Tasks for the classical definition of probability. Fundamentals of Probability for Actuaries

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

From reflections on the eternal to the theory of probability

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

Thanks to the excitement of the Chevalier de Mere, who was equally a gambler and a person who was not indifferent to science, Pascal was forced to find a way to calculate the probability. De Mere was interested in this question: "How many times do you need to throw two dice in pairs so that the probability of getting 12 points exceeds 50%?". The second question that interested the gentleman extremely: "How to divide the bet between the participants in the unfinished game?" Of course, Pascal successfully answered both questions of de Mere, who became the unwitting initiator of the development of the theory of probability. Interestingly, the person of de Mere remained known in this area, and not in literature.

Previously, no mathematician has yet made an attempt to calculate the probabilities of events, since it was believed that this was only a guesswork solution. Blaise Pascal gave the first definition of the probability of an event and showed that this is a specific figure that can be justified mathematically. Probability theory has become the basis for statistics and is widely used in modern science.

What is randomness

If we consider a test that can be repeated an infinite number of times, then we can define a random event. This is one of the possible outcomes of the experience.

Experience is the implementation of specific actions in constant conditions.

In order to be able to work with the results of experience, events are usually denoted by the letters A, B, C, D, E ...

Probability of a random event

To be able to proceed to the mathematical part of probability, it is necessary to define all its components.

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

Probability theory is:

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

Relationships between events

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

In relation to each other, events can be:

• Equally possible.
• compatible.
• Incompatible.
• Opposite (mutually exclusive).
• Dependent.

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

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

If events A and B never occur at the same time in the same experiment, then they are called incompatible. coin toss - good example: the appearance of tails is automatically the non-appearance of heads.

The probability for the sum of such incompatible events consists of the sum of the probabilities of each of the events:

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

If the occurrence of one event makes the occurrence of another impossible, then they are called opposite. Then one of them is designated as A, and the other - Ā (read as "not A"). The occurrence of event A means that Ā did not occur. These two events form full group with the sum of probabilities equal to 1.

Dependent events have mutual influence, decreasing or increasing each other's probability.

Relationships between events. Examples

It is much easier to understand the principles of probability theory and the combination of events using examples.

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

The event is one of possible outcomes of experience - a red ball, a blue ball, a ball with the number six, etc.

Test number 1. There are 6 balls, three of which are blue with odd numbers, and the other three are red with even numbers.

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

Based on this example, we can name combinations:

• Reliable event. In Spanish No. 2, the event "get the blue ball" is reliable, since the probability of its occurrence is 1, since all the balls are blue and there can be no miss. Whereas the event "get the ball with the number 1" is random.
• Impossible event. In Spanish No. 1 with blue and red balls, the event "get the purple ball" is impossible, since the probability of its occurrence is 0.
• Equivalent events. In Spanish No. 1, the events “get the ball with the number 2” and “get the ball with the number 3” are equally likely, and the events “get the ball with an even number” and “get the ball with the number 2” have different probabilities.
• Compatible events. Getting a six in the process of throwing a die twice in a row are compatible events.
• Incompatible events. In the same Spanish No. 1 events "get the red ball" and "get the ball with an odd number" cannot be combined in the same experience.
• opposite events. Most prime example This is coin tossing, when drawing heads is the same as not drawing tails, and the sum of their probabilities is always 1 (full group).
• Dependent events. So, in Spanish No. 1, you can set yourself the goal of extracting a red ball twice in a row. Extracting it or not extracting it the first time affects the probability of extracting it the second time.

It can be seen that the first event significantly affects the probability of the second (40% and 60%).

Event Probability Formula

The transition from fortune-telling to exact data occurs through the translation of the topic into the mathematical plane. That is, judgments about a random event like "high probability" or "minimum probability" can be translated to specific numerical data. It is already permissible to evaluate, compare and introduce such material into more complex calculations.

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

So, the formula for the probability of an event is:

Where m is the number of favorable outcomes for event A, n is the sum of all possible outcomes for this experience. The probability of an event is always between 0 and 1:

0 ≤ P(A) ≤ 1.

Calculation of the probability of an event. Example

Let's take Spanish. No. 1 with balls, which is described earlier: 3 blue balls with numbers 1/3/5 and 3 red balls with numbers 2/4/6.

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

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

As can be seen from the calculations, event C has a higher probability, since the number of possible positive outcomes is higher than in A and B.

Incompatible events

Such events cannot appear simultaneously in the same experience. As in Spanish No. 1, it is impossible to get a blue and a red ball at the same time. That is, you can get either a blue or a red ball. In the same way, an even and an odd number cannot appear in a die at the same time.

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

The sum of several events is an event that implies the occurrence of at least one of them. The product of several events is the joint occurrence of them all.

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

Probability of the sum of incompatible events

If the probability of incompatible events is considered, then the probability of the sum of events is equal to the sum of their probabilities:

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

For example: we calculate the probability that in Spanish. No. 1 with blue and red balls will drop a number between 1 and 4. We will calculate not in one action, but by the sum of the probabilities of the elementary components. So, in such an experiment there are only 6 balls or 6 of all possible outcomes. The numbers that satisfy the condition are 2 and 3. The probability of getting the number 2 is 1/6, the probability of the number 3 is also 1/6. The probability of getting a number between 1 and 4 is:

The probability of the sum of incompatible events of a full group is 1.

So, if in the experiment with a cube we add up the probabilities of getting all the numbers, then as a result we get one.

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

Р(А) + Р(Ā) = 1

Probability of producing incompatible events

Multiplication of probabilities is used when considering the occurrence of two or more incompatible events in one observation. The probability that events A and B will appear in it at the same time is equal to the product of their probabilities, or:

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

For example, the probability that in No. 1 as a result of two attempts, a blue ball will appear twice, equal to

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

Joint Events

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

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

The probability of the sum of joint events. Example

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

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

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

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

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

This formula for the probability of an event can also be applied to incompatible events, where the probability of the joint occurrence of an event P(AB) = 0. This means that the probability of the sum of incompatible events can be considered a special case of the proposed formula.

Probability geometry for clarity

Interestingly, the probability of the sum of joint events can be represented as two areas A and B that intersect with each other. As you can see from the picture, the area of ​​their union is equal to the total area minus the area of ​​their intersection. This geometric explanation makes the seemingly illogical formula more understandable. Note that geometric solutions not uncommon in probability theory.

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

Dependent events

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

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

Example of calculating the probability of dependent events

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

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

1. Tambourine.
2. Another suit.

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

P A (B) \u003d 8 / 35 \u003d 0.23

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

P A (B) \u003d 9/35 \u003d 0.26.

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

Multiplication of dependent events

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

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

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

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

P (AB) \u003d P (A) * P A (B)

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

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

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

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

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

Total probability of an event

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

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

So, the formula for the total probability for event B with a complete group of random events A1, A2, ..., A n is:

A look into the future

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

It can be said that, by recognizing the probability, we somehow take a theoretical step into the future, looking at it through the prism of formulas.

In economics, as well as in other areas human activity or in nature, we constantly have to deal with events that cannot be accurately predicted. Thus, the volume of sales of goods depends on demand, which can vary significantly, and on a number of other factors that are almost impossible to take into account. Therefore, when organizing production and sales, one has to predict the outcome of such activities on the basis of either one's own previous experience, or similar experience of other people, or intuition, which is also largely based on experimental data.

In order to somehow evaluate the event under consideration, it is necessary to take into account or specially organize the conditions in which this event is recorded.

The implementation of certain conditions or actions to identify the event in question is called experience or experiment.

The event is called random if, as a result of experience, it may or may not occur.

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

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

One of the main problems in probability theory is the problem of determining a quantitative measure of the possibility of an event occurring.

Algebra of events

Events are called incompatible if they cannot be observed together in the same experience. Thus, the presence of two and three cars in one store for sale at the same time are two incompatible events.

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

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

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

An event consisting in the appearance of two goods at the same time in the store is a product of events: - the appearance of one product, - the appearance of another product.

Events form a complete group of events if at least one of them necessarily occurs in the experience.

Example. The port has two berths for ships. Three events can be considered: - the absence of vessels at the berths, - the presence of one vessel at one of the berths, - the presence of two vessels at two berths. These three events form a complete group of events.

Opposite two unique possible events that form a complete group are called.

If one of the events that are opposite is denoted by , then the opposite event is usually denoted by .

Classical and statistical definitions of the probability of an event

Each of the equally possible test results (experiments) is called an elementary outcome. They are usually denoted by letters . For example, a dice is thrown. There can be six elementary outcomes according to the number of points on the sides.

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

A quantitative measure of the possibility of occurrence of the event under consideration is the probability.

Two definitions of the probability of an event are most widely used: classic and statistical.

The classical definition of probability is related to the notion of a favorable outcome.

Exodus is called favorable this event, if its occurrence entails the occurrence of this event.

In the given example, the event under consideration is an even number of points on the dropped edge, has three favorable outcomes. In this case, the general
the number of possible outcomes. So, here you can use the classical definition of the probability of an event.

Classic definition equals the ratio of the number of favorable outcomes to the total number of possible outcomes

where is the probability of the event , is the number of favorable outcomes for the event, is the total number of possible outcomes.

In the considered example

The statistical definition of probability is associated with the concept of the relative frequency of occurrence of an event in experiments.

The relative frequency of occurrence of an event is calculated by the formula

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

Statistical definition. The probability of an event is the number relative to which the relative frequency is stabilized (established) with an unlimited increase in the number of experiments.

In practical problems, the relative frequency is taken as the probability of an event at a sufficiently large numbers tests.

From these definitions of the probability of an event, it can be seen that the inequality always holds

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

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

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

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

Relation (1.1)

called relative frequency event A in the series of experiments.

It is easy to verify the validity of the properties:

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

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

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

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

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

This definition is called statistical definition of probability .

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

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

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

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

(1.4)

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

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

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

Indeed, according to (1.4)

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

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

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

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

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

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

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

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

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

The following properties follow from the definition of probability:

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

Indeed, if the event is certain, then each elementary outcome of the test favors the event. In this case m=p, Consequently,

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

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

Indeed, if the event is impossible, then none of the elementary outcomes of the trial favors the event. In this case t= 0, therefore, P(A)=m/n=0/n=0. (1.7)

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

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

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

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

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

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

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

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

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

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

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

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

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 occur. This article will be presented in summary theory of probability formulas and examples of solving problems in the theory of probability, which will be in the 4th task of the exam in mathematics (profile level).

Why do we need the theory of probability

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

Playing cards, dice, roulette created situations where any of a finite number of equally probable events could occur. There was a need to give numerical estimates of the possibility of the occurrence of an 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. Was created modern theory probabilities.

Basic concepts of probability theory

The object of study of probability theory is events and their probabilities. If the 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 happened at the same time.

The product of events A and B is the event C, which consists in the fact that both the event A and the event B happened.

Events A and B are said to be incompatible if they cannot happen at the same time.

An event A is said to be impossible if it cannot happen. Such an event is denoted by the symbol .

An event A is called certain if it will definitely occur. Such an event is denoted by the symbol .

Let each event A be assigned a number P(A). This number P(A) is called the probability of the event A if the following conditions are satisfied with such a correspondence.

An important particular 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 introduced by the formula . The probability introduced in this way is called the classical probability. It can be proved that properties 1-4 hold in this case.

Problems in the theory of probability, which are found on the exam in mathematics, are mainly related to classical probability. Such tasks can be very simple. Particularly simple are problems in probability theory in demo versions. It is easy to calculate the number of favorable outcomes, the number of all outcomes is written directly in the condition.

We get the answer according to the formula.

An example of a task from the exam in mathematics to determine the probability

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

Solution.

There are 20 equiprobable elementary outcomes in total, that is, Marina can take any of the 20 pies. But we need to estimate the probability that Marina will take the rice patty, that is, where A is the choice of the rice patty. This means that we have a total of 8 favorable outcomes (choosing rice pies). Then the probability will be determined by the formula:

Independent, Opposite, and Arbitrary Events

However, in open jar tasks began to meet more complex tasks. Therefore, let us draw the reader's attention to other questions studied in probability theory.

Events A and B are called independent if the probability of each of them does not depend on whether the other event occurred.

Event B consists in the fact that event A did not occur, 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. .

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 product 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.

Not always counting the number of outcomes is so simple. In some cases, it is necessary to use combinatorics formulas. The most important thing is to count the number of events that meet certain conditions. Sometimes such 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 to place the second student. For the third student there are 4 free places, for the fourth - 3, for the fifth - 2, 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 read "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, .

Consider now 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 to place the second student. To find the number of all options, you need to find the product.

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

In our case .

And the last one in this series. How many ways are there to choose 3 students out of 6? The first student can be chosen in 6 ways, the second in 5 ways, and the third in 4 ways. But among these options, the same three students occur 6 times. To find the number of all options, you need to calculate the value: . In the general case, the answer to this question is given by the formula for the number of combinations of elements by elements:

In our case .

Examples of solving problems from the exam in mathematics to determine the probability

Problem 1. From the collection, ed. Yashchenko.

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

.

Answer: 0.3.

Problem 2. From the collection, ed. Yashchenko.

In each batch of 1000 light bulbs, an average of 20 defective ones. Find the probability that a light bulb chosen at random from a batch is good.

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

Answer: 0.98.

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

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

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

Answer: 0.06.

In the geometry exam, the student answers one question from the list of exam questions. The probability that this is a trigonometry question is 0.2. The probability that this is an Outer Corners question is 0.15. There are no questions related to these two topics at the same time. Find the probability that the student will get a question on one of these two topics on 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 in a year is 0.29. Find the probability that at least one lamp does not burn out within a 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 variants of such events. We accept the notation: - the light bulb is on, - the light bulb is burned out. And immediately next we calculate the probability of an event. For example, the probability of an event in which three independent events “light bulb burned out”, “light bulb on”, “light bulb on” occurred: where the probability of the event “light bulb on” is calculated as the probability of an event opposite to the event “light bulb off”, namely .

Note that there are only 7 incompatible events favorable to us. The probability of such events is equal to the sum of the probabilities of each of the events: .

Answer: 0.975608.

You can see another problem in the picture:

Thus, you and I understood what the theory of probability is, formulas and examples of problem solving for which you can meet in the version of the exam.

Knowing that the probability can be measured, let's try to express it in numbers. There are three possible paths.

Rice. 1.1. Measuring Probability

PROBABILITY DETERMINED BY SYMMETRY

There are situations in which the possible outcomes are equally likely. For example, when tossing a coin once, if the coin is standard, the probability of getting heads or tails is the same, i.e. P(heads) = P(tails). Since only two outcomes are possible, then P(heads) + P(tails) = 1, therefore P(heads) = P(tails) = 0.5.

In experiments where outcomes have equal chances of occurring, the probability of the event E, P(E) is:

Example 1.1. The coin is tossed three times. What is the probability of two heads and one tail?

First, let's find all possible outcomes: To make sure that all possible options we have found, we will use a tree diagram (see Chapter 1 section 1.3.1).

So, there are 8 equally likely outcomes, therefore, the probability of them is 1/8. Event E - two "eagles" and "tails" - there were three. That's why:

Example 1.2. A standard die is rolled twice. What is the probability that the sum of the points is 9 or more?

Let's find all possible outcomes.

Table 1.2. The total number of points obtained by rolling a dice twice

So, in 10 out of 36 possible outcomes, the sum of points is 9, or therefore:

EMPIRICALLY DETERMINED PROBABILITY

An example with a coin from Table. 1.1 clearly illustrates the mechanism for determining probabilities.

At total number experiments of which are successful, the probability of the desired result is calculated as follows:

The ratio is the relative frequency of occurrence of a certain result in a sufficiently long experiment. The probability is calculated either on the basis of the data of the experiment, on the basis of past data.

Example 1.3. Of the five hundred electric lamps tested, 415 have worked for more than 1000 hours. Based on the data of this experiment, it can be concluded that the probability of normal operation of a lamp of this type for more than 1000 hours is:

Note. The control is destructive, so not all lamps can be tested. If only one lamp were tested, then the probability would be 1 or 0 (i.e. will it be able to work 1000 hours or not). Hence the need to repeat the experiment.

Example 1.4. In table. 1.3 shows data on the experience of men working in the company:

Table 1.3. Male work experience

What is the probability that the next person hired by the firm will work for at least two years?

Solution.

The table shows that 38 out of 100 employees have been with the company for more than two years. The empirical probability that the next employee will stay with the company for more than two years is:

At the same time, we assume that new employee“Typical, and working conditions are unchanged.

SUBJECTIVE EVALUATION OF PROBABILITY

In business, there are often situations in which there is no symmetry, and there are no experimental data either. Therefore, determining the probability of a favorable outcome under the influence of the views and experience of the researcher is subjective.

Example 1.5.

1. An investment expert believes that the probability of making a profit during the first two years is 0.6.

2. Marketing manager's forecast: the probability of selling 1000 units of a product in the first month after its introduction to the market is 0.4.