Initially, being just a collection of information and empirical observations about the game of dice, the theory of probability became a thorough science. The first to give it a mathematical framework were Fermat and Pascal.

From thinking about the eternal to the theory of probability

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

Thanks to the passion of the Chevalier de Mere, who was equally a gambler and a man not indifferent to science, Pascal was forced to find a way to calculate probability. De Mere was interested in the following 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, which was of great interest to the gentleman: “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 probability theory. It is interesting that the person of de Mere remained known in this area, and not in literature.

Previously, no mathematician had ever attempted to calculate the probabilities of events, since it was believed that this was only a guessing solution. Blaise Pascal gave the first definition of the probability of an event and showed that it 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 likely outcomes of the experiment.

Experience is the implementation of specific actions under constant conditions.

To be able to work with the results of the experiment, events are usually designated by the letters A, B, C, D, E...

Probability of a random event

In order to begin 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 some event (A or B) occurring as a result of an experience. The probability is denoted as P(A) or P(B).

In probability theory they distinguish:

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

Relationships between events

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

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 reduce to zero the probability of the occurrence of event B, then they compatible.

If events A and B never occur simultaneously in the same experience, then they are called incompatible. Tossing a coin is a good example: the appearance of heads 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 the probability of each other.

Relationships between events. Examples

Using examples it is much easier to understand the principles of probability theory and combinations of events.

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

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

Test No. 1. There are 6 balls involved, three of which are blue with odd numbers on them, and the other three are red with even numbers.

Test No. 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 equal to 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 “getting the purple ball” is impossible, since the probability of its occurrence is 0.
• Equally possible events. In Spanish No. 1, the events “get the ball with the number 2” and “get the ball with the number 3” are equally possible, 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 twice in a row while throwing a die is a compatible event.
• Incompatible events. In the same Spanish No. 1, the events “get a red ball” and “get a 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 equivalent to not drawing tails, and the sum of their probabilities is always 1 (full group).
• Dependent Events. So, in Spanish No. 1, you can set the goal of drawing the red ball twice in a row. Whether or not it is retrieved the first time affects the likelihood of being retrieved 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 precise data occurs through the translation of the topic into a mathematical plane. That is, judgments about a random event such as “high probability” or “minimal probability” can be translated into specific numerical data. It is already permissible to evaluate, compare and enter such material into more complex calculations.

From a calculation point of view, determining the probability of an event is the ratio of the number of elementary positive outcomes to the number of all possible outcomes of experience regarding a specific event. Probability is denoted by P(A), where P stands for the word “probabilite”, 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 outcomes possible for this experience. In this case, the probability of an event always lies 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 was described earlier: 3 blue balls with the numbers 1/3/5 and 3 red balls with the numbers 2/4/6.

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

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

As can be seen from the calculations, event C has a higher probability, since the number of probable 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 dice 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 of the occurrence of event A or B, and the product of them AB is the occurrence 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 presupposes the occurrence of at least one of them. The production of several events is the joint occurrence of them all.

In probability theory, as a rule, the use of the conjunction “and” denotes a sum, and the conjunction “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 addition of their probabilities:

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

For example: let's calculate the probability that in Spanish. No. 1 with blue and red balls, a number between 1 and 4 will appear. 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 getting 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 an experiment with a cube we add up the probabilities of all numbers appearing, the result will be one.

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

P(A) + P(Ā) = 1

Probability of incompatible events occurring

Probability multiplication 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 simultaneously is equal to the product of their probabilities, or:

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

For example, the probability that in Spanish 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 to extract balls, only blue balls are 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 occurrence of one of them can coincide with the occurrence of another. 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 appears 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.

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 occurrence (that is, their joint occurrence):

R joint (A+B)=P(A)+P(B)- P(AB)

Let's assume that the probability of hitting the target with one shot is 0.4. Then event A is hitting the target in the first attempt, B - in the second. These events are joint, since it is possible that you can hit the target with both the first and second shots. But events are not dependent. What is the probability of the event of hitting the target with two shots (at least with 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.

Geometry of probability for clarity

Interestingly, the probability of the sum of joint events can be represented as two areas A and B, which intersect with each other. As can be seen 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.

Determining the probability of the sum of many (more than two) joint events is quite cumbersome. To calculate it, you need to use the formulas that are provided for these cases.

Dependent Events

Events are called dependent if the occurrence of one (A) of them affects the probability of the occurrence of another (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). Ordinary probability was denoted as P(B) or the probability of independent events. In the case of dependent events, a new concept is introduced - conditional probability P A (B), which is the probability of a dependent event B, subject to the occurrence of event A (hypothesis), on which it depends.

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

An example of calculating the probability of dependent events

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

Using a deck of 36 cards as an example, let’s look at dependent events. We need to determine the probability that the second card drawn from the deck will be of diamonds if the first card drawn is:

1. Bubnovaya.
2. A different color.

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

R A (B) =8/35=0.23

If the second option is true, then the deck has 35 cards, and the full number of diamonds (9) is still retained, then the probability of the following event B:

R A (B) =9/35=0.26.

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

Multiplying dependent events

Guided by the previous chapter, we accept the first event (A) as a fact, but in essence, it is of a random nature. The probability of this event, namely drawing a diamond from a deck of cards, is equal to:

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

Since the theory does not exist on its own, but is intended to serve for practical purposes, it is fair to note that what is most often needed is the probability of producing dependent events.

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 (dependent on A):

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

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

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

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

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

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

Total probability of an event

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

• 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 equal to:

Looking to the future

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

We can say that by recognizing probability, we in some way take a theoretical step into the future, looking at it through the prism of formulas.

It is unlikely that many people think about whether it is possible to calculate events that are more or less random. In simple terms, is it possible to know which side of the cube will come up next? It was this question that two great scientists asked themselves, who laid the foundation for such a science as the theory of probability, in which the probability of an event is studied quite extensively.

Origin

If you try to define such a concept as probability theory, you will get the following: this is one of the branches of mathematics that studies the constancy of random events. Of course, this concept does not really reveal the whole essence, so it is necessary to consider it in more detail.

I would like to start with the creators of the theory. As mentioned above, there were two of them, and they were one of the first to try to calculate the outcome of this or that event using formulas and mathematical calculations. In general, the beginnings of this science appeared in the Middle Ages. At that time, various thinkers and scientists tried to analyze gambling games, such as roulette, craps, and so on, thereby establishing the pattern and percentage of a particular number falling out. The foundation was laid in the seventeenth century by the above-mentioned scientists.

At first, their works could not be considered great achievements in this field, because all they did were simply empirical facts, and experiments were carried out visually, without using formulas. Over time, it was possible to achieve great results, which appeared as a result of observing the throwing of dice. It was this tool that helped to derive the first intelligible formulas.

Like-minded people

It is impossible not to mention such a person as Christiaan Huygens in the process of studying a topic called “probability theory” (the probability of an event is covered precisely in this science). This person is very interesting. He, like the scientists presented above, tried to derive the pattern of random events in the form of mathematical formulas. It is noteworthy that he did not do this together with Pascal and Fermat, that is, all his works did not intersect with these minds. Huygens deduced

An interesting fact is that his work came out long before the results of the discoverers’ work, or rather, twenty years earlier. Among the identified concepts, the most famous are:

• the concept of probability as the value of chance;
• mathematical expectation for discrete cases;
• theorems of multiplication and addition of probabilities.

It is also impossible not to remember who also made a significant contribution to the study of the problem. Conducting his own tests, independent of anyone, he was able to present a proof of the law of large numbers. In turn, the scientists Poisson and Laplace, who worked at the beginning of the nineteenth century, were able to prove the original theorems. It was from this moment that probability theory began to be used to analyze errors in observations. Russian scientists, or rather Markov, Chebyshev and Dyapunov, could not ignore this science. Based on the work done by great geniuses, they established this subject as a branch of mathematics. These figures worked already at the end of the nineteenth century, and thanks to their contribution, the following phenomena were proven:

• law of large numbers;
• Markov chain theory;
• central limit theorem.

So, with the history of the birth of science and with the main people who influenced it, everything is more or less clear. Now the time has come to clarify all the facts.

Basic Concepts

Before touching on laws and theorems, it is worth studying the basic concepts of probability theory. The event plays a leading role in it. This topic is quite voluminous, but without it it will not be possible to understand everything else.

An event in probability theory is any set of outcomes of an experiment. There are quite a few concepts of this phenomenon. Thus, the scientist Lotman, working in this area, said that in this case we are talking about what “happened, although it might not have happened.”

Random events (the theory of probability pays special attention to them) is a concept that implies absolutely any phenomenon that has the opportunity to occur. Or, conversely, this scenario may not happen if many conditions are met. It is also worth knowing that it is random events that capture the entire volume of phenomena that have occurred. The theory of probability indicates that all conditions can be repeated constantly. It is their conduct that is called “experience” or “test”.

A reliable event is a phenomenon that is one hundred percent likely to happen in a given test. Accordingly, an impossible event is one that will not happen.

The combination of a pair of actions (conditionally, case A and case B) is a phenomenon that occurs simultaneously. They are designated as AB.

The sum of pairs of events A and B is C, in other words, if at least one of them happens (A or B), then C will be obtained. The formula for the described phenomenon is written as follows: C = A + B.

Incongruent events in probability theory imply that two cases are mutually exclusive. Under no circumstances can they happen at the same time. Joint events in probability theory are their antipode. What is meant here is that if A happened, then it does not prevent B in any way.

Opposite events (probability theory considers them in great detail) are easy to understand. The best way to understand them is by comparison. They are almost the same as incompatible events in probability theory. But their difference lies in the fact that one of many phenomena must happen in any case.

Equally probable events are those actions whose repetition is equal. To make it clearer, you can imagine tossing a coin: the loss of one of its sides is equally likely to fall out of the other.

It is easier to consider an auspicious event with an example. Let's say there is an episode B and an episode A. The first is the roll of the dice with an odd number appearing, and the second is the appearance of the number five on the die. Then it turns out that A favors B.

Independent events in probability theory are projected only onto two or more cases and imply the independence of any action from another. For example, A is the loss of heads when tossing a coin, and B is the drawing of a jack from the deck. They are independent events in probability theory. At this point it became clearer.

Dependent events in probability theory are also permissible only for a set of them. They imply the dependence of one on the other, that is, phenomenon B can occur only if A has already happened or, conversely, has not happened, when this is the main condition for B.

The outcome of a random experiment consisting of one component is elementary events. The theory of probability explains that this is a phenomenon that happened only once.

Basic formulas

So, the concepts of “event” and “probability theory” were discussed above; a definition of the basic terms of this science was also given. Now it’s time to get acquainted directly with the important formulas. These expressions mathematically confirm all the main concepts in such a complex subject as probability theory. The probability of an event plays a huge role here too.

It’s better to start with the basic ones. And before you start with them, it’s worth considering what they are.

Combinatorics is primarily a branch of mathematics; it deals with the study of a huge number of integers, as well as various permutations of both the numbers themselves and their elements, various data, etc., leading to the appearance of a number of combinations. In addition to probability theory, this branch is important for statistics, computer science and cryptography.

So, now we can move on to presenting the formulas themselves and their definition.

The first of them will be the expression for the number of permutations, it looks like this:

P_n = n ⋅ (n - 1) ⋅ (n - 2)…3 ⋅ 2 ⋅ 1 = n!

The equation is applied only if the elements differ only in the order of their arrangement.

Now the placement formula will be considered, it looks like this:

A_n^m = n ⋅ (n - 1) ⋅ (n-2) ⋅ ... ⋅ (n - m + 1) = n! : (n - m)!

This expression is applicable not only to the order of placement of the element, but also to its composition.

The third equation from combinatorics, and it is also the last, is called the formula for the number of combinations:

C_n^m = n ! : ((n - m))! :m!

A combination refers to selections that are not ordered; accordingly, this rule applies to them.

It was easy to understand the combinatorics formulas; now you can move on to the classical definition of probabilities. This expression looks like this:

In this formula, m is the number of conditions favorable to event A, and n is the number of absolutely all equally possible and elementary outcomes.

There are a large number of expressions; the article will not cover all of them, but the most important ones will be touched upon, such as, for example, the probability of the sum of events:

P(A + B) = P(A) + P(B) - this theorem is for adding only incompatible events;

P(A + B) = P(A) + P(B) - P(AB) - and this one is for adding only compatible ones.

Probability of events occurring:

P(A ⋅ B) = P(A) ⋅ P(B) - this theorem is for independent events;

(P(A ⋅ B) = P(A) ⋅ P(B∣A); P(A ⋅ B) = P(A) ⋅ P(A∣B)) - and this one is for the dependent.

The list of events will be completed by the formula of events. Probability theory tells us about Bayes' theorem, which looks like this:

P(H_m∣A) = (P(H_m)P(A∣H_m)) : (∑_(k=1)^n P(H_k)P(A∣H_k)),m = 1,..., n

In this formula, H 1, H 2, ..., H n is a complete group of hypotheses.

Examples

If you carefully study any section of mathematics, it is not complete without exercises and sample solutions. So is the theory of probability: events and examples here are an integral component that confirms scientific calculations.

Formula for the number of permutations

Let's say there are thirty cards in a deck of cards, starting with a value of one. Next question. How many ways are there to stack the deck so that cards with value one and two are not next to each other?

The task has been set, now let's move on to solving it. First you need to determine the number of permutations of thirty elements, for this we take the formula presented above, it turns out P_30 = 30!.

Based on this rule, we find out how many options there are to fold the deck in different ways, but we need to subtract from them those in which the first and second cards are next to each other. To do this, let's start with the option when the first is above the second. It turns out that the first card can take up twenty-nine places - from the first to the twenty-ninth, and the second card from the second to the thirtieth, making a total of twenty-nine places for a pair of cards. In turn, the rest can accept twenty-eight places, and in any order. That is, to rearrange twenty-eight cards, there are twenty-eight options P_28 = 28!

As a result, it turns out that if we consider the solution when the first card is above the second, there will be 29 ⋅ 28 extra possibilities! = 29!

Using the same method, you need to calculate the number of redundant options for the case when the first card is under the second. It also turns out to be 29 ⋅ 28! = 29!

It follows from this that there are 2 ⋅ 29 extra options!, while the necessary ways to assemble a deck are 30! - 2 ⋅ 29!. All that remains is to count.

30! = 29! ⋅ 30; 30!- 2 ⋅ 29! = 29! ⋅ (30 - 2) = 29! ⋅ 28

Now you need to multiply all the numbers from one to twenty-nine, and then finally multiply everything by 28. The answer is 2.4757335 ⋅〖10〗^32

Example solution. Formula for placement number

In this problem, you need to find out how many ways there are to put fifteen volumes on one shelf, but provided that there are thirty volumes in total.

The solution to this problem is a little simpler than the previous one. Using the already known formula, it is necessary to calculate the total number of arrangements of thirty volumes of fifteen.

A_30^15 = 30 ⋅ 29 ⋅ 28⋅... ⋅ (30 - 15 + 1) = 30 ⋅ 29 ⋅ 28 ⋅ ... ⋅ 16 = 202 843 204 931 727 360 000

The answer, accordingly, will be equal to 202,843,204,931,727,360,000.

Now let's take a slightly more difficult task. You need to find out how many ways there are to arrange thirty books on two bookshelves, given that one shelf can only hold fifteen volumes.

Before starting the solution, I would like to clarify that some problems can be solved in several ways, and this one has two methods, but both use the same formula.

In this problem, you can take the answer from the previous one, because there we calculated how many times you can fill a shelf with fifteen books in different ways. It turned out A_30^15 = 30 ⋅ 29 ⋅ 28 ⋅ ... ⋅ (30 - 15 + 1) = 30 ⋅ 29 ⋅ 28 ⋅ ...⋅ 16.

We will calculate the second shelf using the permutation formula, because fifteen books can be placed in it, while only fifteen remain. We use the formula P_15 = 15!.

It turns out that the total will be A_30^15 ⋅ P_15 ways, but, in addition to this, the product of all numbers from thirty to sixteen will need to be multiplied by the product of numbers from one to fifteen, in the end you will get the product of all numbers from one to thirty, that is, the answer equals 30!

But this problem can be solved in another way - easier. To do this, you can imagine that there is one shelf for thirty books. All of them are placed on this plane, but since the condition requires that there be two shelves, we saw one long one in half, so we get two of fifteen. From this it turns out that there can be P_30 = 30 options for arrangement!.

Example solution. Formula for combination number

Now we will consider a version of the third problem from combinatorics. It is necessary to find out how many ways there are to arrange fifteen books, provided that you need to choose from thirty absolutely identical ones.

To solve, of course, the formula for the number of combinations will be applied. From the condition it becomes clear that the order of the identical fifteen books is not important. Therefore, initially you need to find out the total number of combinations of thirty books of fifteen.

C_30^15 = 30 ! : ((30-15)) ! : 15! = 155 117 520

That's it. Using this formula, we were able to solve this problem in the shortest possible time; the answer, accordingly, is 155,117,520.

Example solution. Classic definition of probability

Using the formula above, you can find the answer to a simple problem. But this will help to clearly see and track the progress of actions.

The problem states that there are ten absolutely identical balls in the urn. Of these, four are yellow and six are blue. One ball is taken from the urn. You need to find out the probability of getting blue.

To solve the problem, it is necessary to designate getting the blue ball as event A. This experiment can have ten outcomes, which, in turn, are elementary and equally possible. At the same time, out of ten, six are favorable to event A. We solve using the formula:

P(A) = 6: 10 = 0.6

Applying this formula, we learned that the probability of getting the blue ball is 0.6.

Example solution. Probability of the sum of events

An option will now be presented that is solved using the sum-of-events probability formula. So, the condition is given that there are two boxes, the first contains one gray and five white balls, and the second contains eight gray and four white balls. As a result, they took one of them from the first and second boxes. You need to find out what is the chance that the balls you get will be gray and white.

To solve this problem, it is necessary to identify events.

• So, A - took a gray ball from the first box: P(A) = 1/6.
• A’ - took a white ball also from the first box: P(A") = 5/6.
• B - a gray ball was removed from the second box: P(B) = 2/3.
• B’ - took a gray ball from the second box: P(B") = 1/3.

According to the conditions of the problem, it is necessary for one of the phenomena to happen: AB’ or A’B. Using the formula, we get: P(AB") = 1/18, P(A"B) = 10/18.

Now the formula for multiplying the probability has been used. Next, to find out the answer, you need to apply the equation of their addition:

P = P(AB" + A"B) = P(AB") + P(A"B) = 11/18.

This is how you can solve similar problems using the formula.

Bottom line

The article presented information on the topic "Probability Theory", in which the probability of an event plays a vital role. Of course, not everything was taken into account, but, based on the presented text, you can theoretically familiarize yourself with this section of mathematics. The science in question can be useful not only in professional matters, but also in everyday life. With its help, you can calculate any possibility of any event.

The text also touched upon significant dates in the history of the formation of the theory of probability as a science, and the names of the people whose work was invested in it. This is how human curiosity led to the fact that people learned to calculate even random events. Once upon a time they were simply interested in this, but today everyone already knows about it. And no one will say what awaits us in the future, what other brilliant discoveries related to the theory under consideration will be made. But one thing is for sure - research does not stand still!

In the Unified State Examination tasks in mathematics, there are also more complex probability problems (than we considered in Part 1), where we have to apply the rule of addition, multiplication of probabilities, and distinguish between compatible and incompatible events.

So, the theory.

Joint and non-joint events

Events are called incompatible if the occurrence of one of them excludes the occurrence of others. That is, only one specific event or another can happen.

For example, when throwing a die, you can distinguish between events such as getting an even number of points and getting an odd number of points. These events are incompatible.

Events are called joint if the occurrence of one of them does not exclude the occurrence of the other.

For example, when throwing a die, you can distinguish such events as rolling an odd number of points and rolling a number of points that are a multiple of three. When a three is rolled, both events occur.

Sum of events

The sum (or combination) of several events is an event consisting of the occurrence of at least one of these events.

At the same time sum of two incompatible events is the sum of the probabilities of these events:

For example, the probability of getting 5 or 6 points on a die with one throw will be , because both events (rolling 5, rolling 6) are inconsistent and the probability of one or the other event occurring is calculated as follows:

The probability sum of two joint events equal to the sum of the probabilities of these events without taking into account their joint occurrence:

For example, in a shopping center, two identical machines sell coffee. The probability that the machine will run out of coffee by the end of the day is 0.3. The probability that both machines will run out of coffee is 0.12. Let's find the probability that by the end of the day the coffee will run out in at least one of the machines (that is, either one, or the other, or both at once).

The probability of the first event “coffee will run out in the first machine” as well as the probability of the second event “coffee will run out in the second machine” according to the condition is equal to 0.3. Events are collaborative.

The probability of the joint occurrence of the first two events according to the condition is 0.12.

This means that the probability that by the end of the day the coffee will run out in at least one of the machines is

Dependent and independent events

Two random events A and B are called independent if the occurrence of one of them does not change the probability of the occurrence of the other. Otherwise, events A and B are called dependent.

For example, when two dice are rolled simultaneously, one of them, say 1, and the other one, 5, are independent events.

Product of probabilities

The product (or intersection) of several events is an event consisting of the joint occurrence of all these events.

If two occur independent events A and B with probabilities P(A) and P(B) respectively, then the probability of the occurrence of events A and B at the same time is equal to the product of the probabilities:

For example, we are interested in seeing a six appear on a die twice in a row. Both events are independent and the probability of each of them occurring separately is . The probability that both of these events will occur will be calculated using the above formula: .

See a selection of tasks to practice the topic.

In economics, as in other areas of human activity or in nature, we constantly have to deal with events that cannot be accurately predicted. Thus, the sales volume of a product 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 carrying out sales, you have to predict the outcome of such activities on the basis of either your own previous experience, or similar experience of other people, or intuition, which to a large extent also relies on experimental data.

In order to somehow evaluate the event in question, 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 experience it may or may not occur.

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

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

One of the main tasks in probability theory is the task 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.

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

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

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

An event consisting of the appearance of two goods in a store at the same time 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 is sure to occur in experience.

Example. The port has two berths for receiving ships. Three events can be considered: - the absence of ships at the berths, - the presence of one ship at one of the berths, - the presence of two ships 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 is opposite is denoted by , then the opposite event is usually denoted by .

Classical and statistical definitions of event probability

Each of the equally possible results of tests (experiments) is called an elementary outcome. They are usually designated by letters. For example, a die is thrown. There can be a total of six elementary outcomes based on the number of points on the sides.

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

A quantitative measure of the possibility of the occurrence of the event in question is probability.

The most widely used definitions of the probability of an event are: classic And statistical.

The classical definition of probability is associated with the concept of a favorable outcome.

The outcome is called favorable to a given event if its occurrence entails the occurrence of this event.

In the above example, the event in question—an even number of points on the rolled side—has three favorable outcomes. In this case, the general
number of possible outcomes. This means that the classical definition of the probability of an event can be used here.

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 outcomes favorable to 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 using the formula

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

Statistical definition. The probability of an event is the number around which the relative frequency stabilizes (sets) with an unlimited increase in the number of experiments.

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

From these definitions of the probability of an event it is clear that the inequality is always satisfied

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

From a practical point of view, probability of an event is the ratio of the number of those observations in which the event in question occurred to the total number of observations. This interpretation is acceptable in the case of a sufficiently large number of observations or experiments. For example, if about half of the people you meet on the street are women, then you can say that the probability that the person you meet on the street will be a woman is 1/2. In other words, an estimate of the probability of an event can be the frequency of its occurrence in a long series of independent repetitions of a random experiment.

Probability in mathematics

In the modern mathematical approach, classical (that is, not quantum) probability is given by Kolmogorov’s axiomatics. Probability is a measure P, which is defined on the set X, called probability space. This measure must have the following properties:

From these conditions it follows that the probability measure P also has the property additivity: if sets A 1 and A 2 do not intersect, then . To prove you need to put everything A 3 , A 4 , ... equal to the empty set and apply the property of countable additivity.

The probability measure may not be defined for all subsets of the set X. It is enough to define it on a sigma algebra, consisting of some subsets of the set X. In this case, random events are defined as measurable subsets of space X, that is, as elements of sigma algebra.

Probability sense

When we find that the reasons for some possible fact actually occurring outweigh the contrary reasons, we consider that fact probable, otherwise - incredible. This preponderance of positive bases over negative ones, and vice versa, can represent an indefinite set of degrees, as a result of which probability(And improbability) It happens more or less .

Complex individual facts do not allow for an exact calculation of the degrees of their probability, but even here it is important to establish some large subdivisions. So, for example, in the legal field, when a personal fact subject to trial is established on the basis of testimony, it always remains, strictly speaking, only probable, and it is necessary to know how significant this probability is; in Roman law, a quadruple division was adopted here: probatio plena(where the probability practically turns into reliability), further - probatio minus plena, then - probatio semiplena major and finally probatio semiplena minor .

In addition to the question of the probability of the case, the question may arise, both in the field of law and in the moral field (with a certain ethical point of view), of how likely it is that a given particular fact constitutes a violation of the general law. This question, which serves as the main motive in the religious jurisprudence of the Talmud, also gave rise to very complex systematic constructions and a huge literature, dogmatic and polemical, in Roman Catholic moral theology (especially from the end of the 16th century) (see Probabilism).

The concept of probability allows for a certain numerical expression when applied only to such facts that are part of certain homogeneous series. So (in the simplest example), when someone throws a coin a hundred times in a row, we find here one general or large series (the sum of all falls of the coin), consisting of two private or smaller, in this case numerically equal, series (falls " heads" and falls "tails"); The probability that this time the coin will land heads, that is, that this new member of the general series will belong to this of the two smaller series, is equal to the fraction expressing the numerical relationship between this small series and the larger one, namely 1/2, that is, the same probability belongs to one or the other of two particular series. In less simple examples, the conclusion cannot be deduced directly from the data of the problem itself, but requires prior induction. So, for example, the question is: what is the probability for a given newborn to live to be 80 years old? Here there must be a general, or large, series of a certain number of people born in similar conditions and dying at different ages (this number must be large enough to eliminate random deviations, and small enough to maintain the homogeneity of the series, for for a person, born, for example, in St. Petersburg into a wealthy, cultured family, the entire million-strong population of the city, a significant part of which consists of people from various groups who can die prematurely - soldiers, journalists, workers in dangerous professions - represents a group too heterogeneous for a real determination of probability) ; let this general series consist of ten thousand human lives; it includes smaller series representing the number of people living to a particular age; one of these smaller series represents the number of people living to age 80. But it is impossible to determine the number of this smaller series (like all others) a priori; this is done purely inductively, through statistics. Suppose statistical studies have established that out of 10,000 middle-class St. Petersburg residents, only 45 live to be 80; Thus, this smaller series is related to the larger one as 45 is to 10,000, and the probability for a given person to belong to this smaller series, that is, to live to be 80 years old, is expressed as a fraction of 0.0045. The study of probability from a mathematical point of view constitutes a special discipline - probability theory.

See also

Notes

Literature

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