Probability theory formulas and examples of problem solving. Mathematics for programmers: probability theory What is p a in probability theory

“Accidents are not accidental”... It sounds like something a philosopher said, but in fact, studying randomness is the destiny of the great science of mathematics. In mathematics, chance is dealt with by probability theory. Formulas and examples of tasks, as well as the main definitions of this science will be presented in the article.

What is probability theory?

Probability theory is one of the mathematical disciplines that studies random events.

To make it a little clearer, let's give a small example: if you throw a coin up, it can land on heads or tails. While the coin is in the air, both of these probabilities are possible. That is, the probability of possible consequences is 1:1. If one is drawn from a deck of 36 cards, then the probability will be indicated as 1:36. It would seem that there is nothing to explore and predict here, especially with the help of mathematical formulas. However, if you repeat a certain action many times, you can identify a certain pattern and, based on it, predict the outcome of events in other conditions.

To summarize all of the above, probability theory in the classical sense studies the possibility of the occurrence of one of the possible events in a numerical value.

From the pages of history

The theory of probability, formulas and examples of the first tasks appeared in the distant Middle Ages, when attempts to predict the outcome of card games first arose.

Initially, probability theory had nothing to do with mathematics. It was justified by empirical facts or properties of an event that could be reproduced in practice. The first works in this area as a mathematical discipline appeared in the 17th century. The founders were Blaise Pascal and Pierre Fermat. They studied gambling for a long time and saw certain patterns, which they decided to tell the public about.

The same technique was invented by Christiaan Huygens, although he was not familiar with the results of the research of Pascal and Fermat. The concept of “probability theory”, formulas and examples, which are considered the first in the history of the discipline, were introduced by him.

The works of Jacob Bernoulli, Laplace's and Poisson's theorems are also of no small importance. They made probability theory more like a mathematical discipline. Probability theory, formulas and examples of basic tasks received their current form thanks to Kolmogorov’s axioms. As a result of all the changes, probability theory became one of the mathematical branches.

Basic concepts of probability theory. Events

The main concept of this discipline is “event”. There are three types of events:

• Reliable. Those that will happen anyway (the coin will fall).
• Impossible. Events that will not happen under any circumstances (the coin will remain hanging in the air).
• Random. The ones that will happen or won't happen. They can be influenced by various factors that are very difficult to predict. If we talk about a coin, then there are random factors that can affect the result: the physical characteristics of the coin, its shape, its original position, the force of the throw, etc.

All events in the examples are indicated in capital Latin letters, with the exception of P, which has a different role. For example:

• A = “students came to lecture.”
• Ā = “students did not come to the lecture.”

In practical tasks, events are usually written down in words.

One of the most important characteristics of events is their equal possibility. That is, if you toss a coin, all variants of the initial fall are possible until it falls. But events are also not equally possible. This happens when someone deliberately influences an outcome. For example, “marked” playing cards or dice, in which the center of gravity is shifted.

Events can also be compatible and incompatible. Compatible events do not exclude each other's occurrence. For example:

• A = “the student came to the lecture.”
• B = “the student came to the lecture.”

These events are independent of each other, and the occurrence of one of them does not affect the occurrence of the other. Incompatible events are defined by the fact that the occurrence of one excludes the occurrence of another. If we talk about the same coin, then the loss of “tails” makes it impossible for the appearance of “heads” in the same experiment.

Actions on events

Events can be multiplied and added; accordingly, logical connectives “AND” and “OR” are introduced in the discipline.

The amount is determined by the fact that either event A or B, or two, can occur simultaneously. If they are incompatible, the last option is impossible; either A or B will be rolled.

Multiplication of events consists in the appearance of A and B at the same time.

Now we can give several examples to better remember the basics, probability theory and formulas. Examples of problem solving below.

Exercise 1: The company takes part in a competition to receive contracts for three types of work. Possible events that may occur:

• A = “the firm will receive the first contract.”
• A 1 = “the firm will not receive the first contract.”
• B = “the firm will receive a second contract.”
• B 1 = “the firm will not receive a second contract”
• C = “the firm will receive a third contract.”
• C 1 = “the firm will not receive a third contract.”

Using actions on events, we will try to express the following situations:

• K = “the company will receive all contracts.”

In mathematical form, the equation will have the following form: K = ABC.

• M = “the company will not receive a single contract.”

M = A 1 B 1 C 1.

Let’s complicate the task: H = “the company will receive one contract.” Since it is not known which contract the company will receive (first, second or third), it is necessary to record the entire range of possible events:

H = A 1 BC 1 υ AB 1 C 1 υ A 1 B 1 C.

And 1 BC 1 is a series of events where the firm does not receive the first and third contract, but receives the second. Other possible events were recorded using the appropriate method. The symbol υ in the discipline denotes the connective “OR”. If we translate the above example into human language, the company will receive either the third contract, or the second, or the first. In a similar way, you can write down other conditions in the discipline “Probability Theory”. The formulas and examples of problem solving presented above will help you do this yourself.

Actually, the probability

Perhaps, in this mathematical discipline, the probability of an event is the central concept. There are 3 definitions of probability:

• classic;
• statistical;
• geometric.

Each has its place in the study of probability. Probability theory, formulas and examples (9th grade) mainly use the classic definition, which sounds like this:

• The probability of situation A is equal to the ratio of the number of outcomes that favor its occurrence to the number of all possible outcomes.

The formula looks like this: P(A)=m/n.

A is actually an event. If a case opposite to A appears, it can be written as Ā or A 1 .

m is the number of possible favorable cases.

n - all events that can happen.

For example, A = “draw a card of the heart suit.” There are 36 cards in a standard deck, 9 of them are of hearts. Accordingly, the formula for solving the problem will look like:

P(A)=9/36=0.25.

As a result, the probability that a card of the heart suit will be drawn from the deck will be 0.25.

Towards higher mathematics

Now it has become a little known what the theory of probability is, formulas and examples of solving problems that come across in the school curriculum. However, probability theory is also found in higher mathematics, which is taught in universities. Most often they operate with geometric and statistical definitions of the theory and complex formulas.

The theory of probability is very interesting. It is better to start studying formulas and examples (higher mathematics) small - with the statistical (or frequency) definition of probability.

The statistical approach does not contradict the classical one, but slightly expands it. If in the first case it was necessary to determine with what probability an event will occur, then in this method it is necessary to indicate how often it will occur. Here a new concept of “relative frequency” is introduced, which can be denoted by W n (A). The formula is no different from the classic one:

If the classical formula is calculated for prediction, then the statistical one is calculated according to the results of the experiment. Let's take a small task for example.

The technological control department checks products for quality. Among 100 products, 3 were found to be of poor quality. How to find the frequency probability of a quality product?

A = “the appearance of a quality product.”

W n (A)=97/100=0.97

Thus, the frequency of a quality product is 0.97. Where did you get 97 from? Out of 100 products that were checked, 3 were found to be of poor quality. We subtract 3 from 100 and get 97, this is the amount of quality goods.

A little about combinatorics

Another method of probability theory is called combinatorics. Its basic principle is that if a certain choice A can be made in m different ways, and a choice B can be made in n different ways, then the choice of A and B can be made by multiplication.

For example, there are 5 roads leading from city A to city B. There are 4 paths from city B to city C. In how many ways can you get from city A to city C?

It's simple: 5x4=20, that is, in twenty different ways you can get from point A to point C.

Let's complicate the task. How many ways are there to lay out cards in solitaire? There are 36 cards in the deck - this is the starting point. To find out the number of ways, you need to “subtract” one card at a time from the starting point and multiply.

That is, 36x35x34x33x32...x2x1= the result does not fit on the calculator screen, so it can simply be designated 36!. Sign "!" next to the number indicates that the entire series of numbers is multiplied together.

In combinatorics there are such concepts as permutation, placement and combination. Each of them has its own formula.

An ordered set of elements of a set is called an arrangement. Placements can be repeated, that is, one element can be used several times. And without repetition, when elements are not repeated. n are all elements, m are elements that participate in the placement. The formula for placement without repetition will look like:

A n m =n!/(n-m)!

Connections of n elements that differ only in the order of placement are called permutations. In mathematics it looks like: P n = n!

Combinations of n elements of m are those compounds in which it is important what elements they were and what their total number is. The formula will look like:

A n m =n!/m!(n-m)!

Bernoulli's formula

In probability theory, as in every discipline, there are works of outstanding researchers in their field who have taken it to a new level. One of these works is the Bernoulli formula, which allows you to determine the probability of a certain event occurring under independent conditions. This suggests that the occurrence of A in an experiment does not depend on the occurrence or non-occurrence of the same event in earlier or subsequent trials.

Bernoulli's equation:

P n (m) = C n m ×p m ×q n-m.

The probability (p) of the occurrence of event (A) is constant for each trial. The probability that the situation will occur exactly m times in n number of experiments will be calculated by the formula presented above. Accordingly, the question arises of how to find out the number q.

If event A occurs p number of times, accordingly, it may not occur. Unit is a number that is used to designate all outcomes of a situation in a discipline. Therefore, q is a number that denotes the possibility of an event not occurring.

Now you know Bernoulli's formula (probability theory). We will consider examples of problem solving (first level) below.

Task 2: A store visitor will make a purchase with probability 0.2. 6 visitors independently entered the store. What is the likelihood that a visitor will make a purchase?

Solution: Since it is unknown how many visitors should make a purchase, one or all six, it is necessary to calculate all possible probabilities using the Bernoulli formula.

A = “the visitor will make a purchase.”

In this case: p = 0.2 (as indicated in the task). Accordingly, q=1-0.2 = 0.8.

n = 6 (since there are 6 customers in the store). The number m will vary from 0 (not a single customer will make a purchase) to 6 (all visitors to the store will purchase something). As a result, we get the solution:

P 6 (0) = C 0 6 ×p 0 ×q 6 =q 6 = (0.8) 6 = 0.2621.

None of the buyers will make a purchase with probability 0.2621.

How else is Bernoulli's formula (probability theory) used? Examples of problem solving (second level) below.

After the above example, questions arise about where C and r went. Relative to p, a number to the power of 0 will be equal to one. As for C, it can be found by the formula:

C n m = n! /m!(n-m)!

Since in the first example m = 0, respectively, C = 1, which in principle does not affect the result. Using the new formula, let's try to find out what is the probability of two visitors purchasing goods.

P 6 (2) = C 6 2 ×p 2 ×q 4 = (6×5×4×3×2×1) / (2×1×4×3×2×1) × (0.2) 2 × (0.8) 4 = 15 × 0.04 × 0.4096 = 0.246.

The theory of probability is not that complicated. Bernoulli's formula, examples of which are presented above, is direct proof of this.

Poisson's formula

Poisson's equation is used to calculate low probability random situations.

Basic formula:

P n (m)=λ m /m! × e (-λ) .

In this case λ = n x p. Here is a simple Poisson formula (probability theory). We will consider examples of problem solving below.

Task 3: The factory produced 100,000 parts. Occurrence of a defective part = 0.0001. What is the probability that there will be 5 defective parts in a batch?

As you can see, marriage is an unlikely event, and therefore the Poisson formula (probability theory) is used for calculation. Examples of solving problems of this kind are no different from other tasks in the discipline; we substitute the necessary data into the given formula:

A = “a randomly selected part will be defective.”

p = 0.0001 (according to the task conditions).

n = 100000 (number of parts).

m = 5 (defective parts). We substitute the data into the formula and get:

R 100000 (5) = 10 5 /5! X e -10 = 0.0375.

Just like the Bernoulli formula (probability theory), examples of solutions using which are written above, the Poisson equation has an unknown e. In fact, it can be found by the formula:

e -λ = lim n ->∞ (1-λ/n) n .

However, there are special tables that contain almost all values ​​of e.

De Moivre-Laplace theorem

If in the Bernoulli scheme the number of trials is sufficiently large, and the probability of occurrence of event A in all schemes is the same, then the probability of occurrence of event A a certain number of times in a series of tests can be found by Laplace’s formula:

Р n (m)= 1/√npq x ϕ(X m).

X m = m-np/√npq.

To better remember Laplace’s formula (probability theory), examples of problems are below to help.

First, let's find X m, substitute the data (they are all listed above) into the formula and get 0.025. Using tables, we find the number ϕ(0.025), the value of which is 0.3988. Now you can substitute all the data into the formula:

P 800 (267) = 1/√(800 x 1/3 x 2/3) x 0.3988 = 3/40 x 0.3988 = 0.03.

Thus, the probability that the flyer will work exactly 267 times is 0.03.

Bayes formula

The Bayes formula (probability theory), examples of solving problems with the help of which will be given below, is an equation that describes the probability of an event based on the circumstances that could be associated with it. The basic formula is as follows:

P (A|B) = P (B|A) x P (A) / P (B).

A and B are definite events.

P(A|B) is a conditional probability, that is, event A can occur provided that event B is true.

P (B|A) - conditional probability of event B.

So, the final part of the short course “Probability Theory” is the Bayes formula, examples of solutions to problems with which are below.

Task 5: Phones from three companies were brought to the warehouse. At the same time, the share of phones that are manufactured at the first plant is 25%, at the second - 60%, at the third - 15%. It is also known that the average percentage of defective products at the first factory is 2%, at the second - 4%, and at the third - 1%. You need to find the probability that a randomly selected phone will be defective.

A = “randomly picked phone.”

B 1 - the phone that the first factory produced. Accordingly, introductory B 2 and B 3 will appear (for the second and third factories).

As a result we get:

P (B 1) = 25%/100% = 0.25; P(B 2) = 0.6; P (B 3) = 0.15 - thus we found the probability of each option.

Now you need to find the conditional probabilities of the desired event, that is, the probability of defective products in companies:

P (A/B 1) = 2%/100% = 0.02;

P(A/B 2) = 0.04;

P (A/B 3) = 0.01.

Now let’s substitute the data into the Bayes formula and get:

P (A) = 0.25 x 0.2 + 0.6 x 0.4 + 0.15 x 0.01 = 0.0305.

The article presents probability theory, formulas and examples of problem solving, but this is only the tip of the iceberg of a vast discipline. And after everything that has been written, it will be logical to ask the question of whether the theory of probability is needed in life. It’s difficult for an ordinary person to answer; it’s better to ask someone who has used it to win the jackpot more than once.

The mathematics course prepares a lot of surprises for schoolchildren, one of which is a problem on probability theory. Students have problems solving such tasks in almost one hundred percent of cases. To understand and understand this issue, you need to know the basic rules, axioms, and definitions. To understand the text in the book, you need to know all the abbreviations. We offer to learn all this.

Science and its application

Since we are offering a crash course in “probability theory for dummies,” we first need to introduce the basic concepts and letter abbreviations. To begin with, let’s define the very concept of “probability theory”. What kind of science is this and why is it needed? Probability theory is one of the branches of mathematics that studies random phenomena and quantities. She also considers the patterns, properties and operations performed with these random variables. What is it for? Science has become widespread in the study of natural phenomena. Any natural and physical processes cannot do without the presence of chance. Even if the results were recorded as accurately as possible during the experiment, if the same test is repeated, the result will most likely not be the same.

We will definitely look at examples of tasks, you can see for yourself. The outcome depends on many different factors that are almost impossible to take into account or register, but nevertheless they have a huge impact on the outcome of the experiment. Vivid examples include the task of determining the trajectory of the planets or determining the weather forecast, the probability of meeting a familiar person while traveling to work, and determining the height of an athlete’s jump. The theory of probability also provides great assistance to brokers on stock exchanges. A problem in probability theory, the solution of which previously had many problems, will become a mere trifle for you after three or four examples given below.

Events

As stated earlier, science studies events. Probability theory, we will look at examples of problem solving a little later, studies only one type - random. But nevertheless, you need to know that events can be of three types:

• Impossible.
• Reliable.
• Random.

We propose to discuss each of them a little. An impossible event will never happen, under any circumstances. Examples include: freezing water at above-zero temperatures, pulling a cube from a bag of balls.

A reliable event always occurs with a 100% guarantee if all conditions are met. For example: you received a salary for the work done, received a diploma of higher professional education if you studied conscientiously, passed exams and defended your diploma, and so on.

Everything is a little more complicated: during the experiment it can happen or not, for example, pulling an ace from a card deck after making no more than three attempts. You can get the result either on the first try or not at all. It is the probability of the occurrence of an event that science studies.

Probability

This is, in a general sense, an assessment of the possibility of a successful outcome of an experience in which an event occurs. Probability is assessed at a qualitative level, especially if quantitative assessment is impossible or difficult. A problem in probability theory with a solution, or more precisely with an estimate, involves finding that very possible share of a successful outcome. Probability in mathematics is the numerical characteristics of an event. It takes values ​​from zero to one, denoted by the letter P. If P is equal to zero, then the event cannot happen; if it is one, then the event will occur with one hundred percent probability. The more P approaches one, the stronger the probability of a successful outcome, and vice versa, if it is close to zero, then the event will occur with low probability.

Abbreviations

The probability problem you'll soon be faced with may contain the following abbreviations:

• P and P(X);
• A, B, C, etc;

Some others are also possible: additional explanations will be made as necessary. We suggest, first, to clarify the abbreviations presented above. The first one on our list is factorial. To make it clear, we give examples: 5!=1*2*3*4*5 or 3!=1*2*3. Next, given sets are written in curly brackets, for example: (1;2;3;4;..;n) or (10;140;400;562). The following notation is the set of natural numbers, which is quite often found in tasks on probability theory. As mentioned earlier, P is a probability, and P(X) is the probability of the occurrence of an event X. Events are denoted by capital letters of the Latin alphabet, for example: A - a white ball was caught, B - blue, C - red or, respectively, . The small letter n is the number of all possible outcomes, and m is the number of successful ones. From here we get the rule for finding classical probability in elementary problems: P = m/n. The theory of probability “for dummies” is probably limited to this knowledge. Now, to consolidate, let's move on to the solution.

Problem 1. Combinatorics

The student group consists of thirty people, from whom it is necessary to choose a headman, his deputy and a trade union leader. It is necessary to find the number of ways to do this action. A similar task may appear on the Unified State Exam. The theory of probability, the solution of problems of which we are now considering, may include problems from the course of combinatorics, finding classical probability, geometric probability and problems on basic formulas. In this example, we are solving a task from a combinatorics course. Let's move on to the solution. This task is the simplest:

1. n1=30 - possible prefects of the student group;
2. n2=29 - those who can take the post of deputy;
3. n3=28 people apply for the position of trade unionist.

All we have to do is find the possible number of options, that is, multiply all the indicators. As a result, we get: 30*29*28=24360.

This will be the answer to the question posed.

Problem 2. Rearrangement

There are 6 participants speaking at the conference, the order is determined by drawing lots. We need to find the number of possible draw options. In this example, we are considering a permutation of six elements, that is, we need to find 6!

In the abbreviations paragraph, we already mentioned what it is and how it is calculated. In total, it turns out that there are 720 drawing options. At first glance, a difficult task has a very short and simple solution. These are the tasks that probability theory considers. We will look at how to solve higher-level problems in the following examples.

Problem 3

A group of twenty-five students must be divided into three subgroups of six, nine and ten people. We have: n=25, k=3, n1=6, n2=9, n3=10. It remains to substitute the values ​​into the required formula, we get: N25(6,9,10). After simple calculations, we get the answer - 16,360,143,800. If the task does not say that it is necessary to obtain a numerical solution, then it can be given in the form of factorials.

Problem 4

Three people guessed numbers from one to ten. Find the probability that someone's numbers will match. First we must find out the number of all outcomes - in our case it is a thousand, that is, ten to the third power. Now let’s find the number of options when everyone has guessed different numbers, to do this we multiply ten, nine and eight. Where did these numbers come from? The first guesses a number, he has ten options, the second already has nine, and the third needs to choose from the remaining eight, so we get 720 possible options. As we already calculated earlier, there are 1000 options in total, and without repetitions there are 720, therefore, we are interested in the remaining 280. Now we need a formula for finding the classical probability: P = . We received the answer: 0.28.

Events that happen in reality or in our imagination can be divided into 3 groups. These are certain events that will definitely happen, impossible events and random events. Probability theory studies random events, i.e. events that may or may not happen. This article will briefly present the theory of probability formulas and examples of solving problems in probability theory, which will be in task 4 of the Unified State Exam in mathematics (profile level).

Why do we need probability theory?

Historically, the need to study these problems arose in the 17th century in connection with the development and professionalization of gambling and the emergence of casinos. This was a real phenomenon that required its own study and research.

Playing cards, dice, and roulette created situations where any of a finite number of equally possible events could occur. There was a need to give numerical estimates of the possibility of the occurrence of a particular event.

In the 20th century, it became clear that this seemingly frivolous science plays an important role in understanding the fundamental processes occurring in the microcosm. The modern theory of probability was created.

Basic concepts of probability theory

The object of study of probability theory is events and their probabilities. If an event is complex, then it can be broken down into simple components, the probabilities of which are easy to find.

The sum of events A and B is called event C, which consists in the fact that either event A, or event B, or events A and B occurred simultaneously.

The product of events A and B is an event C, which means that both event A and event B occurred.

Events A and B are called incompatible if they cannot occur simultaneously.

An event A is called impossible if it cannot happen. Such an event is indicated by the symbol.

An event A is called certain if it is sure to happen. Such an event is indicated by the symbol.

Let each event A be associated with a number P(A). This number P(A) is called the probability of event A if the following conditions are met with this correspondence.

An important special case is the situation when there are equally probable elementary outcomes, and arbitrary of these outcomes form events A. In this case, the probability can be entered using the formula. Probability introduced in this way is called classical probability. It can be proven that in this case properties 1-4 are satisfied.

Probability theory problems that appear on the Unified State Examination in mathematics are mainly related to classical probability. Such tasks can be very simple. The probability theory problems in the demonstration versions are especially simple. It is easy to calculate the number of favorable outcomes; the number of all outcomes is written right in the condition.

We get the answer using the formula.

An example of a problem from the Unified State Examination in mathematics on determining probability

There are 20 pies on the table - 5 with cabbage, 7 with apples and 8 with rice. Marina wants to take the pie. What is the probability that she will take the rice cake?

Solution.

There are 20 equally probable elementary outcomes, that is, Marina can take any of the 20 pies. But we need to estimate the probability that Marina will take the rice pie, that is, where A is the choice of the rice pie. This means that the number of favorable outcomes (choices of pies with rice) is only 8. Then the probability will be determined by the formula:

Independent, Opposite and Arbitrary Events

However, more complex tasks began to be found in the open task bank. Therefore, let us draw the reader’s attention to other issues studied in probability theory.

Events A and B are said to be independent if the probability of each does not depend on whether the other event occurs.

Event B is that event A did not happen, i.e. event B is opposite to event A. The probability of the opposite event is equal to one minus the probability of the direct event, i.e. .

Probability addition and multiplication theorems, formulas

For arbitrary events A and B, the probability of the sum of these events is equal to the sum of their probabilities without the probability of their joint event, i.e. .

For independent events A and B, the probability of the occurrence of these events is equal to the product of their probabilities, i.e. in this case .

The last 2 statements are called the theorems of addition and multiplication of probabilities.

Counting the number of outcomes is not always so simple. In some cases it is necessary to use combinatorics formulas. The most important thing is to count the number of events that satisfy certain conditions. Sometimes these kinds of calculations can become independent tasks.

In how many ways can 6 students be seated in 6 empty seats? The first student will take any of the 6 places. Each of these options corresponds to 5 ways for the second student to take a place. There are 4 free places left for the third student, 3 for the fourth, 2 for the fifth, and the sixth will take the only remaining place. To find the number of all options, you need to find the product, which is denoted by the symbol 6! and reads "six factorial".

In the general case, the answer to this question is given by the formula for the number of permutations of n elements. In our case.

Let us now consider another case with our students. In how many ways can 2 students be seated in 6 empty seats? The first student will take any of the 6 places. Each of these options corresponds to 5 ways for the second student to take a place. To find the number of all options, you need to find the product.

In general, the answer to this question is given by the formula for the number of placements of n elements over k elements

In our case .

And the last case in this series. In how many ways can you choose three students out of 6? The first student can be selected in 6 ways, the second - in 5 ways, the third - in four ways. But among these options, the same three students appear 6 times. To find the number of all options, you need to calculate the value: . In general, the answer to this question is given by the formula for the number of combinations of elements by element:

In our case .

Examples of solving problems from the Unified State Exam in mathematics to determine probability

Task 1. From the collection edited by. Yashchenko.

There are 30 pies on the plate: 3 with meat, 18 with cabbage and 9 with cherries. Sasha chooses one pie at random. Find the probability that he ends up with a cherry.

.

Answer: 0.3.

Task 2. From the collection edited by. Yashchenko.

In each batch of 1000 light bulbs, on average, 20 are defective. Find the probability that a light bulb taken at random from a batch will be working.

Solution: The number of working light bulbs is 1000-20=980. Then the probability that a light bulb taken at random from a batch will be working:

Answer: 0.98.

The probability that student U will solve more than 9 problems correctly during a math test is 0.67. The probability that U. will correctly solve more than 8 problems is 0.73. Find the probability that U will solve exactly 9 problems correctly.

If we imagine a number line and mark points 8 and 9 on it, then we will see that the condition “U. will solve exactly 9 problems correctly” is included in the condition “U. will solve more than 8 problems correctly”, but does not apply to the condition “U. will solve more than 9 problems correctly.”

However, the condition “U. will solve more than 9 problems correctly” is contained in the condition “U. will solve more than 8 problems correctly.” Thus, if we designate events: “U. will solve exactly 9 problems correctly" - through A, "U. will solve more than 8 problems correctly" - through B, "U. will correctly solve more than 9 problems” through C. That solution will look like this:

Answer: 0.06.

In a geometry exam, a student answers one question from a list of exam questions. The probability that this is a Trigonometry question is 0.2. The probability that this is a question on External Angles is 0.15. There are no questions that simultaneously relate to these two topics. Find the probability that a student will get a question on one of these two topics in the exam.

Let's think about what events we have. We are given two incompatible events. That is, either the question will relate to the topic “Trigonometry” or to the topic “External angles”. According to the probability theorem, the probability of incompatible events is equal to the sum of the probabilities of each event, we must find the sum of the probabilities of these events, that is:

Answer: 0.35.

The room is illuminated by a lantern with three lamps. The probability of one lamp burning out within a year is 0.29. Find the probability that at least one lamp will not burn out during the year.

Let's consider possible events. We have three light bulbs, each of which may or may not burn out independently of any other light bulb. These are independent events.

Then we will indicate the options for such events. Let's use the following notations: - the light bulb is on, - the light bulb is burnt out. And right next to it we will calculate the probability of the event. For example, the probability of an event in which three independent events “the light bulb is burned out”, “the light bulb is on”, “the light bulb is on” occurred: , where the probability of the event “the light bulb is on” is calculated as the probability of the event opposite to the event “the light bulb is not on”, namely: .

Probability theory is a mathematical science that allows, from the probabilities of some random events, to find the probabilities of other random events related in some way to the first.

A statement that an event occurs with probability, equal to, for example, ½, does not yet represent a final value in itself, since we strive for reliable knowledge. The final cognitive value are those results of probability theory that allow us to state that the probability of the occurrence of some event A is very close to unity or (which is the same thing) the probability of the non-occurrence of event A is very small. In accordance with the principle of “neglecting sufficiently small probabilities,” such an event is rightly considered practically certain. Below (in the Limit Theorems section) it is shown that conclusions of this kind that have scientific and practical interest are usually based on the assumption that the occurrence or non-occurrence of event A depends on a large number of random factors that are poorly interrelated with each other. Therefore, we can also say that probability theory is a mathematical science that elucidates the patterns that arise during the interaction of a large number of random factors.

Subject of probability theory.

To describe the natural connection between certain conditions S and the event A, the occurrence or non-occurrence of which under given conditions can be accurately established, natural science usually uses one of the following two schemes:

a) with each fulfillment of conditions S, event A occurs. This form, for example, has all the laws of classical mechanics, which state that given initial conditions and forces acting on a body or system of bodies, movement will occur in a uniquely defined way.

b) Under conditions S, event A has a certain probability P (A / S), equal to p. So, for example, the laws of radioactive radiation state that for each radioactive substance there is a certain probability that from a given amount of substance, a certain number N of atoms will decay in a given period of time.

Let us call the frequency of event A in a given series of n trials (that is, from n repeated implementations of conditions S) the ratio h = m/n of the number m of those trials in which A occurred to their total number n. The presence of event A under conditions S of a certain probability equal to p is manifested in the fact that in almost every sufficiently long series of tests the frequency of event A is approximately equal to p.

Statistical patterns, that is, patterns described by a scheme of type (b), were first discovered in gambling games like dice. Statistical patterns of birth and death have also been known for a very long time (for example, the probability of a newborn being a boy is 0.515). Late 19th century and 1st half of the 20th century. marked by the discovery of a large number of statistical laws in physics, chemistry, biology, etc.

The possibility of applying the methods of probability theory to the study of statistical patterns related to fields of science that are very distant from each other is based on the fact that the probabilities of events always satisfy some simple relationships, which will be discussed below (see the section Basic concepts of probability theory). The study of the properties of event probabilities on the basis of these simple relationships is the subject of probability theory.

Basic concepts of probability theory.

The basic concepts of probability theory as a mathematical discipline are most simply defined within the framework of the so-called elementary probability theory. Each test T, considered in elementary probability theory, is such that it ends with one and only one of the events E1, E2,..., ES (one or the other, depending on the case). These events are called trial outcomes. Each outcome Ek is associated with a positive number pk - the probability of this outcome. The numbers pk must add up to one. Then the events A are considered, consisting in the fact that “either Ei, or Ej,..., or Ek occurs.” Outcomes Ei, Ej,..., Ek are called favorable to A, and by definition, the probability P (A) of event A is assumed to be equal to the sum of the probabilities of outcomes favorable to it:

P (A) = pi + ps + … + pk. (1)

The special case p1 = p2 =... ps = 1/S leads to the formula

P (A) = r/s. (2)

Formula (2) expresses the so-called classical definition of probability, according to which the probability of any event A is equal to the ratio of the number r of outcomes favorable to A to the number s of all “equally possible” outcomes. The classical definition of probability only reduces the concept of “probability” to the concept of “equal possibility”, which remains without a clear definition.

Example. When throwing two dice, each of the 36 possible outcomes can be designated (i, j), where i is the number of points that appears on the first dice, j on the second. Outcomes are assumed to be equally probable. Event A - “the sum of points is 4”, is favored by three outcomes (1; 3), (2; 2), (3; 1). Therefore, P(A) = 3/36 = 1/12.

Based on any given events, two new events can be determined: their union (sum) and combination (product). Event B is called a union of events A 1, A 2,..., Ar,- if it has the form: “either A1, or A2,..., or Ar occurs.”

Event C is called a combination of events A1, A.2,..., Ar if it has the form: “both A1, A2,..., and Ar occur.” The union of events is denoted by the sign È, and the combination by the sign Ç. Thus, they write:

B = A1 È A2 È … È Ar, C = A1 Ç A2 Ç … Ç Ar.

Events A and B are called incompatible if their simultaneous occurrence is impossible, that is, if among the outcomes of the test there is not a single one favorable to both A and B.

The introduced operations of combining and combining events are associated with two main theorems of mathematical theory—the theorems of addition and multiplication of probabilities.

Probability addition theorem. If the events A1, A2,..., Ar are such that every two of them are inconsistent, then the probability of their union is equal to the sum of their probabilities.

So, in the above example with throwing two dice, event B - “the sum of points does not exceed 4”, is the union of three incompatible events A2, A3, A4, which consists in the fact that the sum of points is equal to 2, 3, 4, respectively. The probabilities of these events 1/36; 2/36; 3/36. According to the addition theorem, the probability P (B) is equal to

1/36 + 2/36 + 3/36 = 6/36 = 1/6.

The conditional probability of event B given condition A is determined by the formula

which, as can be shown, is in full accordance with the properties of frequencies. Events A1, A2,..., Ar are called independent if the conditional probability of each of them, provided that any of the others have occurred, is equal to its “unconditional” probability

Probability multiplication theorem. The probability of combining events A1, A2,..., Ar is equal to the probability of event A1, multiplied by the probability of event A2, taken under the condition that A1 has occurred,..., multiplied by the probability of event Ar, provided that A1, A2,.. ., Ar-1 have arrived. For independent events, the multiplication theorem leads to the formula:

P (A1 Ç A2 Ç … Ç Ar) = P (A1) Ї P (A2) Ї … Ї P (Ar), (3)

that is, the probability of combining independent events is equal to the product of the probabilities of these events. Formula (3) remains valid if in both its parts some of the events are replaced with their opposites.

Example. 4 shots are fired at the target with a hit probability of 0.2 per shot. Target hits from different shots are assumed to be independent events. What is the probability of hitting the target exactly three times?

Each test outcome can be indicated by a sequence of four letters [e.g., (y, n, n, y) means that the first and fourth shots hit (success), and the second and third shots did not hit (failure)]. There will be 2Ї2Ї2Ї2 = 16 outcomes. In accordance with the assumption of independence of the results of individual shots, formula (3) and a note to it should be used to determine the probabilities of these outcomes. Thus, the probability of outcome (y, n. n, n) should be set equal to 0.2Ї0.8Ї0.8Ї0.8 = 0.1024; here 0.8 = 1-0.2 is the probability of a miss with a single shot. The event “the target is hit three times” is favored by the outcomes (y, y, y, n), (y, y, n, y), (y, n, y, y). (n, y, y, y), the probability of each is the same:

0.2Ї0.2Ї0.2Ї0.8 =...... =0.8Ї0.2Ї0.2Ї0.2 = 0.0064;

therefore, the required probability is equal to

4Ї0.0064 = 0.0256.

Generalizing the reasoning of the analyzed example, we can derive one of the basic formulas of probability theory: if events A1, A2,..., An are independent and each have a probability p, then the probability of exactly m of them occurring is equal to

Pn (m) = Cnmpm (1 - p) n-m; (4)

here Cnm denotes the number of combinations of n elements of m. For large n, calculations using formula (4) become difficult. Let the number of shots in the previous example be 100, and the question is asked to find the probability x that the number of hits lies in the range from 8 to 32. Application of formula (4) and the addition theorem gives an accurate, but practically unusable expression of the desired probability

The approximate value of the probability x can be found using Laplace's theorem

and the error does not exceed 0.0009. The result found shows that the event 8 £ m £ 32 is almost certain. This is the simplest, but typical example of the use of limit theorems in probability theory.

The basic formulas of elementary probability theory also include the so-called total probability formula: if the events A1, A2,..., Ar are pairwise incompatible and their union is a reliable event, then for any event B its probability is equal to the sum

The probability multiplication theorem is particularly useful when considering compound tests. A trial T is said to be composed of trials T1, T2,..., Tn-1, Tn if each outcome of a trial T is a combination of some outcomes Ai, Bj,..., Xk, Yl of the corresponding trials T1, T2,... , Tn-1, Tn. From one reason or another, the probabilities are often known

Classification of events into possible, probable and random. Concepts of simple and complex elementary events. Operations on events. Classic definition of the probability of a random event and its properties. Elements of combinatorics in probability theory. Geometric probability. Axioms of probability theory.

Event classification

One of the basic concepts of probability theory is the concept of an event. Under event understand any fact that may occur as a result of an experience or test. Under experience, or test, refers to the implementation of a certain set of conditions.

Examples of events:

– hitting the target when firing from a gun (experience - making a shot; event - hitting the target);
– the loss of two emblems when throwing a coin three times (experience - throwing a coin three times; event - the loss of two emblems);
– the appearance of a measurement error within specified limits when measuring the range to a target (experience - range measurement; event - measurement error).

Countless similar examples can be given. Events are indicated by capital letters of the Latin alphabet, etc.

Distinguish joint events And incompatible. Events are called joint if the occurrence of one of them does not exclude the occurrence of the other. Otherwise, the events are called incompatible. For example, two dice are tossed. The event is the loss of three points on the first die, the event is the loss of three points on the second die. and - joint events. Let the store receive a batch of shoes of the same style and size, but different colors. Event - a box taken at random will contain black shoes, an event - the box will contain brown shoes, and - incompatible events.

The event is called reliable, if it is sure to occur under the conditions of a given experiment.

An event is called impossible if it cannot occur under the conditions of a given experience. For example, the event that a standard part will be taken from a batch of standard parts is reliable, but a non-standard part is impossible.

The event is called possible, or random, if as a result of experience it may appear, but it may not appear. An example of a random event could be the identification of product defects during inspection of a batch of finished products, a discrepancy between the size of the processed product and the specified one, or the failure of one of the links in the automated control system.

The events are called equally possible, if, according to the test conditions, none of these events is objectively more possible than the others. For example, let a store be supplied with light bulbs (in equal quantities) by several manufacturing plants. Events involving the purchase of a light bulb from any of these factories are equally possible.

An important concept is full group of events. Several events in a given experiment form a complete group if at least one of them is sure to appear as a result of the experiment. For example, an urn contains ten balls, six of them are red, four are white, and five balls have numbers. - the appearance of a red ball during one draw, - the appearance of a white ball, - the appearance of a ball with a number. Events form a complete group of joint events.

Let us introduce the concept of an opposite, or additional, event. Under opposite An event is understood as an event that must necessarily occur if some event does not occur. Opposite events are incompatible and the only possible ones. They form a complete group of events. For example, if a batch of manufactured products consists of good and defective products, then when one product is removed, it may turn out to be either good - event , or defective - event .

Operations on events

When developing an apparatus and methodology for studying random events in probability theory, the concept of the sum and product of events is very important.

The sum, or union, of several events is an event consisting of the occurrence of at least one of these events.

The sum of events is indicated as follows:

For example, if an event is hitting the target with the first shot, an event - with the second, then the event is hitting the target in general, it does not matter with which shot - the first, second or both.

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

The production of events is indicated

For example, if the event is that the target is hit with the first shot, the event is that the target is hit with the second shot, then the event is that the target was hit with both shots.

The concepts of sum and product of events have a clear geometric interpretation. Let the event consist of a point getting into the region , the event consists of getting into the region , then the event consists of the point getting into the region shaded in Fig. 1, and the event is when a point hits the area shaded in Fig. 2.

Classic definition of the probability of a random event

To quantitatively compare events according to the degree of possibility of their occurrence, a numerical measure is introduced, which is called the probability of an event.

The probability of an event is a number that expresses the measure of the objective possibility of the occurrence of an event.

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

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

This is the classic definition of probability. Thus, to find the probability of an event, it is necessary, having considered the various outcomes of the test, to find a set of uniquely possible, equally possible and incompatible cases, calculate their total number, the number of cases favorable to a given event, and then perform the calculation using formula (1.1).

From formula (1.1) it follows that the probability of an event is a non-negative number and can vary from zero to one depending on the proportion of the favorable number of cases from the total number of cases:

Properties of Probability

Property 1. If all cases are favorable for a given event, then this event is sure to occur. Consequently, the event in question is reliable, and the probability of its occurrence is , since in this case

Property 2. If there is not a single case favorable for a given event, then this event cannot occur as a result of experience. Consequently, the event in question is impossible, and the probability of its occurrence is , since in this case:

Property 3. The probability of the occurrence of events that form a complete group is equal to one.

Property 4. The probability of the occurrence of the opposite event is determined in the same way as the probability of the occurrence of the event:

where is the number of cases favorable to the occurrence of the opposite event. Hence the probability of the opposite event occurring is equal to the difference between unity and the probability of the event occurring:

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

Example 1. While dialing a phone number, the subscriber forgot one digit and dialed it at random. Find the probability that the correct number is dialed.

Solution. Let us denote the event that the required number is dialed. The subscriber could dial any of the 10 digits, so the total number of possible outcomes is 10. These outcomes are the only possible (one of the digits must be dialed) and equally possible (the digit is dialed at random). Only one outcome favors the event (there is only one required number). The required probability is equal to the ratio of the number of outcomes favorable to the event to the number of all outcomes:

Elements of combinatorics

In probability theory, placements, permutations and combinations are often used. If a set is given, then placement (combination) of the elements by is any ordered (unordered) subset of the elements of the set. When placed is called rearrangement from elements.

Let, for example, be given a set. The placements of the three elements of this set of two are , , , , , ; combinations - , , .

Two combinations differ in at least one element, and placements differ either in the elements themselves or in the order in which they appear. The number of combinations of elements by is calculated by the formula

is the number of placements of elements by ; - number of permutations of elements.

Example 2. In a batch of 10 parts there are 7 standard ones. Find the probability that among 6 parts taken at random there are exactly 4 standard ones.

Solution. The total number of possible test outcomes is equal to the number of ways in which 6 parts can be extracted from 10, i.e., equal to the number of combinations of 10 elements of 6. The number of outcomes favorable to the event (among the 6 taken parts there are exactly 4 standard ones) is determined as follows: 4 standard parts can be taken from 7 standard parts in different ways; in this case, the remaining parts must be non-standard; There are ways to take 2 non-standard parts from non-standard parts. Therefore, the number of favorable outcomes is equal to . The initial probability is equal to the ratio of the number of outcomes favorable to the event to the number of all outcomes:

Statistical definition of probability

Formula (1.1) is used to directly calculate the probabilities of events only when experience is reduced to a pattern of cases. In practice, the classical definition of probability is often not applicable for two reasons: first, the classical definition of probability assumes that the total number of cases must be finite. In fact, it is often not limited. Secondly, it is often impossible to present the outcomes of an experiment in the form of equally possible and incompatible events.

The frequency of occurrence of events during repeated Experiments tends to stabilize around some constant value. Thus, a certain constant value can be associated with the event under consideration, around which frequencies are grouped and which is a characteristic of the objective connection between the set of conditions under which experiments are carried out and the event.

The probability of a random event is the number around which the frequencies of this event are grouped as the number of trials increases.

This definition of probability is called statistical.

The advantage of the statistical method of determining probability is that it is based on a real experiment. However, its significant drawback is that to determine the probability it is necessary to perform a large number of experiments, which are very often associated with material costs. The statistical determination of the probability of an event, although it quite fully reveals the content of this concept, does not make it possible to actually calculate the probability.

The classical definition of probability considers the complete group of a finite number of equally possible events. In practice, very often the number of possible test outcomes is infinite. In such cases, the classical definition of probability is not applicable. However, sometimes in such cases you can use another method of calculating probability. For definiteness, we restrict ourselves to the two-dimensional case.

Let a certain region of area , which contains another region of area, be given on the plane (Fig. 3). A dot is thrown into the area at random. What is the probability that a point will fall into the region? It is assumed that a point thrown at random can hit any point in the region, and the probability of hitting any part of the region is proportional to the area of ​​the part and does not depend on its location and shape. In this case, the probability of hitting the area when throwing a point at random into the area is

Thus, in the general case, if the possibility of a random appearance of a point inside a certain area on a line, plane or in space is determined not by the position of this area and its boundaries, but only by its size, i.e. length, area or volume, then the probability of a random point falling inside a certain region is defined as the ratio of the size of this region to the size of the entire region in which a given point can appear. This is the geometric definition of probability.

Example 3. A round target rotates at a constant angular velocity. One fifth of the target is painted green, and the rest is white (Fig. 4). A shot is fired at the target in such a way that hitting the target is a reliable event. You need to determine the probability of hitting the target sector colored green.

Solution. Let’s denote “the shot hit the sector colored green.” Then . The probability is obtained as the ratio of the area of ​​the part of the target painted green to the entire area of ​​the target, since hits on any part of the target are equally possible.

Axioms of probability theory

From the statistical definition of the probability of a random event it follows that the probability of an event is the number around which the frequencies of this event observed experimentally are grouped. Therefore, the axioms of probability theory are introduced so that the probability of an event has the basic properties of frequency.

Axiom 1. Each event corresponds to a certain number that satisfies the condition and is called its probability.