Problems for the classical determination of probability. Examples of solutions. The basic concept of probability theory

"Accidents are not accidental" ... It sounds like a philosopher said, but in fact it is the lot of the great science of mathematics to study randomness. In mathematics, chance theory deals with randomness. 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 flip a coin up, it can fall "heads" or "tails". As long as the coin is in the air, both of these possibilities are possible. That is, the probability of possible consequences is 1: 1. If you pull one out of a deck with 36 cards, then the probability will be denoted as 1:36. It would seem that there is nothing to investigate and predict, especially with the help of mathematical formulas. Nevertheless, if you repeat a certain action many times, then you can identify a certain pattern and, on its basis, predict the outcome of events in other conditions.

To summarize all of the above, the theory of probability in the classical sense studies the possibility 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 were first made to predict the outcome of card games.

Initially, the theory of probability had nothing to do with mathematics. It was based on 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. For a long time they studied gambling and saw certain patterns, which they decided to tell the public about.

The same technique was invented by Christian 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 important. They made probability theory more like a mathematical discipline. The theory of probability, formulas and examples of basic tasks received their present form thanks to Kolmogorov's axioms. As a result of all the changes, the theory of probability has become 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:

• Credible. Those that will happen anyway (the coin will fall).
• Impossible. Events that will not happen in any scenario (the coin will remain hanging in the air).
• Random. Those that will or will not happen. They can be influenced by various factors, which are very difficult to predict. If we talk about a coin, then random factors that can affect the result: the physical characteristics of the coin, its shape, initial position, power of the throw, etc.

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

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

In practical exercises, it is customary to write down events in words.

One of the most important characteristics of events is their equality of opportunity. That is, if you flip a coin, all variants of the initial fall are possible until it falls. But also events are not equally possible. This happens when someone specifically influences the outcome. For example, "marked" playing cards or dice in which the center of gravity is shifted.

Events are also compatible and incompatible. Compatible events do not exclude each other from occurring. For example:

• A = "a student came to the lecture."
• B = "student came to the lecture."

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

Actions on events

Events can be multiplied and added, respectively, 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 at the same time. In the case when they are incompatible, the last option is impossible, either A or B will fall out.

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

Now you can give a few examples to better remember the basics, probability theory and formulas. Examples of problem solving further.

Exercise 1: The firm is participating in a competition for contracts for three types of work. Possible events that can 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."

Let's try to express the following situations using actions on events:

• K = "the firm will receive all contracts."

In mathematical form, the equation will look like this: K = ABC.

• M = "the firm will not receive a single contract."

M = A 1 B 1 C 1.

Complicating the task: H = "the firm will receive one contract." Since it is not known which contract the firm will receive (first, second or third), it is necessary to record the entire series of possible events:

Н = А 1 ВС 1 υ AB 1 С 1 υ А 1 В 1 С.

A 1 BC 1 is a series of events where the firm does not receive the first and third contracts, but receives the second. Other possible events were recorded by the corresponding method. The symbol υ in the discipline denotes the "OR" link. If we translate the given example into human language, then the company will receive either a third contract, or a second, or first. Similarly, you can write down other conditions in the discipline "Theory of Probability". The formulas and examples of problem solving presented above will help you do it 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 probabilities. Probability theory, formulas and examples (grade 9) mainly use the classical 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 there is a case opposite to A, 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 hearts. Accordingly, the formula for solving the problem will look like:

P (A) = 9/36 = 0.25.

As a result, the probability that a heart-suit card is drawn from the deck is 0.25.

Towards higher mathematics

Now it has become a little known what the theory of probability is, formulas and examples of solving tasks 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 learning formulas and examples (higher mathematics) small - with a 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 degree of 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 forecasting, then the statistical one - according to the results of the experiment. Take a small assignment, for example.

The technological control department checks the products for quality. Among 100 products, 3 were found to be of poor quality. How do you find the probability of the frequency 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? Of the 100 items that were checked, 3 were found to be of poor quality. We subtract 3 from 100, we 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 of A can be made in m different ways, and the choice of B 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 ways from city B to city C. How many ways can you get from city A to city C?

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

Let's complicate the task. How many ways are there to play 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 from the starting point and multiply.

That is, 36x35x34x33x32 ... x2x1 = the result does not fit on the calculator screen, so you can simply designate it as 36 !. The sign "!" next to a number indicates that the entire series of numbers is multiplied among themselves.

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

An ordered set of elements in a set is called an arrangement. Placements can be repetitive, that is, one element can be used multiple times. And no repetition, when the elements are not repeated. n are all elements, m are elements that participate in the placement. The formula for placement without repetitions would be:

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

Connections of n elements that differ only in the order of placement are called permutations. In mathematics, this is: P n = n!

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

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

Bernoulli's formula

The theory of probability, as well as in every discipline, has the 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 appearance of A in an experiment does not depend on the appearance or non-appearance of the same event in previous or subsequent tests.

Bernoulli's equation:

P n (m) = C n m × p m × q n-m.

The probability (p) of the occurrence of event (A) is unchanged 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, respectively, it may not occur. One 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 the event not occurring.

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

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

Solution: Since it is not known 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 change from 0 (no 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 a probability of 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 p have gone. With respect to p, the 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 buying 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 a direct proof of this.

Poisson's formula

Poisson's equation is used to calculate unlikely random situations.

Basic formula:

P n (m) = λ m / m! × e (-λ).

Moreover, λ = n x p. Here's a simple Poisson formula (probability theory). We will consider examples of solving problems further.

Assignment 3: The factory produced parts in the amount of 100,000 pieces. Defective part appearance = 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 Poisson's formula (probability theory) is used for the calculation. Examples of solving problems of this kind are no different from other tasks of the discipline, we substitute the necessary data in the given formula:

A = "a randomly selected part will be defective."

p = 0.0001 (according to the condition of the task).

n = 100000 (number of parts).

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

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

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

е -λ = lim n -> ∞ (1-λ / n) n.

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

Moivre-Laplace theorem

If the number of tests in the Bernoulli scheme is large enough, 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 the Laplace formula:

Р n (m) = 1 / √npq x ϕ (X m).

X m = m-np / √npq.

To better remember the Laplace formula (probability theory), examples of problems to help you below.

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

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

So the probability that the flyer will fire exactly 267 times is 0.03.

Bayes formula

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 looks like this:

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

A and B are specific events.

P (A | B) - 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.

Assignment 5: Phones from three companies were brought to the warehouse. At the same time, part of the phones 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 in the first factory is 2%, in the second - 4%, and in the third - 1%. It is necessary to find the probability that a randomly selected phone will turn out to be defective.

A = "randomly picked phone."

B 1 - the phone that was made by the first factory. Accordingly, input 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 firms:

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

P (A / B 2) = 0.04;

P (A / B 3) = 0.01.

Now we plug 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 the theory of probability, formulas and examples of problem solving, but this is only the tip of the iceberg of a vast discipline. And after all that has been written, it will be logical to ask the question of whether the theory of probability is needed in life. It is difficult for an ordinary person to answer, it is better to ask about this from the one who has hit the jackpot more than once with its help.

Brief theory

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

The quantities that determine how significant the objective grounds for expecting the occurrence of an event are, are characterized by the probability of the event. It should be emphasized that the probability is an objective value that exists independently of the knower and is conditioned by the entire set of conditions that contribute to the occurrence of an event.

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

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

Let a reliable event break up into equally possible cases, the sum of which gives an event. That is, the cases from which it splits are called favorable for the event, since the appearance of one of them ensures the offensive.

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

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

This is the classic definition of probability. Thus, to find the probability of an event, it is necessary, after considering the various outcomes of the test, to find a set of the only possible, equally possible and inconsistent cases, to calculate their total number n, the number of cases m, favorable for this event, and then perform the calculation using the above formula.

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

The following properties of probability follow from the definition:

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

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

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

Property 4. The probability of occurrence of events forming a complete group is equal to one.

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

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

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 proceeding from logical reasoning.

When a set of conditions is met, a reliable event will surely happen, and the impossible will not necessarily happen. Among the events that, when creating a complex of conditions, may or may not happen, one can count on the appearance of some with more reason, on the appearance of others with less reason. If, for example, there are more white balls in an urn than black ones, then there is more reason to hope for the appearance of a white ball when taken out of the urn at random than for the appearance of a black ball.

An example of solving the problem

Example 1

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

The solution of the problem

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

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

Seeking probability:

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

Number of outcomes favorable to the event:

Seeking probability:

Let the event- there is at least one red and 1 white ball

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

Number of outcomes favorable to the event:

Seeking probability:

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

Example 2

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

Decision

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

Let's use the classical definition of probability:

Total number of possible trial outcomes

The number of trials favorable to the event of interest

One point, two points ..., six points may appear on the rolled edge of the first dice. similarly, six outcomes are possible on the second die roll. Each of the outcomes of throwing the first die can be combined with each of the outcomes of the second. Thus, the total number of possible elementary test outcomes is equal to the number of placements with repetitions (choice with placements of 2 elements from a set of 6):

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

The following combinations of dropped points will favor the event:

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

Probability theory is a fairly extensive independent branch of mathematics. In the school course, the theory of probability is considered very superficially, however, in the exam and the GIA there are tasks on this topic. However, solving the problems of the school course is not so difficult (at least as far as arithmetic operations are concerned) - here you do not need to count derivatives, take integrals and solve complex trigonometric transformations - the main thing is to be able to handle prime numbers and fractions.

Probability theory - basic terms

The main terms of the theory of probability are trial, outcome, and random event. A test in the theory of probability is an experiment - toss a coin, draw a card, draw lots - all these are tests. The result of the test, you guessed it, is called the outcome.

And what is the randomness of an event? In the theory of probability, it is assumed that the test is carried out more than once and there are many outcomes. Many outcomes of a trial are called a random event. For example, if you flip a coin, two random events can happen - heads or tails.

Do not confuse the concepts of an outcome and a random event. The outcome is one result of one trial. A random event is a multitude of possible outcomes. By the way, there is such a term as an impossible event. For example, the "number 8" event on a standard game die is not possible.

How do you find the probability?

We all roughly understand what probability is, and quite often we use this word in our vocabulary. In addition, we can even draw some conclusions regarding the likelihood of this or that event, for example, if there is snow outside the window, we can most likely say that it is not summer now. However, how can this assumption be expressed numerically?

In order to introduce a formula for finding the probability, we introduce one more concept - a favorable outcome, that is, an outcome that is favorable for a particular event. The definition is rather ambiguous, of course, however, according to the condition of the problem, it is always clear which of the outcomes is favorable.

For example: There are 25 people in the class, three of them are Katya. The teacher appoints Olya on duty, and she needs a partner. What is the probability that Katya will become a partner?

In this example, a favorable outcome is partner Katya. We will solve this problem a little later. But first, with the help of an additional definition, we introduce a formula for finding the probability.

• P = A / N, where P is the probability, A is the number of favorable outcomes, N is the total number of outcomes.

All school problems revolve around this one formula, and the main difficulty usually lies in finding the outcomes. Sometimes it is easy to find them, sometimes it is not very good.

How to solve probabilities?

Problem 1

So now let's solve the problem posed above.

The number of favorable outcomes (the teacher will choose Katya) is three, because there are three Katya in the class, and there are 24 overall outcomes (25-1, because Olya has already been selected). Then the probability is: P = 3/24 = 1/8 = 0.125. Thus, the probability that Katya will be Olya's partner is 12.5%. Not difficult, right? Let's look at something a little more complicated.

Task 2

The coin was thrown twice, what is the probability of the combination: one heads and one tails?

So, consider the overall outcomes. How can coins fall - heads / heads, tails / tails, heads / tails, tails / heads? This means that the total number of outcomes is 4. How many favorable outcomes? Two - heads / tails and tails / heads. Thus, the probability of getting a heads / tails combination is:

• P = 2/4 = 0.5 or 50 percent.

Now let's consider the following problem. Masha has 6 coins in her pocket: two - 5 rubles and four - 10 rubles. Masha put 3 coins in another pocket. What is the likelihood that 5-ruble coins end up in different pockets?

For simplicity, let's designate coins with numbers - 1,2 - five-ruble coins, 3,4,5,6 - ten-ruble coins. So how can coins be in your pocket? There are 20 combinations in total:

• 123, 124, 125, 126, 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456.

At first glance, it may seem that some combinations have disappeared, for example, 231, but in our case the combinations 123, 231 and 321 are equivalent.

Now we count how many favorable outcomes we have. For them we take those combinations in which there is either the number 1 or the number 2: 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256. There are 12 of them. Thus, the probability is:

• P = 12/20 = 0.6 or 60%.

The problems in probability theory presented here are pretty simple, but don't think that probability theory is a simple branch of mathematics. If you decide to continue your education at a university (with the exception of humanitarian specialties), you will definitely have pairs in higher mathematics, where you will be introduced to more complex terms of this theory, and the problems there will be much more difficult.

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

From thinking about the eternal to probability theory

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

Thanks to the excitement of the cavalier de Mere, who was equally a player and a person who was not indifferent to science, Pascal was forced to find a way to calculate the probability. De Mere was interested in the following question: "How many times do you need to throw two dice in pairs in order for the probability of getting 12 points to exceed 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 de Mere, who became the unwitting pioneer of the development of the theory of probability. Interestingly, the persona de Mere remained famous in this field, and not in the 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 this is a specific figure that can be substantiated 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 experience.

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 ...

The probability of a random event

In order to be able to start the mathematical part of probability, it is necessary to give definitions to all its components.

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

In the theory of probability, the following are distinguished:

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

Relationships between events

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

In relation to each other, events can be:

• Equally possible.
• Compatible.
• Incompatible.
• Opposite (mutually exclusive).
• Addicted.

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

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

If events A and B never occur simultaneously in the same experience, then they are called incompatible... Tossing a coin is a good example: tails are automatically not 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 onset of one event makes the onset of another impossible, then they are called opposite. Then one of them is designated as A, and the other - Ā (read as "not A"). Occurrence of event A means that Ā did not happen. These two events form a complete group with the sum of probabilities equal to 1.

Dependent events have a mutual influence, decreasing or increasing the likelihood of each other.

Relationships between events. Examples of

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

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

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

Test No. 1. 6 balls participate, three of which are colored blue with odd numbers, and three others are red with even numbers.

Test number 2. 6 balls of blue color with numbers from one to six are participating.

Based on this example, you can name combinations:

• A credible event. In isp. No. 2, the event “to get the blue ball” is reliable, since the probability of its occurrence is 1, since all the balls are blue and there can be no miss. Whereas the event “to get the ball with the number 1” is random.
• Impossible event. In isp. №1 with blue and red balls, the event "to get the purple ball" is impossible, since the probability of its occurrence is equal to 0.
• Equally possible events. In isp. No. 1 of 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 in a row twice in a row are compatible events.
• Incompatible events. In the same isp. No. 1, the events "get a red ball" and "get a ball with an odd number" cannot be combined in the same experiment.
• Opposite events. The most striking example of this is a coin toss where drawing heads is tantamount to not drawing tails, and the sum of their probabilities is always 1 (full group).
• Dependent events... So, in isp. # 1, you can set a goal to extract the red ball twice in a row. It is retrieved or not retrieved the first time affects the likelihood of retrieving it a 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 thoughts to accurate data occurs by translating the topic into a mathematical plane. That is, judgments about a random event like "high probability" or "minimum probability" can be translated to specific numerical data. Such material is already permissible to evaluate, compare and enter into more complex calculations.

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

So, the formula for the probability of an event:

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. Ball # 1 as described earlier: 3 blue balls with numbers 1/3/5 and 3 red balls with numbers 2/4/6.

Several different tasks can be considered based on this test:

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

As can be seen from the calculations, event C has a high 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 isp. No. 1 it is impossible to reach the blue and red ball at the same time. That is, you can get either a blue or a red ball. Likewise, an even and an odd number cannot appear on a die at the same time.

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

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 appearance of all of them.

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

The probability of the sum of inconsistent events

If the probability of inconsistent 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 isp. No. 1 with blue and red balls will drop a number between 1 and 4. Let's calculate not in one action, but the sum of the probabilities of elementary components. So, in such an experience there are only 6 balls or 6 of all possible outcomes. The numbers that satisfy the condition are 2 and 3. The probability of getting the number 2 is 1/6, the probability of the number 3 is also 1/6. The probability that a number between 1 and 4 will be dropped is:

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

So, if, in the experiment with a cube, add up the probabilities of falling out of all numbers, then the result will be one.

This is also true for opposite events, for example, in the experience with a coin, where one side of it is event A, and the other is the opposite event Ā, as you know,

P (A) + P (Ā) = 1

Probability of producing inconsistent events

Probability multiplication is used when considering the appearance 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 isp. №1 as a result of two attempts, a blue ball will appear twice, equal to

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

Joint events

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

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

The probability of the sum of joint events. Example

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

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

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

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

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

This formula for the probability of an event can also be applied to inconsistent events, where the probability of the joint occurrence of an event P (AB) = 0. This means that the probability of the sum of inconsistent 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 in the form of two regions A and B, which intersect with each other. As you can see from the picture, the area of ​​their union is equal to the total area minus the area of ​​their intersection. These geometrical explanations make the formula, illogical at first glance, clearer. Note that geometric solutions are not uncommon in probability theory.

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

Dependent events

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

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

An example of calculating the probability of dependent events

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

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

1. Diamonds.
2. Another suit.

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) in the deck and 1 tambourine (8) less, the probability of event B is:

P A (B) = 8/35 = 0.23

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

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

It can be seen that if event A is agreed that the first card is a tambourine, then the probability of event B decreases, and vice versa.

Multiplication of dependent events

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

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

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

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

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

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

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

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

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

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

Full probability of the 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, ..., And n, .. forms a complete group of events under the condition:

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

So, the formula for the total probability for event B with a full group of random events A1, A2, ..., And n is equal to:

A look into 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 have a probabilistic nature, special methods of work are needed. Probability theory can be used in any technological field as a way to determine the possibility of error or malfunction.

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

Everything in the world happens deterministically or by chance ...
Aristotle

Probability: basic rules

Probability theory calculates the probabilities of various events. The basic concept of probability theory is the concept of a random event.

For example, you toss a coin, it randomly falls on the coat of arms or tails. You don't know in advance which side the coin will fall on. You conclude an insurance contract, you do not know in advance whether or not payments will be made.

In actuarial calculations, you need to be able to estimate the likelihood of various events, so the theory of probability plays a key role. No other area of ​​mathematics can deal with the probabilities of events.

Let's take a closer look at the coin toss. There are 2 mutually exclusive outcomes: a coat of arms or a tails. The outcome of the throw is random, since the observer cannot analyze and take into account all the factors that affect the result. What is the probability of a coat of arms falling? Most will answer ½, but why?

Let formally BUT denotes the fall of the coat of arms. Let the coin toss n time. Then the probability of the event BUT can be defined as the proportion of those throws that result in the coat of arms:

Where n the total number of throws, n (A) number of coat of arms falls.

The relation (1) is called frequency events BUT in a long series of tests.

It turns out that in various series of tests the corresponding frequency at large n grouped around some constant value P (A)... This quantity is called probability of event BUT and denoted by the letter R- shorthand for the English word probability - probability.

Formally, we have:

(2)

This law is called the law of large numbers.

If the coin is correct (symmetrical), then the probability of getting the coat of arms is equal to the probability of falling heads and is equal to ½.

Let be BUT and IN some events, for example, whether an insured event has occurred or not. The combination of two events is an event consisting in the execution of an event BUT, events IN, or both events together. The intersection of two events BUT and IN called an event consisting in implementation as an event BUT and events IN.

Fundamental rules calculus of probabilities of events are as follows:

1. The probability of any event is between zero and one:

2. Let A and B be two events, then:

It reads like this: the probability of combining two events is equal to the sum of the probabilities of these events minus the probability of intersecting events. If the events are inconsistent or disjoint, then the probability of combining (the sum) of the two events is equal to the sum of the probabilities. This law is called law additions probabilities.

We say that an event is reliable if its probability is 1. When analyzing certain phenomena, the question arises of how the occurrence of an event affects IN on the onset of the event BUT... For this, conditional probability :

(4)

It reads like this: probability of occurrence BUT given that IN equals the probability of crossing BUT and IN divided by the probability of the event IN.
In formula (4), it is assumed that the probability of an event IN Above zero.

Formula (4) can also be written as:

(5)

This is the formula multiplication of probabilities.

The conditional probability is also called a posteriori probability of event BUT- probability of occurrence BUT after the onset IN.

In this case, the probability itself is called a priori probability. There are several other important formulas that are heavily used in actuarial calculations.

Total Probability Formula

Let us assume that an experiment is being carried out, the conditions of which can be made in advance. mutually mutually exclusive assumptions (hypotheses):

We assume that there is either a hypothesis, or ... or. The probabilities of these hypotheses are known and equal:

Then the following formula holds: complete probabilities :

(6)

The probability of the event BUT equal to the sum of the products of the probability of occurrence BUT for each hypothesis on the probability of this hypothesis.

Bayes formula

Bayes formula allows you to recalculate the probability of hypotheses in the light of new information given by the result BUT.

Bayes' formula is in a sense the inverse of the total probability formula.

Consider the following practical task.

Problem 1

Suppose there is a plane crash and experts are busy investigating its causes. 4 reasons for the catastrophe are known in advance: either the reason, or, or, or. According to available statistics, these reasons have the following probabilities:

When inspecting the crash site, traces of fuel ignition were found, according to statistics, the probability of this event for one reason or another is as follows:

Question: what is the most probable cause of the disaster?

Let us calculate the probabilities of the causes under the condition of the occurrence of the event BUT.

This shows that the first reason is the most probable, since its probability is maximal.

Task 2

Consider a plane landing at an airfield.

When landing, the weather conditions can be as follows: there is no low cloudiness (), there is a low cloudiness (). In the first case, the probability of a successful landing is P1... In the second case - P2... It's clear that P1> P2.

Blind-landing devices have the likelihood of trouble-free operation R... If there is low cloud cover and the blind landing devices have failed, the probability of a successful landing is P3, and P3<Р2 ... It is known that for a given aerodrome the proportion of days in a year with low cloud cover is equal to.

Find the probability of a safe landing.

We need to find the probability.

There are two mutually exclusive options: the blind landing devices are working, the blind landing devices have failed, so we have:

Hence, according to the formula of total probability:

Problem 3

The insurance company deals with life insurance. 10% of the insured in this company are smokers. If the insured does not smoke, the probability of his death during the year is 0.01 If he is a smoker, then this probability is 0.05.

What is the proportion of smokers among those insured who died during the year?

Answer options: (A) 5%, (B) 20%, (C) 36%, (D) 56%, (E) 90%.

Decision

Let's introduce events:

The condition of the problem means that

In addition, since events and form a complete group of pairwise incompatible events, then.
The probability we are interested in is this.

Using Bayes' formula, we have:

therefore, the correct option is ( IN).

Problem 4

The insurance company sells life insurance contracts in three categories: standard, privileged and ultra-privileged.

50% of all insured are standard, 40% are privileged and 10% are ultra privileged.

The probability of dying within a year for the standard insured is 0.010, for the privileged one is 0.005, and for the ultra privileged one is 0.001.

What is the likelihood that the deceased insured is ultra-privileged?

Decision

Let's consider the following events:

In terms of these events, the probability we are interested in is this. By condition:

Since the events,, form a complete group of pairwise incompatible events, using the Bayes formula, we have:

Random variables and their characteristics

Let some random variable, for example, fire damage or the amount of insurance payments.
A random variable is fully characterized by its distribution function.

Definition. Function called distribution function random variable ξ .

Definition. If there is a function such that for arbitrary a done