According to the theory of probability. Types of events, direct calculation of the probability of occurrence of an event
The doctrine of the laws to which the so-called. random events. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910 ... Dictionary of foreign words of the Russian language
probability theory- - [L.G. Sumenko. English Russian Dictionary of Information Technologies. M.: GP TsNIIS, 2003.] Topics information technology in general EN probability theorytheory of chancesprobability calculation ... Technical Translator's Handbook
Probability theory- there is a part of mathematics that studies the relationships between the probabilities (see Probability and Statistics) of various events. We list the most important theorems related to this science. The probability of occurrence of one of several incompatible events is equal to ... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
PROBABILITY THEORY- mathematical a science that allows, according to the probabilities of some random events (see), to find the probabilities of random events associated with k. l. way with the first. Modern TV based on the axiomatics (see Axiomatic method) of A. N. Kolmogorov. On the… … Russian sociological encyclopedia
Probability theory- a branch of mathematics in which, according to the given probabilities of some random events, the probabilities of other events are found, related in some way to the first. Probability theory also studies random variables and random processes. One of the main… … Concepts of modern natural science. Glossary of basic terms
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Probability Theory- ... Wikipedia
Probability theory- a mathematical discipline that studies the patterns of random phenomena ... Beginnings of modern natural science
PROBABILITY THEORY- (probability theory) see Probability ... Big explanatory sociological dictionary
Probability theory and its applications- (“Probability Theory and Its Applications”), a scientific journal of the Department of Mathematics of the USSR Academy of Sciences. Publishes original articles and short communications on the theory of probability, general questions of mathematical statistics and their applications in the natural sciences and ... ... Great Soviet Encyclopedia
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Probability theory is a branch of mathematics that studies the patterns of random phenomena: random events, random variables, their properties and operations on them.
For a long time, the theory of probability did not have a clear definition. It was formulated only in 1929. The emergence of probability theory as a science is attributed to the Middle Ages and the first attempts at the mathematical analysis of gambling (toss, dice, roulette). The French mathematicians of the 17th century Blaise Pascal and Pierre de Fermat discovered the first probabilistic patterns that arise when throwing dice while studying the prediction of winnings in gambling.
The theory of probability arose as a science from the belief that certain regularities underlie massive random events. Probability theory studies these patterns.
Probability theory deals with the study of events, the occurrence of which is not known for certain. It allows you to judge the degree of probability of the occurrence of some events compared to others.
For example: it is impossible to unambiguously determine the result of a coin tossing heads or tails, but with repeated tossing, approximately the same number of heads and tails falls out, which means that the probability that heads or tails will fall ", is equal to 50%.
test in this case, the implementation of a certain set of conditions is called, that is, in this case, the tossing of a coin. The challenge can be played an unlimited number of times. In this case, the complex of conditions includes random factors.
The test result is event. The event happens:
- Reliable (always occurs as a result of testing).
- Impossible (never happens).
- Random (may or may not occur as a result of the test).
For example, when tossing a coin, an impossible event - the coin will end up on the edge, a random event - the loss of "heads" or "tails". The specific test result is called elementary event. As a result of the test, only elementary events occur. The totality of all possible, different, specific test outcomes is called elementary event space.
Basic concepts of the theory
Probability- the degree of possibility of the occurrence of the event. When the reasons for some possible event to actually occur outweigh the opposite reasons, then this event is called probable, otherwise - unlikely or improbable.
Random value- this is a value that, as a result of the test, can take one or another value, and it is not known in advance which one. For example: the number of fire stations per day, the number of hits with 10 shots, etc.
Random variables can be divided into two categories.
- Discrete random variable such a quantity is called, which, as a result of the test, can take on certain values with a certain probability, forming a countable set (a set whose elements can be numbered). This set can be either finite or infinite. For example, the number of shots before the first hit on the target is a discrete random variable, because this value can take on an infinite, although countable, number of values.
- Continuous random variable is a quantity that can take any value from some finite or infinite interval. Obviously, the number of possible values of a continuous random variable is infinite.
Probability space- the concept introduced by A.N. Kolmogorov in the 1930s to formalize the concept of probability, which gave rise to the rapid development of probability theory as a rigorous mathematical discipline.
The probability space is a triple (sometimes framed in angle brackets: , where
This is an arbitrary set, the elements of which are called elementary events, outcomes or points;
- sigma-algebra of subsets called (random) events;
- probabilistic measure or probability, i.e. sigma-additive finite measure such that .
De Moivre-Laplace theorem- one of the limiting theorems of probability theory, established by Laplace in 1812. She states that the number of successes in repeating the same random experiment with two possible outcomes is approximately normally distributed. It allows you to find an approximate value of the probability.
If, for each of the independent trials, the probability of the occurrence of some random event is equal to () and is the number of trials in which it actually occurs, then the probability of the validity of the inequality is close (for large ) to the value of the Laplace integral.
Distribution function in probability theory- a function characterizing the distribution of a random variable or a random vector; the probability that a random variable X will take on a value less than or equal to x, where x is arbitrary real number. Under certain conditions, it completely determines a random variable.
Expected value- the average value of a random variable (this is the probability distribution of a random variable, considered in probability theory). In English literature, it is denoted by, in Russian -. In statistics, the notation is often used.
Let a probability space and a random variable defined on it be given. That is, by definition, a measurable function. Then, if there exists a Lebesgue integral of over space , then it is called the mathematical expectation, or mean value, and is denoted by .
Variance of a random variable- a measure of the spread of a given random variable, i.e. its deviation from the mathematical expectation. Designated in Russian literature and in foreign. In statistics, the designation or is often used. The square root of the variance is called the standard deviation, standard deviation, or standard spread.
Let be a random variable defined on some probability space. Then
where the symbol denotes the mathematical expectation.
In probability theory, two random events are called independent if the occurrence of one of them does not change the probability of the occurrence of the other. Similarly, two random variables are called dependent if the value of one of them affects the probability of the values of the other.
The simplest form of the law of large numbers is Bernoulli's theorem, which states that if the probability of an event is the same in all trials, then as the number of trials increases, the frequency of the event tends to the probability of the event and ceases to be random.
The law of large numbers in probability theory states that the arithmetic mean of a finite sample from a fixed distribution is close to the theoretical mean of that distribution. Depending on the type of convergence, a weak law of large numbers is distinguished, when convergence in probability takes place, and a strong law of large numbers, when convergence almost certainly takes place.
The general meaning of the law of large numbers is that the joint action of a large number of identical and independent random factors leads to a result that, in the limit, does not depend on chance.
Methods for estimating probability based on the analysis of a finite sample are based on this property. A good example is the prediction of election results based on a survey of a sample of voters.
Central limit theorems- a class of theorems in probability theory stating that the sum of a sufficiently large number of weakly dependent random variables that have approximately the same scale (none of the terms dominates, does not make a decisive contribution to the sum) has a distribution close to normal.
Since many random variables in applications are formed under the influence of several weakly dependent random factors, their distribution is considered normal. In this case, the condition must be observed that none of the factors is dominant. Central limit theorems in these cases justify the application of the normal distribution.
When a coin is tossed, it can be said that it will land heads up, or probability of this is 1/2. Of course, this does not mean that if a coin is tossed 10 times, it will necessarily land on heads 5 times. If the coin is "fair" and if it is tossed many times, then heads will come up very close half the time. Thus, there are two kinds of probabilities: experimental and theoretical .
Experimental and theoretical probability
If we toss a coin a large number of times - say 1,000 - and count how many 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 up:
503/1000, or 0.503.
it experimental definition of probability. This definition of probability stems from observation and study of data and is quite common and very useful. For example, here are some probabilities that were determined experimentally:
1. The chance of a woman developing breast cancer is 1/11.
2. If you kiss someone who has a cold, then the probability 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 the toss of a coin and taking into account that it is equally likely to come up heads or tails, we can calculate the probability of coming up heads: 1 / 2. This is the theoretical definition of probability. Here are some other probabilities that have been theoretically determined 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 a trip, you meet someone and during the course of the conversation you discover that you have a mutual acquaintance. Typical reaction: "That can't be!" In fact, this phrase does not fit, because the probability of such an event is quite high - just over 22%.
Therefore, the experimental probability is 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?" Actually, there is none. It is experimentally possible to determine the probabilities within certain limits. They may or may not coincide with the probabilities that we obtain theoretically. There are situations in which it is much easier to define one type of probability than another. For example, it would be sufficient to find the probability of catching a cold using theoretical probability.
Calculation of experimental probabilities
Consider first the experimental definition of probability. The basic principle 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 conducted to determine the number of left-handers, right-handers and people in whom both hands are equally developed. The results are shown in the graph.
a) Determine the probability that the person is right-handed.
b) Determine the probability that the person is left-handed.
c) Determine the probability that the person is equally fluent in both hands.
d) Most PBA tournaments have 120 players. Based on 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-handers 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 with both hands is P, where
P = 1/100 or 0.01 or 1%.
d) 120 bowlers 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 hire quality control inspectors to ensure this process. The goal is to release the minimum possible number of defective products. But since the company produces thousands of items every day, it cannot afford to inspect each item to determine if it is defective or not. To find out what percentage of products are defective, the company tests far fewer products.
The USDA requires that 80% of the seeds that growers sell germinate. To determine the quality of the seeds that the agricultural company produces, 500 seeds are planted from those that have been produced. After that, it was calculated that 417 seeds germinated.
a) What is the probability that the seed will germinate?
b) Do the seeds meet government standards?
Solution a) We know that out of 500 seeds that were planted, 417 sprouted. The probability of seed germination P, and
P = 417/500 = 0.834, or 83.4%.
b) Since the percentage of germinated seeds exceeded 80% on demand, the seeds meet the state standards.
Example 3 TV ratings. According to statistics, there are 105,500,000 TV households in the United States. Every week, information about viewing programs is collected and processed. Within one week, 7,815,000 households were tuned in to CBS' hit comedy series Everybody Loves Raymond and 8,302,000 households were tuned in to NBC's hit Law & Order (Source: Nielsen Media Research). What is the probability that one home's TV is tuned to "Everybody Loves Raymond" during a given week? to "Law & Order"?
Solution The probability that the TV in one household is set to "Everybody Loves Raymond" is P, and
P = 7.815.000/105.500.000 ≈ 0.074 ≈ 7.4%.
The possibility that the household TV was set 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 doing an experiment, such as tossing a coin or dart, drawing a card from a deck, or testing items on an assembly line. Each possible outcome of such an experiment is called Exodus . The set of all possible outcomes is called outcome space . Event it is a set of outcomes, that is, a subset of the space of outcomes.
Example 4 Throwing darts. Suppose that in the "throwing darts" experiment, the dart hits the target. Find each of the following:
b) Outcome space
Solution
a) Outcomes are: hitting black (H), hitting red (K) and hitting white (B).
b) There is an outcome space (hit black, hit red, hit white), which can be written simply as (B, R, B).
Example 5 Throwing dice.
A die is a cube with six sides, each of which has one to six dots. 
Suppose we are throwing a die. Find
a) Outcomes
b) Outcome space
Solution
a) Outcomes: 1, 2, 3, 4, 5, 6.
b) Outcome space (1, 2, 3, 4, 5, 6).
We denote the probability that an event E occurs as P(E). For example, "the coin will land on tails" can be denoted by H. Then P(H) is the probability that the coin will land on tails. When all 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 shown below. 
For target A, black, red, and white hit events are equally likely, since black, red, and white sectors are the same. However, for target B, the zones with these colors are not the same, that is, hitting them is not equally likely.
Principle P (Theoretical)
If an event E can happen in m ways out of n possible equiprobable outcomes from the outcome space S, then theoretical probability
event, P(E) is
P(E) = m/n.
Example 6 What is the probability of rolling a 3 by rolling a die?
Solution There are 6 equally likely outcomes on the die and there is only one possibility of throwing the number 3. Then the probability P will be P(3) = 1/6.
Example 7 What is the probability of rolling an even number on the die?
Solution The event is the throwing of an even number. This can happen in 3 ways (if you roll 2, 4 or 6). The number of equiprobable outcomes is 6. Then the probability P(even) = 3/6, or 1/2.
We will be using a number of examples related to a standard 52-card deck. Such a deck consists of the cards shown in the figure below. 
Example 8 What is the probability of drawing an ace from a well-shuffled deck of cards?
Solution There are 52 outcomes (the number of cards in the deck), they are equally likely (if the deck is well mixed), and there are 4 ways to draw an ace, so according to the P principle, the probability
P(drawing an ace) = 4/52, or 1/13.
Example 9 Suppose we choose without looking one marble from a bag of 3 red marbles and 4 green marbles. What is the probability of choosing a red ball?
Solution There are 7 equally likely outcomes to get any ball, and since the number of ways to draw a red ball is 3, we get
P(choosing a red ball) = 3/7.
The following statements are results from the P principle.
Probability Properties
a) If the event E cannot happen, then P(E) = 0.
b) If the event E is bound to happen 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 tossing a coin, the event that the coin lands on its edge has zero probability. The probability that a coin is either heads or tails has a probability of 1.
Example 10 Suppose that 2 cards are drawn from a deck with 52 cards. What is the probability that both of them are spades?
Solution The number of ways n of drawing 2 cards from a well-shuffled 52-card deck is 52 C 2 . Since 13 of the 52 cards are spades, the number m of ways to draw 2 spades is 13 C 2 . Then,
P(stretching 2 peaks) \u003d m / n \u003d 13 C 2 / 52 C 2 \u003d 78/1326 \u003d 1/17.
Example 11 Suppose 3 people are randomly selected from a group of 6 men and 4 women. What is the probability that 1 man and 2 women will be chosen?
Solution Number of ways to choose three people from a group of 10 people 10 C 3 . One man can be chosen in 6 C 1 ways and 2 women can be chosen in 4 C 2 ways. According to the fundamental principle of counting, the number of ways to choose the 1st man and 2 women is 6 C 1 . 4C2. Then, the probability that 1 man and 2 women will be chosen is
P = 6 C 1 . 4 C 2 / 10 C 3 \u003d 3/10.
Example 12 Throwing dice. What is the probability of throwing a total of 8 on two dice?
Solution There are 6 possible outcomes on each dice. The outcomes are doubled, that is, there are 6.6 or 36 possible ways in which the numbers on two dice can fall. (It's better if the cubes are different, say one is red and the other is 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 the sum equal to 8, hence the probability is 5/36.
INTRODUCTION
Many things are incomprehensible to us, not because our concepts are weak;
but because these things do not enter the circle of our concepts.
Kozma Prutkov
The main goal of studying mathematics in secondary specialized educational institutions is to give students a set of mathematical knowledge and skills necessary for studying other program disciplines that use mathematics to one degree or another, for the ability to perform practical calculations, for the formation and development of logical thinking.
In this paper, all the basic concepts of the section of mathematics "Fundamentals of Probability Theory and Mathematical Statistics", provided for by the program and the State Educational Standards of Secondary Vocational Education (Ministry of Education of the Russian Federation. M., 2002), are consistently introduced, the main theorems are formulated, most of which are not proved . 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.
Methodical 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, skills and abilities. In addition, the manual will be useful for undergraduate students as a reference tool that allows you to quickly restore in memory what was previously studied.
At the end of the work, examples and tasks are given that students can perform in self-control mode.
Methodological instructions are intended for students of correspondence and full-time forms of education.
BASIC CONCEPTS
Probability theory studies the objective regularities of mass random events. It is a theoretical basis for mathematical statistics, dealing with the development of methods for collecting, describing and processing the results of observations. Through observations (tests, experiments), i.e. experience in the broad sense of the word, there is a knowledge of the phenomena of the real world.
In our practical activities, we often encounter phenomena, the outcome of which cannot be predicted, the result of which depends on chance.
A random phenomenon can be characterized by the ratio of the number of its occurrences to the number of trials, in each of which, under the same conditions of all trials, it could occur or not occur.
Probability theory is a branch of mathematics in which random phenomena (events) are studied and regularities are revealed during their mass repetition.
Mathematical statistics is a branch of mathematics that has as its subject the study of methods for collecting, systematizing, processing and using statistical data to obtain scientifically sound conclusions and make decisions.
At the same time, statistical data is understood as a set of numbers that represent the quantitative characteristics of the features of the studied objects that are of interest to us. Statistical data are obtained as a result of specially designed experiments and observations.
Statistical data in its essence depend on many random factors, so mathematical statistics is closely related to probability theory, which is its theoretical basis.
I. PROBABILITY. THEOREMS OF ADDITION AND PROBABILITY MULTIPLICATION
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 elements of these sets. For example, if we take 10 different numbers 0, 1, 2, 3,:, 9 and make combinations of them, 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 others also differ in the number of digits (for example, 143 and 43).
Thus, the obtained combinations satisfy various conditions.
Depending on the compilation rules, three types of combinations can be distinguished: permutations, placements, combinations.
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) ; in) .
Solution. a) .
b) as well as 
, then you can take it out of brackets
Then we get
in) 
.
Permutations.
A combination of n elements that differ from each other only in the order of the elements is called a permutation.
Permutations are denoted by the symbol P n , where n is the number of elements in each permutation. ( R- the first letter of the French word permutation- permutation).
The number of permutations can be calculated using the formula
or with factorial:
Let's remember that 0!=1 and 1!=1.
Example 2. In how many ways can six different books be arranged on one shelf?
Solution. The desired number of ways is equal to the number of permutations of 6 elements, i.e.
Accommodations.
Placements 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 from the location.
Locations are denoted by the symbol , where m is the number of all available elements, n is the number of elements in each combination. ( BUT- first letter of the French word arrangement, which means "placement, putting in order").
At the same time, it is assumed that nm.
The number of placements can be calculated using the formula
,
those. the number of all possible placements from m elements by n is equal to the product n consecutive integers, of which the greater is m.
We write this formula in factorial form:
Example 3. How many options for the distribution of three vouchers to a sanatorium of various profiles can be made for five applicants?
Solution. The desired number of options 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, which differ from each other by at least one element (here m and n- natural numbers, and n m).
Number of combinations from m elements by n are denoted ( FROM- the first letter of the French word combination- combination).
In general, the number of m elements by n equal to the number of placements from m elements by n divided by the number of permutations from n elements:
Using factorial formulas for placement and permutation numbers, we get:
Example 4. In a team of 25 people, you need to allocate four to work in a certain area. In how many ways can this be done?
Solution. Since the order of the chosen four people does not matter, this can be done in ways.
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, and are assumed);
.
1.2. Solving combinatorial problems
Task 1. 16 subjects are studied at the faculty. On Monday, you need to put 3 subjects in the schedule. In how many ways can this be done?
Solution. There are as many ways to schedule three items out of 16 as there are placements of 16 elements of 3 each.
Task 2. Out of 15 objects, 10 objects must be selected. In how many ways can this be done?
Task 3. Four teams participated in the competition. How many options for the distribution of seats between them are possible?
.
Problem 4. In how many ways can a patrol of three soldiers and one officer be formed if there are 80 soldiers and 3 officers?
Solution. Soldier on patrol can be selected
ways, and officers ways. Since any officer can go with each team of soldiers, there are only ways.
Task 5. Find if it is known that .
Since , we get
,
,
By definition of combination it follows that , . That. .
1.3. The concept of a random event. Event types. 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 given conditions can occur or not occur, then it is called random . In the event that an event must certainly occur, it is called reliable , and in the case when it certainly cannot happen, - impossible.
The events are called incompatible if only one of them can appear each time.
The events are called joint if, under the given conditions, the occurrence of one of these events does not exclude the occurrence of the other in the same test.
The events are called opposite , if under the test conditions they, being its only outcomes, are incompatible.
Events are usually denoted by capital letters of the Latin alphabet: A, B, C, D, : .
A complete system of events A 1 , A 2 , A 3 , : , A n is a set of incompatible events, the occurrence of at least one of which is mandatory for a given test.
If a complete system consists of two incompatible events, then such events are called opposite and are denoted by A and .
Example. There are 30 numbered balls in a box. Determine which of the following events are impossible, certain, opposite:
got a numbered ball (BUT);
draw an even numbered ball (AT);
drawn a ball with an odd number (FROM);
got a ball without a number (D).
Which of them form a complete group?
Solution . BUT- certain event; D- impossible event;
In and FROM- opposite events.
The complete group of events is BUT and D, V and FROM.
The probability of an event is considered as a measure of the objective possibility of the occurrence of a random event.
1.4. The classical definition of probability
The number, which is an expression of the measure of the objective possibility of the occurrence of an event, is called probability this event and is denoted by the symbol P(A).
Definition. Probability of an event BUT is the ratio of the number of outcomes m that favor the occurrence of a given event BUT, to the number n all outcomes (incompatible, unique and equally possible), i.e. .
Therefore, in order to find the probability of an event, it is necessary, after considering the various outcomes of the test, to calculate all possible incompatible 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 trial is a non-negative number not exceeding one.
Indeed, the number m of the desired events lies within . Dividing both parts into n, we get
2. The probability of a certain event is equal to one, because .
3. The probability of an impossible event is zero because .
Problem 1. There are 200 winners out of 1000 tickets in the lottery. One ticket is drawn at random. What is the probability that this ticket wins?
Solution. The total number of different outcomes is n=1000. The number of outcomes favoring the winning is m=200. According to the formula, we get
.
Task 2. In a batch of 18 parts, there are 4 defective ones. 5 pieces are chosen at random. Find the probability that two out of these 5 parts are defective.
Solution. Number of all equally possible independent outcomes n is equal to the number of combinations from 18 to 5 i.e.
Let's calculate the number m that favor event A. Among the 5 randomly selected parts, there should be 3 high-quality and 2 defective ones. 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 quality parts from 14 available quality parts is equal to
.
Any group of quality parts can be combined with any group of defective parts, so the total number of combinations m is
The desired probability of the event A is equal to the ratio of the number of outcomes m that favor 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 symbol A 1 +A 2 + : +A n .
The theorem of addition of 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 and is equal to one.
.
Problem 1. There are 100 lottery tickets. It is known that 5 tickets get a win of 20,000 rubles, 10 - 15,000 rubles, 15 - 10,000 rubles, 25 - 2,000 rubles. and nothing for the rest. Find the probability that the purchased ticket will win at least 10,000 rubles.
Solution. Let A, B, and C be events consisting in the fact that a prize equal to 20,000, 15,000 and 10,000 rubles falls on the purchased ticket. since the events A, B and C are incompatible, then
Task 2. The correspondence department of the technical school receives tests in mathematics from cities A, B and FROM. The probability of receipt of control work from the city BUT equal to 0.6, from the city AT- 0.1. Find the probability that the next control work will come from the city FROM.
What is a probability?
Faced with this term for the first time, I would not understand what it is. So I'll try to explain in an understandable way.
Probability is the chance that the desired event will occur.
For example, you decided to visit a friend, remember the entrance and even the floor on which he lives. But I forgot the number and location of the apartment. And now you are standing on the stairwell, and in front of you are the doors to choose from.
What is the chance (probability) that if you ring the first doorbell, your friend will open it for you? Whole apartment, and a friend lives only behind one of them. With equal chance, we can choose any door.
But what is this chance?
Doors, the right door. Probability of guessing by ringing the first door: . That is, one time out of three you will guess for sure.
We want to know by calling once, how often will we guess the door? Let's look at all the options:
- you called to 1st Door
- you called to 2nd Door
- you called to 3rd Door
And now consider all the options where a friend can be:
a. Per 1st door
b. Per 2nd door
in. Per 3rd door
Let's compare all the options in the form of a table. A tick indicates the options when your choice matches the location of a friend, a cross - when it does not match.
How do you see everything Maybe options friend's location and your choice of which door to ring.
BUT favorable outcomes of all . That is, you will guess the times from by ringing the door once, i.e. .
This is the probability - the ratio of a favorable outcome (when your choice coincided with the location of a friend) to the number of possible events.
The definition is the formula. Probability is usually denoted p, so:
It is not very convenient to write such a formula, so let's take for - the number of favorable outcomes, and for - the total number of outcomes.
The probability can be written as a percentage, for this you need to multiply the resulting result by:
Probably, the word “outcomes” caught your eye. Since mathematicians call various actions (for us, such an action is a doorbell) experiments, it is customary to call the result of such experiments an outcome.
Well, the outcomes are favorable and unfavorable.
Let's go back to our example. Let's say we rang at one of the doors, but a stranger opened it for us. We didn't guess. What is the probability that if we ring one of the remaining doors, our friend will open it for us?
If you thought that, then this is a mistake. Let's figure it out.
We have two doors left. So we have possible steps:
1) Call to 1st Door
2) Call 2nd Door
A friend, with all this, is definitely behind one of them (after all, he was not behind the one we called):
a) a friend 1st door
b) a friend for 2nd door
Let's draw the table again:
As you can see, there are all options, of which - favorable. That is, the probability is equal.
Why not?
The situation we have considered is example of dependent events. The first event is the first doorbell, the second event is the second doorbell.
And they are called dependent because they affect the following actions. After all, if a friend opened the door after the first ring, what would be the probability that he was behind one of the other two? Correctly, .
But if there are dependent events, then there must be independent? True, there are.
A textbook example is tossing a coin.
- We toss a coin. What is the probability that, for example, heads will come up? That's right - because the options for everything (either heads or tails, we will neglect the probability of a coin to stand on edge), but only suits us.
- But the tails fell out. Okay, let's do it again. What is the probability of coming up heads now? Nothing has changed, everything is the same. How many options? Two. How much are we satisfied with? One.
And let tails fall out at least a thousand times in a row. The probability of falling heads at once will be the same. There are always options, but favorable ones.
Distinguishing dependent events from independent events is easy:
- If the experiment is carried out once (once a coin is tossed, the doorbell rings once, etc.), then the events are always independent.
- If the experiment is carried out several times (a coin is tossed once, the doorbell is rung several times), then the first event is always independent. And then, if the number of favorable or the number of all outcomes changes, then the events are dependent, and if not, they are independent.
Let's practice a little to determine the probability.
Example 1
The coin is tossed twice. What is the probability of getting heads up twice in a row?
Solution:
Consider all possible options:
- eagle eagle
- tails eagle
- tails-eagle
- Tails-tails
As you can see, all options. Of these, we are satisfied only. That is the probability:
If the condition asks simply to find the probability, then the answer must be given as a decimal fraction. If it were indicated that the answer must be given as a percentage, then we would multiply by.
Answer:
Example 2
In a box of chocolates, all candies are packed in the same wrapper. However, from sweets - with nuts, cognac, cherries, caramel and nougat.
What is the probability of taking one candy and getting a candy with nuts. Give your answer in percentage.
Solution:
How many possible outcomes are there? .
That is, taking one candy, it will be one of those in the box.
And how many favorable outcomes?
Because the box contains only chocolates with nuts.
Answer:
Example 3
In a box of balls. of which are white and black.
- What is the probability of drawing a white ball?
- We added more black balls to the box. What is the probability of drawing a white ball now?
Solution:
a) There are only balls in the box. of which are white.
The probability is:
b) Now there are balls in the box. And there are just as many whites left.
Answer:
Full Probability
| The probability of all possible events is (). |
For example, in a box of red and green balls. What is the probability of drawing a red ball? Green ball? Red or green ball?
Probability of drawing a red ball
Green ball:
Red or green ball:
As you can see, the sum of all possible events is equal to (). Understanding this point will help you solve many problems.
Example 4
There are felt-tip pens in the box: green, red, blue, yellow, black.
What is the probability of drawing NOT a red marker?
Solution:
Let's count the number favorable outcomes.
NOT a red marker, that means green, blue, yellow, or black.
Probability of all events. And the probability of events that we consider unfavorable (when we pull out a red felt-tip pen) is .
Thus, the probability of drawing NOT a red felt-tip pen is -.
Answer:
| The probability that an event will not occur is minus the probability that the event will occur. |
Rule for multiplying the probabilities of independent events
You already know what independent events are.
And if you need to find the probability that two (or more) independent events will occur in a row?
Let's say we want to know what is the probability that by tossing a coin once, we will see an eagle twice?
We have already considered - .
What if we toss a coin? What is the probability of seeing an eagle twice in a row?
Total possible options:
- Eagle-eagle-eagle
- Eagle-head-tails
- Head-tails-eagle
- Head-tails-tails
- tails-eagle-eagle
- Tails-heads-tails
- Tails-tails-heads
- Tails-tails-tails
I don't know about you, but I made this list wrong once. Wow! And only option (the first) suits us.
For 5 rolls, you can make a list of possible outcomes yourself. But mathematicians are not as industrious as you.
Therefore, they first noticed, and then proved, that the probability of a certain sequence of independent events decreases each time by the probability of one event.
In other words,
Consider the example of the same, ill-fated, coin.
Probability of coming up heads in a trial? . Now we are tossing a coin.
What is the probability of getting tails in a row?
This rule does not only work if we are asked to find the probability that the same event will occur several times in a row.
If we wanted to find the TAILS-EAGLE-TAILS sequence on consecutive flips, we would do the same.
The probability of getting tails - , heads - .
The probability of getting the sequence TAILS-EAGLE-TAILS-TAILS:
You can check it yourself by making a table.
The rule for adding the probabilities of incompatible events.
So stop! New definition.
Let's figure it out. Let's take our worn out coin and flip it once.
Possible options:
- Eagle-eagle-eagle
- Eagle-head-tails
- Head-tails-eagle
- Head-tails-tails
- tails-eagle-eagle
- Tails-heads-tails
- Tails-tails-heads
- Tails-tails-tails
So here are incompatible events, this is a certain, given sequence of events. are incompatible events.
If we want to determine what is the probability of two (or more) incompatible events, then we add the probabilities of these events.
You need to understand that the loss of an eagle or tails is two independent events.
If we want to determine what is the probability of a sequence) (or any other) falling out, then we use the rule of multiplying probabilities.
What is the probability of getting heads on the first toss and tails on the second and third?
But if we want to know what is the probability of getting one of several sequences, for example, when heads come up exactly once, i.e. options and, then we must add the probabilities of these sequences.
Total options suits us.
We can get the same thing by adding up the probabilities of occurrence of each sequence:
Thus, we add probabilities when we want to determine the probability of some, incompatible, sequences of events.
There is a great rule to help you not get confused when to multiply and when to add:
Let's go back to the example where we tossed a coin times and want to know the probability of seeing heads once.
What is going to happen?
Should drop:
(heads AND tails AND tails) OR (tails AND heads AND tails) OR (tails AND tails AND heads).
And so it turns out:
Let's look at a few examples.
Example 5
There are pencils in the box. red, green, orange and yellow and black. What is the probability of drawing red or green pencils?
Solution:
What is going to happen? We have to pull out (red OR green).
Now it’s clear, we add up the probabilities of these events:
Answer:
Example 6
A die is thrown twice, what is the probability that a total of 8 will come up?
Solution.
How can we get points?
(and) or (and) or (and) or (and) or (and).
The probability of falling out of one (any) face is .
We calculate the probability:
Answer:
Workout.
I think now it has become clear to you when you need to how to count the probabilities, when to add them, and when to multiply them. Is not it? Let's get some exercise.
Tasks:
Let's take a deck of cards in which the cards are spades, hearts, 13 clubs and 13 tambourines. From to Ace of each suit.
- What is the probability of drawing clubs in a row (we put the first card drawn back into the deck and shuffle)?
- What is the probability of drawing a black card (spades or clubs)?
- What is the probability of drawing a picture (jack, queen, king or ace)?
- What is the probability of drawing two pictures in a row (we remove the first card drawn from the deck)?
- What is the probability, taking two cards, to collect a combination - (Jack, Queen or King) and Ace The sequence in which the cards will be drawn does not matter.
Answers:
- In a deck of cards of each value, it means:
- The events are dependent, since after the first card drawn, the number of cards in the deck has decreased (as well as the number of “pictures”). Total jacks, queens, kings and aces in the deck initially, which means the probability of drawing the “picture” with the first card:
Since we are removing the first card from the deck, it means that there is already a card left in the deck, of which there are pictures. Probability of drawing a picture with the second card:
Since we are interested in the situation when we get from the deck: “picture” AND “picture”, then we need to multiply the probabilities:
Answer:
- After the first card is drawn, the number of cards in the deck will decrease. Thus, we have two options:
1) With the first card we take out Ace, the second - jack, queen or king
2) With the first card we take out a jack, queen or king, the second - an ace. (ace and (jack or queen or king)) or ((jack or queen or king) and ace). Don't forget about reducing the number of cards in the deck!
If you were able to solve all the problems yourself, then you are a great fellow! Now tasks on the theory of probability in the exam you will click like nuts!
PROBABILITY THEORY. AVERAGE LEVEL
Consider an example. Let's say we throw a die. What kind of bone is this, do you know? This is the name of a cube with numbers on the faces. How many faces, so many numbers: from to how many? Before.
So we roll a die and want it to come up with an or. And we fall out.
In probability theory they say what happened favorable event(not to be confused with good).
If it fell out, the event would also be auspicious. In total, only two favorable events can occur.
How many bad ones? Since all possible events, then the unfavorable of them are events (this is if it falls out or).
Definition:
Probability is the ratio of the number of favorable events to the number of all possible events.. That is, the probability shows what proportion of all possible events are favorable.
The probability is denoted by a Latin letter (apparently, from English word probability - probability).
It is customary to measure the probability as a percentage (see topics and). To do this, the probability value must be multiplied by. In the dice example, probability.
And in percentage: .
Examples (decide for yourself):
- What is the probability that the toss of a coin will land on heads? And what is the probability of a tails?
- What is the probability that an even number will come up when a dice is thrown? And with what - odd?
- In a drawer of plain, blue and red pencils. We randomly draw one pencil. What is the probability of pulling out a simple one?
Solutions:
- How many options are there? Heads and tails - only two. And how many of them are favorable? Only one is an eagle. So the probability
Same with tails: .
- Total options: (how many sides a cube has, so many different options). Favorable ones: (these are all even numbers :).
Probability. With odd, of course, the same thing. - Total: . Favorable: . Probability: .
Full Probability
All pencils in the drawer are green. What is the probability of drawing a red pencil? There are no chances: probability (after all, favorable events -).
Such an event is called impossible.
What is the probability of drawing a green pencil? There are exactly as many favorable events as there are total events (all events are favorable). So the probability is or.
Such an event is called certain.
If there are green and red pencils in the box, what is the probability of drawing a green or a red one? Yet again. Note the following thing: the probability of drawing green is equal, and red is .
In sum, these probabilities are exactly equal. That is, the sum of the probabilities of all possible events is equal to or.
Example:
In a box of pencils, among them are blue, red, green, simple, yellow, and the rest are orange. What is the probability of not drawing green?
Solution:
Remember that all probabilities add up. And the probability of drawing green is equal. This means that the probability of not drawing green is equal.
Remember this trick: The probability that an event will not occur is minus the probability that the event will occur.
Independent events and the multiplication rule
You flip a coin twice and you want it to come up heads both times. What is the probability of this?
Let's go through all the possible options and determine how many there are:
Eagle-Eagle, Tails-Eagle, Eagle-Tails, Tails-Tails. What else?
The whole variant. Of these, only one suits us: Eagle-Eagle. So, the probability is equal.
Good. Now let's flip a coin. Count yourself. Happened? (answer).
You may have noticed that with the addition of each next throw, the probability decreases by a factor. The general rule is called multiplication rule:
The probabilities of independent events change.
What are independent events? Everything is logical: these are those that do not depend on each other. For example, when we toss a coin several times, each time a new toss is made, the result of which does not depend on all previous tosses. With the same success, we can throw two different coins at the same time.
More examples:
- A die is thrown twice. What is the probability that it will come up both times?
- A coin is tossed times. What is the probability of getting heads first and then tails twice?
- The player rolls two dice. What is the probability that the sum of the numbers on them will be equal?
Answers:
- The events are independent, which means that the multiplication rule works: .
- The probability of an eagle is equal. Tails probability too. We multiply:
- 12 can only be obtained if two -ki fall out: .
Incompatible events and the addition rule
Incompatible events are events that complement each other to full probability. As the name implies, they cannot happen at the same time. For example, if we toss a coin, either heads or tails can fall out.
Example.
In a box of pencils, among them are blue, red, green, simple, yellow, and the rest are orange. What is the probability of drawing green or red?
Solution .
The probability of drawing a green pencil is equal. Red - .
Auspicious events of all: green + red. So the probability of drawing green or red is equal.
The same probability can be represented in the following form: .
This is the addition rule: the probabilities of incompatible events add up.
Mixed tasks
Example.
The coin is tossed twice. What is the probability that the result of the rolls will be different?
Solution .
This means that if heads come up first, tails should be second, and vice versa. It turns out that there are two pairs of independent events here, and these pairs are incompatible with each other. How not to get confused about where to multiply and where to add.
There is a simple rule for such situations. Try to describe what should happen by connecting the events with the unions "AND" or "OR". For example, in this case:
Must roll (heads and tails) or (tails and heads).
Where there is a union "and", there will be multiplication, and where "or" is addition:
Try it yourself:
- What is the probability that when you toss a coin twice, the same side will come up both times?
- A die is thrown twice. What is the probability that the sum will drop points?
Solutions:
- (Heads up and heads up) or (tails up and tails up): .
- What are the options? and. Then:
Rolled (and) or (and) or (and): .
Another example:
We toss a coin once. What is the probability that heads will come up at least once?
Solution:
Oh, how I don’t want to sort through the options ... Head-tails-tails, Eagle-heads-tails, ... But you don’t have to! Let's talk about full probability. Remembered? What is the probability that the eagle will never drop? It's simple: tails fly all the time, that means.
PROBABILITY THEORY. BRIEFLY ABOUT THE MAIN
Probability is the ratio of the number of favorable events to the number of all possible events.
Independent events
Two events are independent if the occurrence of one does not change the probability of the other occurring.
Full Probability
The probability of all possible events is ().
The probability that an event will not occur is minus the probability that the event will occur.
Rule for multiplying the probabilities of independent events
The probability of a certain sequence of independent events is equal to the product of the probabilities of each of the events
Incompatible events
Incompatible events are events that cannot occur simultaneously as a result of an experiment. A number of incompatible events form a complete group of events.
The probabilities of incompatible events add up.
Having described what should happen, using the unions "AND" or "OR", instead of "AND" we put the sign of multiplication, and instead of "OR" - addition.
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