Since in this case m=9, n=90, then

Where A- event, "a number with the same digits."

Example 3.3. There are 7 standard parts in a lot of 10 parts. Find the probability that there are 4 standard parts among six randomly selected parts.

The total number of possible elementary outcomes of the test is equal to the number of ways in which 6 parts can be extracted from 10, i.e., the number of combinations of 10 elements of 6 elements. Determine the number of outcomes that favor the event of interest to us A(among the six parts taken, 4 are standard). Four standard parts can be taken from seven standard parts in ways; at the same time, the remaining 6-4=2 parts must be non-standard, but you can take two non-standard parts from 10-7=3 non-standard parts in different ways. Therefore, the number of favorable outcomes is .

Then the desired probability is equal to

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

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 it means

3. The probability of a 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. In this case< m< n, means 0 < m/n < 1, i.e. 0< P(A) < 1. Итак, вероятность любого события удовлетворяет двойному неравенству


The construction of a logically complete probability theory is based on the axiomatic definition of a random event and its probability. In the system of axioms proposed by A. N. Kolmogorov, undefined concepts are an elementary event and probability. Here are the axioms that define the probability:

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

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 and the relationships between them are derived as theorems.

Questions for self-examination

1. What is the name of the numerical characteristic of the possibility of an event?

2. What is called the probability of an event?

3. What is the probability of a certain event?

4. What is the probability of an impossible event?

5. What are the limits of the probability of a random event?

6. What are the limits of the probability of any event?

7. What definition of probability is called classical?

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 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 at the same time, 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:

  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 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.

RUSSIAN ACADEMY OF THE NATIONAL ECONOMY AND PUBLIC SERVICE UNDER THE PRESIDENT OF THE RUSSIAN FEDERATION

OREL BRANCH

Department of Sociology and Information Technology

Typical calculation No. 1

in the discipline "Probability Theory and Mathematical Statistics"

on the topic "Fundamentals of Probability Theory"

Eagle - 2016.

Goal of the work: consolidation of theoretical knowledge on the topic of the foundations of the theory of probability, by solving typical problems. Mastering the concepts of the main types of random events and developing the skills of algebraic operations on events.

Job submission requirements: the work is done in handwritten form, the work must contain all the necessary explanations and conclusions, the formulas must contain a decoding of the accepted designations, the pages must be numbered.

Variant number corresponds to the student's serial number in the group list.

Basic theoretical information

Probability theory- a branch of mathematics that studies the patterns of random phenomena.

The concept of an event. Event classification.

One of the basic concepts of probability theory is the concept of an event. Events are indicated in capital Latin letters. A, IN, WITH,…

Event- this is a possible result (outcome) of a test or experience.

Testing is understood as any purposeful action.

Example : The shooter shoots at the target. A shot is a test, hitting a target is an event.

The event is called random , if under the conditions of a given experiment it can both occur and not occur.

Example : Shot from a gun - test

Inc. A- hitting the target

Inc. IN– miss – random events.

The event is called reliable if as a result of the test it must necessarily occur.

Example : Drop no more than 6 points when throwing a dice.

The event is called impossible if, under the conditions of the given experiment, it cannot occur at all.

Example : More than 6 points rolled when throwing a die.

The events are called incompatible if the occurrence of one of them precludes the occurrence of any other. Otherwise, the events are called joint.

Example : A dice is thrown. A roll of 5 eliminates a roll of 6. These are incompatible events. A student receiving “good” and “excellent” grades in exams in two different disciplines is a joint event.

Two incompatible events, of which one must necessarily occur, are called opposite . Event opposite to event A designate Ā .

Example : The appearance of the "coat of arms" and the appearance of "tails" when tossing a coin are opposite events.

Several events in this experience are called equally possible if there is reason to believe that none of these events is more possible than the others.

Example : drawing ace, tens, queens from a deck of cards - events are equally likely.

Several events form full group if, as a result of the test, one and only one of these events must necessarily occur.

Example : Dropping the number of points 1, 2, 3, 4, 5, 6 when throwing a die.

The classic definition of the probability of an event. Probability Properties

For practical activities, it is important to be able to compare events according to the degree of possibility of their occurrence.

Probability An event is a numerical measure of the degree of objective possibility of an event occurring.

Let's call elementary outcome each of the equally likely test results.

Exodus is called favorable (favorable) event A, if its occurrence entails the occurrence of an event A.

Classic definition : event probability A is equal to the ratio of the number of outcomes favorable for a given event to the total number of possible outcomes.

(1)where P(A) is the probability of an event A,

m- the number of favorable outcomes,

n is the number of all possible outcomes.

Example : There are 1000 tickets in the lottery, of which 700 are not winning. What is the probability of winning on one purchased ticket.

Event A- purchased a winning ticket

Number of possible outcomes n=1000 is the total number of lottery tickets.

Number of outcomes favoring the event A is the number of winning tickets, i.e., m=1000-700=300.

According to the classical definition of probability:

Answer:
.

Note event probability properties:

1) The probability of any event is between zero and one, i.e. 0≤ P(A)≤1.

2) The probability of a certain event is 1.

3) The probability of an impossible event is 0.

In addition to the classical, there are also geometric and statistical definitions of probability.

Elements of combinatorics.

Combinatorics formulas are widely used to calculate the number of outcomes favorable to the event in question or the total number of outcomes.

Let there be a set N from n various elements.

Definition 1: Combinations, each of which includes all n elements and which differ from each other only by the order of the elements are called permutations from n elements.

P n=n! (2), where n! (n-factorial) - product n the first numbers of the natural series, i.e.

n! = 1∙2∙3∙…∙(n–1)∙n

So, for example, 5!=1∙2∙3∙4∙5 = 120

Definition 2: m elements ( m≤n) and differing from each other either in the composition of the elements or their order are called placements from n By m elements.

(3) 
Definition 3: Combinations, each containing m elements ( m≤n) and differing from each other only in the composition of the elements are called combinations from n By m elements.


(4)
Comment: changing the order of elements within the same combination does not result in a new combination.

We formulate two important rules that are often used in solving combinatorial problems

Sum rule: if object A can be chosen m ways, and the object IN – n ways, then the choice is either A or IN can be done m+n ways.

Product rule: if object A can be chosen m ways, and the object IN after each such choice, one can choose n ways, then a pair of objects A And IN can be selected in that order. m ∙ n ways.

Probability shows the possibility of an event with a certain number of repetitions. This is the number of possible outcomes with one or more outcomes divided by the total number of possible events. The probability of several events is calculated by dividing the problem into separate probabilities and then multiplying these probabilities.

Steps

Probability of a single random event

  1. Choose an event with mutually exclusive results. Probability can only be calculated if the event in question either occurs or does not occur. It is impossible to get any event and the opposite result at the same time. An example of such events is rolling a 5 on a game die or winning a certain horse in a race. Five will either come up or not; a certain horse will either come first or not.

    • For example, it is impossible to calculate the probability of such an event: in one roll of the die, 5 and 6 will roll at the same time.

  2. Identify all possible events and outcomes that could occur. Suppose we need to determine the probability that a three of a kind will come up when a dice with 6 numbers is rolled. "Three of a kind" is an event, and since we know that any of the 6 numbers can come up, the number of possible outcomes is six. Thus, we know that in this case there are 6 possible outcomes and one event whose probability we want to determine. Below are two more examples.

    • Example 1. In this case, the event is "selecting a day that falls on the weekend", and the number of possible outcomes is equal to the number of days of the week, that is, seven.
    • Example 2. The event is "draw the red ball", and the number of possible outcomes is equal to the total number of balls, that is, twenty.

  3. Divide the number of events by the number of possible outcomes. This way you determine the probability of a single event. If we consider the case of a 3 on a die roll, the number of events is 1 (three is only on one side of the die), and the total number of outcomes is 6. The result is a ratio of 1/6, 0.166, or 16.6%. The probability of an event for the two examples above is found as follows:

    • Example 1. What is the probability that you will randomly choose a day that falls on a weekend? The number of events is 2, since there are two days off in one week, and the total number of outcomes is 7. Thus, the probability is 2/7. The result obtained can also be written as 0.285 or 28.5%.
    • Example 2. A box contains 4 blue, 5 red and 11 white balls. If you draw a random ball from the box, what is the probability that it will be red? The number of events is 5, since there are 5 red balls in the box, and the total number of outcomes is 20. Find the probability: 5/20 = 1/4. The result obtained can also be written as 0.25 or 25%.

  4. Add up the probabilities of all possible events and see if the total is 1. The total probability of all possible events should be 1, or 100%. If you don't get 100%, chances are you made a mistake and missed one or more possible events. Check your calculations and make sure you account for all possible outcomes.

    • For example, the probability of rolling a 3 when rolling a die is 1/6. In this case, the probability of any other number falling out of the remaining five is also equal to 1/6. As a result, we get 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6, that is, 100%.
    • If, for example, you forget about the number 4 on the die, adding the probabilities will give you only 5/6, or 83%, which is not equal to one and indicates an error.

  5. Express the probability of an impossible outcome as 0. This means that the given event cannot happen and its probability is 0. This way you can account for impossible events.

    • For example, if you were calculating the probability that Easter falls on a Monday in 2020, you would get 0, since Easter is always celebrated on Sunday.

    Probability of several random events

    1. When considering independent events, calculate each probability separately. Once you determine what the probabilities of events are, they can be calculated separately. Let's say we want to know the probability of rolling a die twice in a row with a 5. We know that the probability of rolling one five is 1/6, and the probability of rolling the second five is also 1/6. The first outcome is not related to the second.

      • Several rolls of fives are called independent events, because what happens the first time does not affect the second event.

    2. Consider the influence of previous outcomes when calculating the probability for dependent events. If the first event affects the probability of the second outcome, it is said to calculate the probability dependent events. For example, if you choose two cards from a deck of 52 cards, after the first card is drawn, the composition of the deck changes, which affects the choice of the second card. To calculate the probability of the second of two dependent events, subtract 1 from the number of possible outcomes when calculating the probability of the second event.

      • Example 1. Consider the following event: Two cards are randomly drawn from the deck one after the other. What is the probability that both cards will have a club suit? The probability that the first card will have a club suit is 13/52, or 1/4, since there are 13 cards of the same suit in the deck.
        • After that, the probability that the second card will be a club suit is 12/51, since there is no longer one club card. This is because the first event affects the second. If you draw the 3 of clubs and don't put it back, there will be one less card in the deck (51 instead of 52).
      • Example 2. There are 4 blue, 5 red and 11 white balls in a box. If three balls are drawn at random, what is the probability that the first is red, the second is blue, and the third is white?
        • The probability that the first ball will be red is 5/20, or 1/4. The probability that the second ball will be blue is 4/19, since there is one less ball left in the box, but still 4 blue ball. Finally, the probability that the third ball is white is 11/18 since we have already drawn two balls.

    3. Multiply the probabilities of each individual event. Regardless of whether you are dealing with independent or dependent events, as well as the number of outcomes (there can be 2, 3 or even 10), you can calculate the overall probability by multiplying the probabilities of all the events in question by each other. As a result, you will get the probability of several events, the following one by one. For example, the task is Find the probability of rolling a dice twice in a row with a 5.. These are two independent events, the probability of each of which is 1/6. Thus, the probability of both events is 1/6 x 1/6 = 1/36, that is, 0.027, or 2.7%.

      • Example 1. Two cards are drawn at random from the deck, one after the other. What is the probability that both cards will have a club suit? The probability of the first event is 13/52. The probability of the second event is 12/51. We find the total probability: 13/52 x 12/51 = 12/204 = 1/17, that is, 0.058, or 5.8%.
      • Example 2. A box contains 4 blue, 5 red and 11 white balls. If three balls are randomly drawn from the box, one after the other, what is the probability that the first is red, the second is blue, and the third is white? The probability of the first event is 5/20. The probability of the second event is 4/19. The probability of the third event is 11/18. So the total probability is 5/20 x 4/19 x 11/18 = 44/1368 = 0.032, or 3.2%.

For practical activity, it is necessary to be able to compare events according to the degree of possibility of their occurrence. Let's consider the classical case. An urn contains 10 balls, 8 of which are white and 2 are black. Obviously, the event “a white ball will be drawn from the urn” and the event “a black ball will be drawn from the urn” have different degrees of possibility of their occurrence. Therefore, to compare events, a certain quantitative measure is needed.

A quantitative measure of the possibility of an event occurring is probability . The most widely used are two definitions of the probability of an event: classical and statistical.

Classic definition probability is related to the notion of a favorable outcome. Let's dwell on this in more detail.

Let the outcomes of some test form a complete group of events and be equally probable, i.e. are uniquely possible, inconsistent and equally possible. Such outcomes are called elementary outcomes, or cases. It is said that the test is reduced to case chart or " urn scheme”, because any probabilistic problem for such a test can be replaced by an equivalent problem with urns and balls of different colors.

Exodus is called favorable event A if the occurrence of this case entails the occurrence of the event A.

According to the classical definition event probability A is equal to the ratio of the number of outcomes that favor this event to the total number of outcomes, i.e.

, (1.1)

Where P(A)- the probability of an event A; m- the number of cases favorable to the event A; n is the total number of cases.

Example 1.1. When throwing a dice, six outcomes are possible - a loss of 1, 2, 3, 4, 5, 6 points. What is the probability of getting an even number of points?

Solution. All n= 6 outcomes form a complete group of events and are equally probable, i.e. are uniquely possible, inconsistent and equally possible. Event A - "the appearance of an even number of points" - is favored by 3 outcomes (cases) - loss of 2, 4 or 6 points. According to the classical formula for the probability of an event, we obtain

P(A) = = . ◄

Based on the classical definition of the probability of an event, we note its properties:

1. The probability of any event lies between zero and one, i.e.

0 ≤ R(A) ≤ 1.

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

3. The probability of an impossible event is zero.

As mentioned earlier, the classical definition of probability is applicable only for those events that can appear as a result of trials that have symmetry of possible outcomes, i.e. reducible to the scheme of cases. However, there is a large class of events whose probabilities cannot be calculated using the classical definition.

For example, if we assume that the coin is flattened, then it is obvious that the events “appearance of a coat of arms” and “appearance of tails” cannot be considered equally possible. Therefore, the formula for determining the probability according to the classical scheme is not applicable in this case.

However, there is another approach to assessing the probability of events, based on how often a given event will occur in the tests performed. In this case, the statistical definition of probability is used.

Statistical Probabilityevent A is the relative frequency (frequency) of the occurrence of this event in n tests performed, i.e.

, (1.2)

Where R * (A) is the statistical probability of an event A; w(A) is the relative frequency of the event A; m is the number of trials in which the event occurred A; n is the total number of trials.

Unlike mathematical probability P(A) considered in the classical definition, the statistical probability R * (A) is a characteristic experienced, experimental. In other words, the statistical probability of an event A the number is called, relative to which the relative frequency is stabilized (established) w(A) with an unlimited increase in the number of tests carried out under the same set of conditions.

For example, when they say about a shooter that he hits a target with a probability of 0.95, this means that out of a hundred shots fired by him under certain conditions (the same target at the same distance, the same rifle, etc. .), on average there are about 95 successful ones. Naturally, not every hundred will have 95 successful shots, sometimes there will be fewer, sometimes more, but on average, with repeated repetition of shooting under the same conditions, this percentage of hits will remain unchanged. The number 0.95, which serves as an indicator of the skill of the shooter, is usually very stable, i.e. the percentage of hits in most shootings will be almost the same for a given shooter, only in rare cases deviating in any significant way from its average value.

Another disadvantage of the classical definition of probability ( 1.1 ), which limits its application is that it assumes a finite number of possible test outcomes. In some cases, this shortcoming can be overcome by using the geometric definition of probability, i.e. finding the probability of hitting a point in a certain area (segment, part of a plane, etc.).

Let a flat figure g forms part of a flat figure G(Fig. 1.1). On the figure G a dot is thrown at random. This means that all points in the area G"equal" in relation to hitting it with a thrown random point. Assuming that the probability of an event A- hitting a thrown point on a figure g- proportional to the area of ​​\u200b\u200bthis figure and does not depend on its location relative to G, neither from the form g, find

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