The classical definition of probability is based on the concept probabilistic experience, or a probabilistic experiment. Its result is one of several possible outcomes, called elementary outcomes, and there is no reason to expect that any elementary outcome will appear more often than others when repeating a probabilistic experiment. For example, consider a probabilistic experiment on throwing a dice. The result of this experience is one of 6 points drawn on the sides of the die.

Thus, there are 6 elementary outcomes in this experiment:

and each of them is equally expectant.

Event in the classical probabilistic experiment is an arbitrary subset of the set of elementary outcomes. In the considered example of throwing a dice, the event is, for example, an even number of points, which consists of elementary outcomes.

The probability of an event is a number:

where is the number of elementary outcomes that make up the event (sometimes they say that this is the number of elementary outcomes that favor the occurrence of an event), and is the number of all elementary outcomes.

In our example:

Combinatorial elements.

When describing many probabilistic experiments, elementary outcomes can be identified with one of the following objects of combinatorics (the science of finite sets).

Permutation from numbers is called an arbitrary ordered record of these numbers without repetitions. For example, for a set of three numbers, there are 6 different permutations:

, , , , , .

For an arbitrary number of permutations is

(product of consecutive numbers of natural numbers, starting from 1).

A combination of software is called an arbitrary unordered collection of any elements of the set. For example, for a set of three numbers, there are 3 different combinations of 3 to 2:

For an arbitrary pair,, the number of combinations from and is equal to

For example,

Hypergeometric distribution.

Consider the following probabilistic experience. There is a black box containing white and black balls. The balls are the same size and are indistinguishable by touch. The experiment consists in taking out balls at random. The event, the probability of which must be found, is that of these balls are white, and the rest are black.

Let's enumerate all balls with numbers from 1 to. Let numbers 1, ¼, correspond to white balls, and numbers, ¼, to black balls. The elementary outcome in this experience is an unordered collection of elements from a set, that is, a combination of by. Therefore, there are all elementary outcomes.

Let's find the number of elementary outcomes that favor the occurrence of the event. The matching sets are made up of “white” and “black” numbers. You can choose numbers from the "white" numbers in ways, and numbers from the "black" - in ways. White and black sets can be combined arbitrarily, so there are only elementary outcomes favorable to the event.

The probability of an event is

The resulting formula is called the hypergeometric distribution.

Task 5.1. The box contains 55 conditional and 6 defective parts of the same type. What is the probability that among the three randomly chosen parts there will be at least one defective one?

Solution. There are 61 details in total, we take 3. An elementary outcome is a combination of 61 to 3. The number of all elementary outcomes is equal. Favorable outcomes are divided into three groups: 1) these are outcomes in which 1 part is defective, and 2 are good; 2) 2 parts are defective, and 1 is good; 3) all 3 parts are defective. The number of sets of the first type is equal, the number of sets of the second type is equal, the number of sets of the third type is. Consequently, elementary outcomes favor the occurrence of an event. The probability of an event is

Algebra of events

The space of elementary events is called the set of all elementary outcomes related to a given experience.

The sum two events is called an event, which consists of elementary outcomes belonging to the event or event.

By product two events is called an event consisting of elementary outcomes belonging simultaneously to events and.

Events and are called inconsistent if.

The event is called opposite event, if the event is favored by all those elementary outcomes that do not belong to the event. In particular, , .

Sum theorem.

In particular, .

Conditional probability events, provided that the event happened, is called the ratio of the number of elementary outcomes belonging to the intersection to the number of elementary outcomes belonging to. In other words, the conditional probability of an event is determined by the classical probability formula, in which the new probability space is. The conditional probability of the event is denoted through.

THEOREM about the product. ...

Events are called independent, if . For independent events, the product theorem gives a relation.

The following two formulas are a consequence of the sum and product theorems.

Formula of total probability. The full group of hypotheses is an arbitrary set of incompatible events,, ¼,, in the sum making up the entire probabilistic space:

In this situation, for an arbitrary event, a formula is valid, called the formula of total probability,

where is the Laplace function,,. The Laplace function is tabulated, and its values ​​for a given one can be found in any textbook on probability theory and mathematical statistics.

Task 5.3. It is known that in a large batch of parts there are 11% defective parts. 100 parts are selected for verification. What is the probability that among them there are no more than 14 defective ones? Estimate the answer using the Moivre-Laplace theorem.

Solution. We are dealing with a Bernoulli test, where,,. Success is defined as finding a defective part, and the number of successes satisfies the inequality. Hence,

Direct calculation gives:

, , , , , , , , , , , , , , .

Hence, . Now we will apply the integral theorem of Moivre-Laplace. We get:

Using the table of values ​​of the function, taking into account the oddness of the function, we obtain

The approximate calculation error does not exceed.

Random variables

A random variable is a numerical characteristic of probabilistic experience, which is a function of elementary outcomes. If,, ¼, there is a set of elementary outcomes, then the random variable is a function. It is more convenient, however, to characterize a random variable by listing all its possible values ​​and the probabilities with which it takes this value.

Such a table is called the law of distribution of a random variable. Since the events form a complete group, the law of probabilistic normalization is fulfilled

The mathematical expectation, or mean value, of a random variable is a number equal to the sum of the products of the values ​​of the random variable by the corresponding probabilities.

The variance (the degree of spread of values ​​around the mathematical expectation) of a random variable is the mathematical expectation of a random variable,

It can be shown that

The magnitude

is called the mean square deviation of the random variable.

The distribution function for a random variable has the probability of getting into the set, that is

It is a non-negative, non-decreasing function taking values ​​from 0 to 1. For a random variable with a finite set of values, it is a piecewise constant function with discontinuities of the second kind at the points of states. In this case, it is continuous on the left and.

Task 5.4. Two dice are thrown in succession. If one, three or five points fall out on one dice, the player loses 5 rubles. When two or four points are dropped, the player receives 7 rubles. When six points are dropped, the player loses 12 rubles. Random value x there is a player's gain on two throwing of the dice. Find the distribution law x, plot the distribution function, find the mathematical expectation and variance x.

Solution. Let us first consider what the player's payoff is in one roll of the dice. Let the event consist of 1, 3 or 5 points. Then, and the winnings will be rubles. Let the event consist of 2 or 4 points. Then, and the winnings will be rubles. Finally, let the event signify a drop of 6 points. Then the prize is equal to rubles.

Now we will consider all possible combinations of events, and with two throws of the dice, and determine the values ​​of the winnings for each such combination.

If an event has occurred, then, at the same time.

If an event has occurred, then, at the same time.

Similarly, for we obtain,.

All found states and the total probabilities of these states are written into the table:

We check the fulfillment of the law of probabilistic normalization: on the real line, you need to be able to determine the probability of a random variable falling into this interval 1) and rapidly decreasing for, ¼,

When a coin is tossed, we can say that it will fall heads up, or probability this is 1/2. Of course, this does not mean that if a coin is tossed 10 times, it will necessarily land up heads 5 times. If the coin is fair and if it is flipped many times, it will land very close half the time. Thus, there are two types of probabilities: experimental and theoretical .

Experimental and theoretical probability

If we flip a coin a large number of times - say 1000 - and count the number of times it comes up heads, we can determine the probability that it will come up heads. If heads come up 503 times, we can calculate the probability of it coming out:
503/1000, or 0.503.

it experimental determination of probability. This definition of probability stems from observation and examination of data and is quite common and very useful. For example, here are some of the probabilities that have been determined experimentally:

1. The probability that a woman will develop breast cancer is 1/11.

2. If you are kissing someone who has a cold, the chance that you will also get a cold is 0.07.

3. A person who has just been released from prison has an 80% chance of going back to prison.

If we consider tossing a coin and taking into account that it is just as likely to come up heads or tails, we can calculate the probability of hitting heads: 1 / 2. This is a theoretical definition of probability. Here are some other probabilities that have been determined theoretically using mathematics:

1. If there are 30 people in a room, the probability that two of them have the same birthday (excluding the year) is 0.706.

2. During the trip, you meet someone, and during the conversation you discover that you have a mutual acquaintance. Typical reaction: "It can't be!" In fact, this phrase does not fit, because the probability of such an event is quite high - just over 22%.

Thus, experimental probabilities are determined by observation and data collection. Theoretical probabilities are determined by mathematical reasoning. Examples of experimental and theoretical probabilities, such as those discussed above, and especially those that we do not expect, lead us to the importance of studying probability. You may ask, "What is true probability?" In fact, there is none. Experimentally, you can determine the probabilities within certain limits. They may or may not coincide with the probabilities that we get theoretically. There are situations in which it is much easier to identify one type of probability than another. For example, it would be enough to find the probability of catching a cold using the theoretical probability.

Calculating Experimental Probabilities

Consider first the experimental definition of probability. The basic principle that we use to calculate such probabilities is as follows.

Principle P (experimental)

If in an experiment in which n observations are made, the situation or event E occurs m times in n observations, then the experimental probability of the event is said to be P (E) = m / n.

Example 1 Sociological survey. An experimental study was carried out to determine the number of left-handers, right-handers and people in whom both arms are equally developed.The results are shown in the graph.

a) Determine the likelihood that the person is right-handed.

b) Determine the likelihood that the person is left-handed.

c) Determine the likelihood that the person is equally fluent in both hands.

d) Most of the tournaments run by the Professional Bowling Association have 120 players. Based on the data from this experiment, how many players can be left-handed?

Solution

a) The number of people who are right-handed is 82, the number of left-handed people is 17, and the number of those who are equally fluent in both hands is 1. The total number of observations is 100. Thus, the probability that a person is right-handed is P
P = 82/100, or 0.82, or 82%.

b) The probability that a person is left-handed is P, where
P = 17/100, or 0.17, or 17%.

c) The probability that a person is equally fluent in both hands is P, where
P = 1/100, or 0.01, or 1%.

d) 120 bowling players, and from (b) we can expect 17% to be left-handed. From here
17% of 120 = 0.17. 120 = 20.4,
that is, we can expect about 20 players to be left-handed.

Example 2 Quality control ... It is very important for a manufacturer to keep the quality of their products at a high level. In fact, companies employ quality control inspectors to ensure this process. The goal is to produce as few defective items as possible. But since the company produces thousands of pieces every day, it cannot afford to check every piece to determine if it is defective or not. To find out what percentage of the products are defective, the company checks far fewer products.
The USDA requires 80% of the seeds that growers sell to germinate. To determine the quality of the seeds that the agricultural company produces, 500 seeds are planted out of those that were produced. After that, it was calculated that 417 seeds germinated.

a) What is the likelihood that the seed will germinate?

b) Do the seeds meet government standards?

Solution a) We know that out of 500 seeds that have been planted, 417 have sprouted. The probability of seed germination is P, and
P = 417/500 = 0.834, or 83.4%.

b) Since the percentage of germinated seeds has exceeded 80% on demand, the seeds meet government standards.

Example 3 TV ratings. According to statistics, there are 105.5 million households with televisions in the United States. Every week, information about the viewing of programs is collected and processed. Within one week, 7,815,000 households tuned in to the hit comedy Everybody Loves Raymond on CBS and 8302,000 households were tuned in to the hit Law & Order series on NBC (Source: Nielsen Media Research). What is the likelihood that one home's television set is tuned to "Everybody Loves Raymond" for a given week? To "Law & Order"?

Solutionn The probability that the television in one household is set to "Everyone loves Raymond" is P, and
P = 7,815,000 / 105,500,000 ≈ 0.074 ≈ 7.4%.
The possibility that the household's television was tuned to Law & Order is P, and
P = 8,302,000 / 105,500,000 ≈ 0.079 ≈ 7.9%.
These percentages are called ratings.

Theoretical probability

Suppose we are conducting an experiment such as tossing a coin or darts, pulling a card out of a deck, or checking items for quality on an assembly line. Every possible outcome of such an experiment is called Exodus ... The set of all possible outcomes is called space of outcomes . Event it is a set of outcomes, that is, a subset of the outcome space.

Example 4 Throwing darts. Suppose that in a dart throwing experiment, the dart hits the target. Find each of the following:

b) Outcome space

Solution
a) The outcomes are: hitting black (C), hitting red (C) and hitting white (B).

b) There is an outcome space (hitting black, hitting red, hitting white), which can be written simply as (H, K, B).

Example 5 Throwing the dice. A dice is a cube with six faces, each of which has one to six dots drawn on it.


Suppose we are rolling the die. Find
a) Outcomes
b) Outcome space

Solution
a) Outcomes: 1, 2, 3, 4, 5, 6.
b) Space of outcomes (1, 2, 3, 4, 5, 6).

We denote the probability that event E occurs as P (E). For example, "a coin will come up tails" could be H. Then P (H) represents the probability that the coin will come up tails. When all the outcomes of an experiment have the same probability of occurring, they are said to be equally likely. To see the difference between events that are equally likely and events that are not equally likely, consider the target below.

For target A, the events of hitting black, red and white are equally probable, since the black, red and white sectors are the same. However, for target B, the zones with these colors are not the same, that is, they are not equally likely to hit them.

Principle P (Theoretical)

If event E can happen m paths out of n possible equiprobable outcomes from the outcome space S, then theoretical probability events, P (E) is
P (E) = m / n.

Example 6 What is the probability of rolling a 3 on the dice?

Solution On the dice there are 6 equally probable outcomes and there is only one possibility of throwing the number 3. Then the probability P is P (3) = 1/6.

Example 7 What is the probability of throwing an even number on the dice?

Solution An event is the throwing of an even digit. This can happen in 3 ways (if the roll is 2, 4, or 6). The number of equally probable outcomes is 6. Then the probability P (even) = 3/6, or 1/2.

We will use a number of examples related to a standard 52-card deck. Such a deck consists of the cards shown in the picture below.

Example 8 What is the probability of drawing an ace from a well-mixed deck of cards?

Solution There are 52 outcomes (the number of cards in the deck), they are equally probable (if the deck is well mixed), and there are 4 ways to draw an Ace, so according to the P principle, the probability
P (pulling an ace) = 4/52, or 1/13.

Example 9 Suppose we are choosing without looking, one ball from a bag with 3 red balls and 4 green balls. What is the probability of choosing a red ball?

Solution There are 7 equally probable outcomes of getting any ball, and since the number of ways to draw a red ball is 3, we get
P (choice of red bead) = 3/7.

The following statements are results from Principle P.

Probability properties

a) If event E cannot happen, then P (E) = 0.
b) If event E occurs without fail then P (E) = 1.
c) The probability that event E will occur is a number between 0 and 1: 0 ≤ P (E) ≤ 1.

For example, in a coin toss, the event that the coin hits the edge has zero probability. The probability that a coin is either heads or tails has a probability of 1.

Example 10 Suppose you are drawing 2 cards from a 52-card deck. What is the likelihood that both of them are peaks?

Solution The number of paths n of drawing 2 cards from a well-mixed 52-card deck is 52 C 2. Since 13 out of 52 cards are spades, the number of ways m of drawing 2 spades is 13 C 2. Then,
P (pulling 2 peaks) = m / n = 13 C 2/52 C 2 = 78/1326 = 1/17.

Example 11 Suppose 3 people are randomly selected from a group of 6 men and 4 women. What is the likelihood that 1 man and 2 women will be selected?

Solution The number of ways to select three people from a group of 10 people 10 C 3. One man can be selected in 6 C 1 ways and 2 women can be selected in 4 C 2 ways. According to the fundamental principle of counting, the number of ways to choose 1 man and 2 women is 6 C 1. 4 C 2. Then, the probability that 1 man and 2 women will be selected is
P = 6 C 1. 4 C 2/10 C 3 = 3/10.

Example 12 Throwing dice. What is the probability of rolling a total of 8 on two dice?

Solution Each dice has 6 possible outcomes. The outcomes are doubled, that is, there are 6.6 or 36 possible ways in which the numbers on two dice can fall. (It is better if the cubes are different, say one red and the other blue - this will help visualize the result.)

Pairs of numbers that add up to 8 are shown in the figure below. There are 5 possible ways to get a total of 8, hence the probability is 5/36.

Mathematics for Programmers: Probability Theory

Ivan Kamyshan

Some programmers, after working in the field of developing conventional commercial applications, are thinking about mastering machine learning and becoming a data analyst. They often don't understand why certain methods work, and most machine learning methods seem like magic. In fact, machine learning is based on mathematical statistics, which in turn is based on probability theory. Therefore, in this article we will pay attention to the basic concepts of probability theory: we will touch on the definitions of probability, distribution and analyze a few simple examples.

Perhaps you know that the theory of probability is conventionally divided into 2 parts. Discrete probability theory studies phenomena that can be described by a distribution with a finite (or countable) number of possible behaviors (throwing dice, coins). Continuous probability theory studies phenomena distributed over some dense set, for example, on a segment or in a circle.

You can consider the subject of probability theory with a simple example. Imagine yourself as a shooter developer. An integral part of the development of games in this genre is the mechanics of shooting. It is clear that a shooter in which all weapons shoot absolutely accurately will be of little interest to players. Therefore, it is imperative to add scatter to the weapon. But simple randomization of the weapon's hit points will not allow fine tuning, therefore, adjusting the game balance will be difficult. At the same time, using random variables and their distributions, you can analyze how the weapon will work with a given spread, and help make the necessary adjustments.

The space of elementary outcomes

Suppose, from some random experiment that we can repeat many times (for example, tossing a coin), we can extract some formalized information (heads or tails). This information is called an elementary outcome, while it is advisable to consider the set of all elementary outcomes, often denoted by the letter Ω (Omega).

The structure of this space depends entirely on the nature of the experiment. For example, if we consider shooting at a sufficiently large circular target, the space of elementary outcomes will be a circle, for convenience placed with the center at zero, and the outcome will be a point in this circle.

In addition, many elementary outcomes are considered - events (for example, getting into the top ten is a concentric circle of a small radius with a target). In the discrete case, everything is quite simple: we can get any event, including or excluding elementary outcomes, in a finite time. In the continuous case, however, everything is much more complicated: we need some fairly good family of sets for consideration, called algebra by analogy with prime real numbers that can be added, subtracted, divided and multiplied. Sets in algebra can be intersected and united, and the result of the operation will be in algebra. This is a very important property for the mathematics behind all these concepts. The minimal family consists of only two sets - the empty set and the space of elementary outcomes.

Measure and probability

Probability is a way to draw conclusions about the behavior of very complex objects without delving into how they work. Thus, probability is defined as a function of an event (from that very good family of sets) that returns a number - some characteristic of how often such an event can occur in reality. For definiteness, mathematicians agreed that this number should lie between zero and one. In addition, requirements are imposed on this function: the probability of an impossible event is zero, the probability of the entire set of outcomes is one, and the probability of combining two independent events (disjoint sets) is equal to the sum of the probabilities. Another name for probability is a probability measure. Most often, the Lebesgue measure is used, generalizing the concepts of length, area, volume to any dimensions (n ​​-dimensional volume), and thus it is applicable for a wide class of sets.

Together, the set of a set of elementary outcomes, a family of sets, and a probability measure is called probabilistic space... Consider how you can build a probability space for an example of shooting at a target.

Consider shooting a large round target of radius R that cannot be missed. We put a set of elementary events a circle centered at the origin of coordinates of radius R. Since we are going to use area (Lebesgue measure for two-dimensional sets) to describe the probability of an event, we will use a family of measurable (for which this measure exists) sets.

Note In fact, this is a technical matter and in simple tasks the process of determining a measure and a family of sets does not play a special role. But it is necessary to understand that these two objects exist, because in many books on the theory of probability, theorems begin with the words: “ Let (Ω, Σ, P) be a probability space ...».

As mentioned above, the probability of the entire space of elementary outcomes should be equal to one. The area (two-dimensional Lebesgue measure, which we will denote λ 2 (A), where A is an event) of a circle, according to a well-known formula from school, is equal to π * R 2. Then we can introduce the probability P (A) = λ 2 (A) / (π * R 2), and this value will already lie between 0 and 1 for any event A.

If we assume that hitting any point on the target is equally probable, the search for the probability of the shooter hitting some area of ​​the target is reduced to finding the area of ​​this set (from this we can conclude that the probability of hitting a specific point is zero, because the area of ​​the point is zero).

For example, we want to know what is the probability that the shooter will get into the top ten (event A - the shooter got into the required set). In our model, "ten" is represented by a circle with center at zero and radius r. Then the probability of getting into this circle is P (A) = λ 2 / (A) π * R 2 = π * r 2 / (π R 2) = (r / R) 2.

This is one of the simplest types of problems for "geometric probability" - most of these problems require finding an area.

Random variables

A random variable is a function that converts elementary outcomes into real numbers. For example, in the considered problem, we can introduce a random value ρ (ω) - the distance from the point of impact to the center of the target. The simplicity of our model allows us to explicitly define the space of elementary outcomes: Ω = (ω = (x, y) numbers such that x 2 + y 2 ≤ R 2). Then the random variable ρ (ω) = ρ (x, y) = x 2 + y 2.

Means of abstraction from probabilistic space. Distribution function and density

It is good when the structure of space is well known, but in reality this is not always the case. Even if the structure of the space is known, it can be complex. To describe random variables, if their expression is unknown, there is the concept of a distribution function, which is denoted by F ξ (x) = P (ξ< x) (нижний индекс ξ здесь означает случайную величину). Т.е. это вероятность множества всех таких элементарных исходов, для которых значение случайной величины ξ на этом событии меньше, чем заданный параметр x .

The distribution function has several properties:

1. First, it is between 0 and 1.
2. Second, it does not decrease when its x argument increases.
3. Third, when -x is very large, the distribution function is close to 0, and when x itself is large, the distribution function is close to 1.

Probably, the meaning of this construction on the first reading is not very clear. One of the useful properties is the distribution function allows you to search for the probability that a value takes a value from an interval. So, P (the random variable ξ takes values ​​from the interval) = F ξ (b) -F ξ (a). Based on this equality, we can investigate how this value changes if the boundaries a and b of the interval are close.

Let d = b-a, then b = a + d. Therefore, F ξ (b) -F ξ (a) = F ξ (a + d) - F ξ (a). For small values ​​of d, the above difference is also small (if the distribution is continuous). It makes sense to consider the ratio p ξ (a, d) = (F ξ (a + d) - F ξ (a)) / d. If for sufficiently small values ​​of d this ratio differs little from some constant p ξ (a), independent of d, then at this point the random variable has a density equal to p ξ (a).

Note Readers who have previously encountered the concept of a derivative may notice that p ξ (a) is the derivative of the function F ξ (x) at the point a. Anyway, you can explore the concept of a derivative in a related article on the Mathprofi website.

Now the meaning of the distribution function can be defined as follows: its derivative (density p ξ, which we defined above) at point a describes how often the random variable will fall into a small interval centered at point a (neighborhood of point a) compared to the neighborhoods of other points ... In other words, the faster the distribution function grows, the more likely such a value will appear in a random experiment.

Let's go back to the example. We can calculate the distribution function for a random variable, ρ (ω) = ρ (x, y) = x 2 + y 2, which denotes the distance from the center to the point of random hitting the target. By definition, F ρ (t) = P (ρ (x, y)< t) . т.е. множество {ρ(x,y) < t)} – состоит из таких точек (x,y) , расстояние от которых до нуля меньше, чем t . Мы уже считали вероятность такого события, когда вычисляли вероятность попадания в «десятку» - она равна t 2 /R 2 . Таким образом, Fρ(t) = P(ρ(x,y) < t) = t 2 /R 2 , для 0

We can find the density p ρ of this random variable. Note right away that it is zero outside the interval, because the distribution function over this interval is unchanged. At the ends of this interval, the density is not determined. It can be found within an interval using a table of derivatives (for example, from the Mathprofi website) and elementary rules of differentiation. The derivative of t 2 / R 2 is equal to 2t / R 2. This means that we found the density on the entire axis of real numbers.

Another useful property of density is the probability that a function takes a value from an interval is calculated using the integral of the density over this interval (you can find out what it is in the articles about your own, improper, indefinite integrals on the Mathprofi website).

At the first reading, the integral over the interval of the function f (x) can be thought of as the area of ​​a curvilinear trapezoid. Its sides are a fragment of the Ox axis, an interval (horizontal coordinate axis), vertical segments connecting points (a, f (a)), (b, f (b)) on a curve with points (a, 0), (b, 0 ) on the Ox axis. The last side is a fragment of the graph of the function f from (a, f (a)) to (b, f (b)). We can talk about the integral over the interval (-∞; b], when, for sufficiently large negative values, and the value of the integral over the interval will change negligibly compared to the change in the number a. The integral over the intervals is determined in a similar way. theorytheory of chancesprobability calculation ... Technical translator's guide

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INTRODUCTION

Many things are incomprehensible to us, not because our concepts are weak;
but because these things are not included in the range of our concepts.
Kozma Prutkov

The main goal of studying mathematics in secondary specialized educational institutions is to provide students with a set of mathematical knowledge and skills necessary to study other program disciplines that use mathematics to some extent, for the ability to perform practical calculations, for the formation and development of logical thinking.

This work consistently introduces all the basic concepts of the section of mathematics "Fundamentals of the theory of probability and mathematical statistics" provided by the program and the State educational standards of secondary vocational education (Ministry of Education of the Russian Federation. M., 2002), formulates the main theorems, most of which are not proven ... The main tasks and methods for their solution and technologies for applying these methods to solving practical problems are considered. The presentation is accompanied by detailed comments and numerous examples.

Methodological instructions can be used for initial acquaintance with the studied material, when taking notes of lectures, for preparing for practical exercises, for consolidating the acquired knowledge, abilities and skills. In addition, the manual will be useful for senior students as a reference tool that allows you to quickly recall what was studied earlier.

At the end of the work, examples and assignments are given that students can perform in self-control mode.

Methodical instructions are intended for students of part-time and full-time forms of education.

BASIC CONCEPTS

Probability theory studies the objective laws of mass random events. It is a theoretical basis for mathematical statistics, engaged in the development of methods for collecting, describing and processing observation results. Through observations (tests, experiments), i.e. experience in the broad sense of the word, cognition of the phenomena of the real world takes place.

In our practice, we often come across phenomena, the outcome of which cannot be predicted, the result of which depends on the case.

A random phenomenon can be characterized by the ratio of the number of its advances to the number of trials, in each of which, under the same conditions of all trials, it could or may not have occurred.

Probability theory is a branch of mathematics in which random phenomena (events) are studied and patterns are revealed during their massive repetition.

Mathematical statistics is a branch of mathematics that has as its subject of study methods of collecting, systematizing, processing and using statistical data to obtain scientifically based conclusions and decision-making.

In this case, statistical data is understood as a set of numbers that represent the quantitative characteristics of the features of the objects of interest to us. Statistical data are obtained as a result of specially set experiments and observations.

Statistical data inherently depends on many random factors, therefore, mathematical statistics is closely related to the theory of probability, which is its theoretical basis.

I. PROBABILITY. ADDITION AND MULTIPLICATION OF PROBABILITIES

1.1. Basic concepts of combinatorics

In the section of mathematics called combinatorics, some problems are solved related to the consideration of sets and the compilation of various combinations of the elements of these sets. For example, if we take 10 different numbers 0, 1, 2, 3,:, 9 and make combinations from them, then we will get different numbers, for example, 143, 431, 5671, 1207, 43, etc.

We see that some of these combinations differ only in the order of the digits (for example, 143 and 431), others in the numbers included in them (for example, 5671 and 1207), and still others differ in the number of digits (for example, 143 and 43).

Thus, the combinations obtained satisfy various conditions.

Three types of combinations can be distinguished depending on the rules of composition: rearrangement, placement, combination.

Let's first get acquainted with the concept factorial.

The product of all natural numbers from 1 to n inclusive is called n-factorial and write.

Calculate: a); b); v) .

Solution. a) .

b) Since and , then you can take out the brackets

Then we get

v) .

Permutations.

A combination of n elements that differ from each other only in the order of the elements are called permutations.

Permutations are indicated by the symbol P n , where n is the number of elements included in each permutation. ( R- the first letter of a French word permutation- permutation).

The number of permutations can be calculated by the formula

or using factorial:

Remember that 0! = 1 and 1! = 1.

Example 2. In how many ways can six different books be arranged on one shelf?

Solution. The required number of ways is equal to the number of permutations of 6 elements, i.e.

Accommodation.

Accommodations from m elements in n in each such compounds are called that differ from each other either by the elements themselves (at least one), or by the order of the arrangement.

Placements are indicated by the symbol, where m- the number of all available elements, n- the number of elements in each combination. ( A- first letter of a French word arrangement, which means "placement, putting in order").

Moreover, it is believed that nm.

The number of placements can be calculated using the formula

,

those. the number of all possible placements from m elements by n equal to product n consecutive integers, of which the greater is m.

Let's write this formula in factorial form:

Example 3. How many options for the distribution of three vouchers in sanatoriums of various profiles can be made for five applicants?

Solution. The required number of variants is equal to the number of placements of 5 elements by 3 elements, i.e.

.

Combinations.

Combinations are all possible combinations of m elements by n that differ from each other by at least one element (here m and n- natural numbers, and n m).

Number of combinations of m elements by n are denoted ( WITH-first letter of a French word combination- combination).

In general, a number from m elements by n is equal to the number of placements from m elements by n divided by the number of permutations from n elements:

Using factorial formulas for the numbers of placements and permutations, we get:

Example 4. In a team of 25 people, you need to allocate four to work on a specific site. How many ways can this be done?

Solution. Since the order of the selected four people does not matter, there are several ways to do this.

We find by the first formula

.

In addition, when solving problems, the following formulas are used that express the main properties of combinations:

(by definition, it is assumed and);

.

1.2. Solving combinatorial problems

Task 1. 16 subjects are studied at the faculty. On Monday, you need to schedule 3 items. How many ways can you do this?

Solution. There are as many ways to schedule three items out of 16 as you can make placements from 16 items of 3 each.

Problem 2. From 15 objects it is necessary to select 10 objects. How many ways can this be done?

Problem 3. Four teams took part in the competition. How many options for the distribution of seats between them are possible?

.

Problem 4. In how many ways can you create a patrol of three soldiers and one officer, if there are 80 soldiers and 3 officers?

Solution. You can choose a soldier on patrol

in ways, and officers in ways. Since any officer can go with each team of soldiers, there are only ways.

Problem 5. Find, if it is known that.

Since, we get

,

,

By the definition of a combination it follows that,. That. ...

1.3. The concept of a random event. Types of events. Event probability

Any action, phenomenon, observation with several different outcomes, realized under a given set of conditions, will be called test.

The result of this action or observation is called event .

If an event under the given conditions may or may not occur, then it is called random ... In the case when an event must certainly happen, it is called reliable , and in the case when it obviously cannot happen, - impossible.

Events are called inconsistent if only one of them may appear at a time.

Events are called joint if under the given conditions the occurrence of one of these events does not exclude the occurrence of another during the same test.

Events are called opposite if, under the conditions of the test, they, being its only outcomes, are incompatible.

Events are usually designated by capital letters of the Latin alphabet: A, B, C, D, : .

The complete system of events А 1, А 2, А 3,:, А n is a set of incompatible events, the onset of at least one of which is obligatory for a given test.

If the complete system consists of two incompatible events, then such events are called opposite and are designated A and.

Example. The box contains 30 numbered balls. Establish which of the following events are impossible, reliable, opposite:

got a numbered ball (A);

got a ball with an even number (V);

got an odd-numbered ball (WITH);

got a ball without a number (D).

Which ones make up a complete group?

Solution ... A- a reliable event; D- an impossible event;

In and WITH- opposite events.

The full group of events consists of A and D, B and WITH.

The probability of an event is considered as a measure of the objective possibility of the occurrence of a random event.

1.4. Classical definition of probability

A number that is an expression of a measure of the objective possibility of an event occurring is called probability this event and is indicated by the symbol P (A).

Definition. Probability of the event A is the ratio of the number of outcomes m, favorable to the onset of a given event A, to the number n all outcomes (inconsistent, unique and equally possible), i.e. ...

Therefore, to find the probability of an event, it is necessary, after considering the various outcomes of the trial, to calculate all possible inconsistent outcomes. n, choose the number of outcomes we are interested in m and calculate the ratio m To n.

The following properties follow from this definition:

The probability of any test is a non-negative number not exceeding one.

Indeed, the number m of the desired events is within the limits. Dividing both parts into n, we get

2. The probability of a reliable event is equal to one, since ...

3. The probability of an impossible event is zero, because.

Problem 1. In the lottery of 1000 tickets, there are 200 winning. Take out one ticket at random. What is the probability that this ticket is a winner?

Solution. The total number of different outcomes is n= 1000. The number of outcomes favorable to getting a win is m = 200. According to the formula, we get

.

Problem 2. There are 4 defective parts in a batch of 18 parts. 5 parts are chosen at random. Find the probability that out of these 5 parts, two will turn out to be defective.

Solution. The number of all equally possible independent outcomes n is equal to the number of combinations from 18 to 5 i.e.

Let's count the number m, favorable for event A. Among 5 parts taken at random, there should be 3 high-quality and 2 defective. The number of ways to select two defective parts from 4 available defective parts is equal to the number of combinations from 4 to 2:

The number of ways to select three high-quality parts from 14 available high-quality parts is

.

Any group of quality parts can be combined with any group of defective parts, therefore the total number of combinations m is

The sought probability of event A is equal to the ratio of the number of outcomes m, favorable to this event, to the number n of all equally possible independent outcomes:

.

The sum of a finite number of events is an event consisting in the occurrence of at least one of them.

The sum of two events is denoted by the symbol A + B, and the sum n events by the symbol А 1 + А 2 +: + А n.

The addition theorem for probabilities.

The probability of the sum of two incompatible events is equal to the sum of the probabilities of these events.

Corollary 1. If the event А 1, А 2,:, А n form a complete system, then the sum of the probabilities of these events is equal to one.

Corollary 2. The sum of the probabilities of opposite events is equal to one.

.

Problem 1. There are 100 lottery tickets. It is known that 5 tickets will receive a prize of 20,000 rubles each, 10 tickets - 15,000 rubles each, 15 tickets - 10,000 rubles each, 25 - 2,000 rubles each. and nothing for the rest. Find the probability that a prize of at least 10,000 rubles will be received on the purchased ticket.

Solution. Let A, B, and C be the events consisting in the fact that a prize falls on the purchased ticket, equal to 20,000, 15,000 and 10,000 rubles, respectively. since events A, B and C are inconsistent, then

Task 2. The correspondence department of the technical school receives tests in mathematics from cities A, B and WITH... Probability of receipt of test work from the city A equals 0.6, from city V- 0.1. Find the probability that the next test will come from the city WITH.

The simplest example of a connection between two events is a causal relationship, when the onset of one of the events necessarily leads to the onset of the other, or vice versa, when the onset of one excludes the possibility of the onset of the other.

To characterize the dependence of some events on others, the concept is introduced conditional probability.

Definition. Let be A and V- two random events of the same challenge. Then the conditional probability of the event A or the probability of event A, provided that event B has occurred, is a number.

Denoting the conditional probability, we obtain the formula

, .

Problem 1. Calculate the probability that a second boy will be born in a family with one child, a boy.

Solution. Let the event A is that there are two boys in the family, and the event V- that one boy.

Consider all possible outcomes: boy and boy; boy and girl; girl and boy; girl and girl.

Then, and by the formula we find

.

Event A called independent from the event V if the occurrence of the event V has no effect on the likelihood of an event occurring A.

Probability multiplication theorem

The probability of the simultaneous occurrence of two independent events is equal to the product of the probabilities of these events:

The probability of occurrence of several events, independent in the aggregate, is calculated by the formula

Problem 2. The first urn contains 6 black and 4 white balls, the second - 5 black and 7 white balls. One ball is removed from each urn. What is the probability that both balls will turn out to be white?

And V there is an event AB... Hence,

b) If the first element works, then an event takes place (opposite to the event A- failure of this element); if the second element works - event V. Let's find the probabilities of events and:

Then the event that both elements will work is and, therefore,