Looking at the topic of "Fate" and some other topics somehow related to the concept of randomness or determinism, I had a desire to briefly explain some of the common mistakes or misunderstandings that many people make about certain things. I will try to make this post as short as possible and not go into too much detail.
First, let's understand that the idea of determinism (the idea of a universe where all events develop according to one scenario and are completely dependent on the past), if you look at it objectively, is no more natural than the idea of indeterminism (the idea of a universe where "fates" does not exist, it is impossible in principle to predict the future regardless of the amount of knowledge about this universe, since an inevitable random factor takes place in the development of “fate”).
The idea of a universe where everything is predetermined took root in people's minds, mainly thanks to Newtonian physics, which was ultra-precise and gave almost perfect results in calculations and their correspondence to reality. Any inaccuracies in the results could be explained by the inaccuracy of the initial measurements, and, in fact, this was actually the case. Thanks to these truly outstanding results of Newtonian physics, the idea of a “mechanical” universe arose, which develops with the precision of a clock and in which there is no place for accidents, there is only a place for circumstances unknown to us.
However, there are a couple of things that currently refute not Newtonian physics itself, but the idea of determinism. The first is probability theory - a mathematical discipline that developed after the advent of Newtonian physics and about which nothing was known at the time when this physics appeared and experienced its golden era. The second is the emergence of quantum physics, a branch of physics that deals with the fundamental laws of our universe and which is very difficult to understand at a conceptual level.
Unfortunately, on the one hand, Newtonian physics was so deeply rooted in the minds of many scientists of the early 20th century that until the end of their days they did not recognize the role of probability in the laws of the universe. The most a shining example such a scientist is Albert Einstein. On the other hand, until now in schools they study mainly Newtonian physics alone, as for quantum physics, in my opinion it is not normally taught in any form at all, so people have an instinctive desire to present it as a “superstructure” or “ model" on top of Newtonian physics.
To begin with, very briefly about quantum physics. This is not a “mathematical model”, not a “model” or a “superstructure” over Newtonian physics. In general, it’s better to throw these words out of your head. Although in fact, yes, quantum physics is really a mathematical model. But we don’t know what exactly this model is. We only know that this is not a model on top of Newtonian physics.
Roughly speaking, the role of probabilities in quantum physics is a fundamental property of quantum objects. This is NOT the result of inaccuracies in measurements or an attempt to bring these inaccuracies into some framework. Inaccuracies in measurements are a separate line in the results that are not related to the laws of physics.
There are people who believe that instead of quantum physics with probabilities, there should be some theory that will get rid of them and allow, say, to predict which atom will decay at a specific moment in time, say, in one gram of uranium. Most of these people are considered freaks and there is even a special Quantum Randi challenge: http://www.science20.com/alpha_meme/official_quantum_randi_challenge-80168 which, by analogy with the usual Randi challenge, should bring them to clean water. The reason why most scientists feel so bad about this idea is because of Bell's theorem, a very complex theorem that states that such a theory cannot exist in principle.
This theorem has been proven mathematically and all experiments so far confirm it.
Having dealt with quantum physics, let's move on to a world more familiar to us. The world around us is governed primarily by Newtonian physics. Almost all people would agree that the results of Newton's experiment can be predicted with 100% accuracy even before it is carried out. Does this mean that our “macroscopic” physical world is deterministic and there is no chance for chance to play a role in it?
To reformulate the question from the other side: is it possible to stage an experiment in the world of Newtonian physics that would demonstrate the laws of probability and the specific result of which would be impossible to predict? The answer to this question is unequivocal - yes. And here is an example of such an experience:
This video demonstrates the operation of a typical "probability machine". All balls are assumed to be the same weight, and all sticks are also the same. Despite this, the path of each individual ball, as well as the exact final result, cannot be predicted. In the end, however, the balls will line up in a normal distribution, as they should according to probability theory.
The specific path of the ball is constantly subject to Newton's laws. I have a feeling that someone will definitely think, “It’s because we don’t know all the factors! If we knew every factor with 100% accuracy, we could accurately predict the path.”
Let's take a closer look at these factors. When we're talking about about such phenomena, every little detail can play a decisive role in where exactly the ball ends up. It's not just the weight of the balls and the microscopic shape of the sticks - after all, the same ball will take a different path each time. A huge number of factors play a role, down to the specific numerical value of gravity in this place at this moment in time and the specific arrangement of atoms in the ball and stick. In turn, each of these factors depends on a huge number of other factors. It can be said, with a certain degree of confidence, that the specific path of the ball depends on the specific state of the universe at that moment. And yet, if we knew everything about this condition, could we predict this path?
Let me express a seditious and shocking thought - what if the specific “decision” about where the ball will fall is “made” at the moment of direct contact of the ball with the stick, and not before? After all, the values of all the decisive factors at this moment also change and the moment of contact does not occur at any specific point in time, such that one can unambiguously divide the time strip into “before and after,” but itself takes a certain time. We should not forget that in Newtonian physics, time and space are not discrete, but extended, they can be endlessly divided into small parts. Quantum physics is discrete, but it is precisely in it that the laws of probability operate.
There is no definitive answer to this question. But I personally believe that this decision is actually made at the moment of contact. In this case, the laws of probability also apply here, and at the “non-quantum” level the universe is also indeterministic.
Ultimately, the very fact of having probability theory leads us to the idea that this is also one of the fundamental laws of the universe, like the indeterminism that follows from it.
Although everyone can give their own answer to this question, fortunately nothing has been proven yet. Everyone can decide for themselves what seems more likely and more natural to them personally.
In the “many-worlds” quantum interpretation (more precisely, there are many of them, these interpretations, which are united under this name), most often the probability is presented very roughly, to the point that throwing an ordinary six-sided die is a random process. Of course, you can learn to throw a die with a certain result, but when it is thrown at random, then under certain conditions, we can assume that the probability of each side being thrown is 1/6. This happens because this is a generally uncontrollable process which, when approached, can be reduced to the same points of contact as in the staged experiment presented above. In real conditions, of course, it is very difficult to find these points or to draw boundaries that establish what information about the process can in principle be obtained and what can be learned from this information.
According to this interpretation, the universe is divided into several universes, in each of which one of the probabilities is realized. The same thing happens with any other probabilistic process (that is, in the experiment above, two universes after each “solution” of the ball’s path). The moment of division occurs not at the moment when the cube shows a certain number, but at the moment when it becomes certain that the cube will show this particular number. This point is quite difficult to highlight.
The "many-worlds" interpretation allows us to solve certain paradoxes that arise when trying to interpret quantum physics, for example, the presence of objects that can simultaneously be in two mutually exclusive states (this is the same “alive and dead at the same time” Schrödinger’s cat, although we are talking about quantum objects). Although from the point of view of, let’s say, everyday experience, this interpretation seems completely fantastic.
In addition to the probabilistic movement of objects, there are a number of other phenomena that are considered indeterministic, in particular, the behavior of people, although these phenomena are described by the theory of probability. However, predicting people's behavior is probably impossible in principle. Although it has now been established that behavior is largely determined by subconscious factors, this does not mean the absence of free will, which can determine a lot. In addition, these subconscious factors themselves can also be determined by some kind of chance, which is sometimes even more difficult to predict than a more or less conscious choice.
Based on all these factors, I personally decided for myself that the universe as a whole is indeterministic. This is where the scientific evidence seems to be leading us. It seems to me that this is much more natural than a “deterministic” universe where everything literally depends on the moment of its origin, but at the same time, in order to predict something, you need to have knowledge about the entire universe. Which in itself means the need to have, in fact, a copy of this universe, but at the same time we know that this copy will not be identical (after all, it must also contain quantum processes). In my opinion this is absurd.
Moreover, our world seems to me to be a typically chaotic system. We are simply accustomed to not noticing all this chaos that is happening around us.
Maybe it's for the better. Living in a free world, the future of which neither we nor “he himself” knows, is still much more interesting.
Does anything surprise you?
It amazes me. The data is stable from year to year.
For 7 years range from 14 to 19 thousand dead.
Think about it, a fire is a random event. But it is possible to predict with great accuracy how many people will die in a fire next year (~ 14-19 thousand).
If you look at the statistics of crime in Russia, some indicators will also vary within a certain range.
|
Crimes registered- total |
1839,5 |
2755,7 |
2952,4 |
2968,3 |
2526,3 |
2756,4 |
2893,8 |
3554,7 |
3855,4 |
3582,5 |
3209,9 |
murder and attempt |
15,6 |
31,7 |
31,8 |
33,6 |
32,3 |
31,6 |
31,6 |
30,8 |
27,5 |
22,2 |
20,1 |
intentional causing |
41,0 |
61,7 |
49,8 |
55,7 |
58,5 |
57,1 |
57,4 |
57,9 |
51,4 |
47,3 |
45,4 |
rape and attempted murder |
15,0 |
12,5 |
|||||||||
robbery |
83,3 |
140,6 |
132,4 |
148,8 |
167,3 |
198,0 |
251,4 |
344,4 |
357,3 |
295,1 |
244,0 |
robbery |
16,5 |
37,7 |
39,4 |
44,8 |
47,1 |
48,7 |
55,4 |
63,7 |
59,8 |
45,3 |
35,4 |
Theft |
913,1 |
1367,9 |
1310,1 |
1273,2 |
926,8 |
1150,8 |
1276,9 |
1573,0 |
1677 |
1567 |
1326,3 |
crimes related to |
16,3 |
79,9 |
243,6 |
241,6 |
189,6 |
181,7 |
150,1 |
175,2 |
212,0 |
231,2 |
232,6 |
traffic violations |
96,3 |
50,0 |
52,7 |
54,5 |
56,8 |
53,6 |
26,5 |
26,6 |
26,3 |
25,6 |
24,3 |
of which entailed |
15,9 |
14,4 |
15,4 |
15,5 |
16,1 |
17,6 |
16,0 |
15,7 |
15,8 |
15,5 |
13,6 |
corrupt practices |
11,1 |
11,6 |
12,5 |
In a stable system, the probability of events occurring is maintained from year to year. That is, from a person’s point of view, a random event happened to him. And from the point of view of the system, it was predetermined.
Reasonable man must strive to think based on the laws of probability (statistics). But in life, few people think about probability. Decisions are made emotionally.
People are afraid to fly by plane. Meanwhile, the most dangerous thing about flying on an airplane is the road to the airport by car. But try to explain to someone that a car is more dangerous than an airplane.
According to research: in the United States, in the first 3 months after the terrorist attacks of September 11, 2001, another thousand people died... indirectly. ABOUT Not in fear, they stopped flying by plane and began moving around the country in cars. And since it is more dangerous, the number of deaths has increased.
They scare people on television: bird and swine flus, terrorism..., but the likelihood of these events is negligible compared to real threats. It is more dangerous to cross a zebra crossing than to fly in an airplane. Falling coconuts kill ~150 people a year. This is ten times more than from a shark bite. But the film "Killer Coconut" has not yet been made.
The world is ruled by probability and we need to remember this.
I recommend books by Nassim Taleb:
Fooled by chance
Black swan
They will help you see the world from a chance perspective..
P.S.
An anecdote on topic.
Mathematics professors ask:
- Will you go to vote in the elections?
- No
- Why, professor?
- According to probability theory, my vote will not affect anything
- But professor, what if everyone turns out to be just as “smart”?
- According to the same probability theory, everyone will not turn out to be smart...
Best wishes,
Vladimir Nikonov, website author:
koob.ru - electronic library
b17.ru - psychologists
- articles and programs for self-development
mindmachine.ru - store of devices for brain training
1.2. Areas of application of probability theory
Methods of probability theory are widely used in various branches of natural science and technology:
in reliability theory,
queuing theory,
theoretical physics,
geodesy,
astronomy,
shooting theory,
theories of observational errors,
theories of automatic control,
general theory of communications and in many other theoretical and applied sciences.
Probability theory also serves to substantiate mathematical and applied statistics, which in turn is used in planning and organizing production, in the analysis of technological processes, preventive and acceptance control of product quality, and for many other purposes.
In recent years, methods of probability theory have increasingly penetrated into various fields of science and technology, contributing to their progress.
1.3. Brief historical background
The first works in which the basic concepts of probability theory arose were attempts to create a theory of gambling (Cardano, Huygens, Pascal, Fermat and others in the 16th-17th centuries).
The next stage in the development of probability theory is associated with the name of Jacob Bernoulli (1654 – 1705). The theorem he proved, which later became known as the “Law of Large Numbers,” was the first theoretical justification of the previously accumulated facts.
Probability theory owes further successes to Moivre, Laplace, Gauss, Poisson and others. The new, most fruitful period is associated with the names of P. L. Chebyshev (1821 - 1894) and his students A. A. Markov (1856 - 1922) and A. M. . Lyapunova (1857 – 1918). During this period, probability theory becomes a harmonious mathematical science. Its subsequent development is due primarily to Russian and Soviet mathematicians (S.N. Bernstein, V.I. Romanovsky, A.N. Kolmogorov, A.Ya. Khinchin, B.V. Gnedenko, N.V. Smirnov, etc. ).
1.4. Tests and events. Types of events
The basic concepts of probability theory are the concept of an elementary event and the concept of the space of elementary events. Above, an event is called random if, under the implementation of a certain set of conditions S it can either happen or not happen. In the future, instead of saying “a set of conditions S carried out”, let’s say briefly: “the test was carried out”. Thus, the event will be considered as the result of the test.
Definition. Random event refers to any fact that may or may not occur as a result of experience.
Moreover, one or another experimental result can be obtained with varying degrees of possibility. That is, in some cases we can say that one event will almost certainly happen, while another will almost never happen.
Definition. Space of elementary outcomesΩ is the set containing all possible outcomes of a given random experiment, of which exactly one occurs in the experiment. The elements of this set are called elementary outcomes and are designated by the letter ω (“omega”).
Then events are called subsets of the set Ω. An event A Ω is said to have occurred as a result of an experiment if one of the elementary outcomes included in the set A occurred in the experiment.
For simplicity, we will assume that the number of elementary events is finite. A subset of the space of elementary events is called a random event. This event may or may not happen as a result of the test (getting three points when throwing a dice, calling the phone at the moment, etc.).
Example 1. The shooter shoots at a target divided into four areas. The shot is a test. Hitting a certain area of the target is an event.
Example 2. The urn contains colored balls. One ball is taken at random from the urn. Retrieving a ball from an urn is a test. The appearance of a ball of a certain color is an event.
In a mathematical model, one can accept the concept of an event as initial, which is not given a definition and is characterized only by its properties. Based on real meaning concept of an event, it is possible to define different types of events.
Definition. A random event is called reliable, if it is certain to happen (rolling from one to six points when throwing a dice), and impossible, if it obviously cannot happen as a result of experience (rolling seven points when throwing a dice). In this case, a reliable event contains all points of the space of elementary events, and an impossible event does not contain a single point of this space.
Definition. Two random events are called incompatible, if they cannot occur simultaneously for the same test outcome. In general, any number of events are called incompatible, if the appearance of one of them excludes the appearance of others.
A classic example of incompatible events is the result of tossing a coin - the loss of the front side of the coin excludes the loss of the reverse side (in the same experiment).
Another example is when a part is randomly pulled out of a parts box. The appearance of a standard part eliminates the appearance of a non-standard part. The events “a standard part appeared” and “a non-standard part appeared” are incompatible.
Definition. Several events form full group, if at least one of them appears as a result of the test.
In other words, the occurrence of at least one of the events of the complete group is a reliable event. In particular, if the events forming full group, are pairwise incompatible, then as a result of the test one and only one of these events will appear. This special case is of greatest interest because it is used further.
Example. Two cash and clothing lottery tickets were purchased. One and only one of the following events will definitely happen: “the winnings fell on the first ticket and did not fall on the second”, “the winnings did not fall on the first ticket and fell on the second”, “the winnings fell on both tickets”, “there were no winnings on both tickets” fell out." These events form a complete group of pairwise incompatible events.
Example. The shooter fired at the target. One of the following two events will definitely happen: hit, miss. These two incompatible events form a complete group.
Example. If one ball is drawn at random from a box containing only red and green balls, then the appearance of a white one among the drawn balls is an impossible event. The appearance of the red and the appearance of the green balls form a complete group of events.
Definition. Events are said to be equally possible if there is reason to believe that none of them is more possible than the other.
Example. The appearance of a “coat of arms” and the appearance of an inscription when throwing a coin are equally possible events. Indeed, it is assumed that the coin is made of a homogeneous material, has a regular cylindrical shape, and the presence of minting does not affect the loss of one side or another of the coin.
Example. The appearance of one or another number of points on a thrown dice are equally possible events. Indeed, it is assumed that the die is made of a homogeneous material, has the shape of a regular polyhedron, and the presence of points does not affect the loss of any face.
In the ball example above, the appearance of red and green balls are equally likely events if there are equal numbers of red and green balls in the box. If there are more red balls in the box than green ones, then the appearance of a green ball is a less probable event than the appearance of a red one.
| " |
The text of the work is posted without images and formulas.
The full version of the work is available in the "Work Files" tab in PDF format
Introduction
Probability theory is a mathematical science that studies mathematical models random phenomena, calculates the probabilities of the occurrence of certain events.
The basics of probability theory are taught in the mathematics curriculum of every school. In addition, tasks in this discipline are a mandatory part of the OGE in grades 9 and 11.
One of the most important areas of application of probability theory is economics. Currently, it is impossible to imagine the study and forecasting of economic phenomena without the use of economic modeling, regression analysis, trend and smoothing models and other methods based on patterns, which are studied in courses in probability theory and mathematical statistics.
Also, probability theory has wide application in such areas as weather forecasting in a specific period. Therefore, there is a desire to practically check whether this science will help for the purposes whose solution is necessary in everyday life.
The purpose of this work is to studying the features of the application of probability theory in life and analyzing data obtained during a practical experiment;
Research objectives:
Study and analyze the necessary literature on the research topic;
Solve a number of problems on the classical determination of probability.
Experimentally test the use of probability in everyday life.
This work consists of two parts: “Chapter 1. Theoretical part”, “Chapter 2. Experimental part”, each of which is divided into separate paragraphs.
Object of study: application of probability theory in life;
Subject of research: basics of probability theory;
Probabilistic ideas today stimulate the development of the entire complex of knowledge, from the sciences of inanimate nature to the sciences of society. Progress modern natural science inseparable from the use and development of probabilistic ideas and methods. Nowadays, it is difficult to name any area of research where probabilistic methods are not used.
Research hypothesis: in-depth study this topic will allow us to be competent in exams of grades 9 and 11;
Practical significance: The material examined during the study enriches life experience methods for solving standard and non-standard tasks according to probability theory.
Chapter 1 Theoretical part 1.1 History of the emergence of probability theory
A French nobleman, a certain Monsieur de Mere, was a dice gambler and passionately wanted to get rich. He spent a lot of time discovering the secret of dice. He invented various options for the game, assuming that in this way he would acquire a large fortune. So, for example, he suggested throwing one die 4 times in turn and convinced his partner that at least once a six would come up. If a six did not come out in 4 throws, then the opponent won.
At that time, the branch of mathematics that today we call probability theory did not yet exist, and therefore, in order to make sure whether his assumptions were correct, Mr. Mere turned to his friend, the famous mathematician and philosopher B. Pascal, with a request that he study two famous questions , the first of which he tried to solve himself. The questions were:
How many times must two dice be thrown so that the number of times two sixes are thrown at once is more than half of the total number of throws?
How to fairly divide the money bet by two players if for some reason they stopped the game prematurely?
Pascal not only became interested in this, but also wrote a letter to the famous mathematician P. Fermat, which provoked him to do it general laws dice game and probability of winning.
Thus, the excitement and thirst to get rich gave impetus to the emergence of a new extremely significant mathematical discipline: probability theory. Mathematicians of such stature as Pascal and Fermat, Huygens (1629–1695), who wrote the treatise “On Calculations in Gambling,” Jacob Bernoulli (1654–1705), Moivre (1667–1754), Laplace ( 1749-1827), Gauss (1777-1855) and Poisson (1781-1840). Nowadays, probability theory is used in almost all branches of knowledge: statistics, forecasters (weather forecast), biology, economics, technology, construction, etc.
1.2 The concept of probability theory
Probability theory is the science of the patterns of random events. In probability theory, a random event is understood as any phenomenon that may or may not occur (randomly) when a certain set of conditions are met. Each such implementation is called a test, experience or experiment.
Events can be divided into reliable, impossible and random.
Reliable An event that is sure to occur during a test is called. Impossible An event that is known to not occur during testing is called. Random is an event that, as a result of an experiment, can either occur or not occur (depending on random circumstances).
The subject of probability theory are the patterns of mass random events, where by mass we mean repeated repetition.
Let's look at a few events:
the appearance of the coat of arms when throwing a coin;
the appearance of three coats of arms when a coin is tossed three times;
hitting the target when fired;
winnings from a cash lottery ticket.
Obviously, each of these events has some degree of possibility. In order to quantitatively compare events with each other according to the degree of possibility, you need to associate a certain number with each event.
Probability of event is a numerical measure of the degree of objective possibility of this event. The probability of a reliable event is taken as the unit of measurement of probability. The probability of an impossible event is zero. The probability of any random event is denoted by P and ranges from zero to one: 0 ≤ P ≤ 1.
The probability of a random event is the ratio of the number n of incompatible equally probable elementary events that make up the event to the number of all possible elementary events N:
The emergence of probability theory as a science dates back to the Middle Ages and the first attempts mathematical analysis gambling (toss, dice). Initially, its basic concepts did not have a strictly mathematical form; they could be treated as some empirical facts, as properties real events, and they were formulated in visual representations.
1.3 Application of probability theory in life
We all use probability theory to one degree or another, based on the analysis of events that have occurred in our lives. We know that death from a car accident is more likely than from a lightning strike because the former, unfortunately, happens so often. One way or another, we pay attention to the likelihood of things in order to predict our behavior. But unfortunately, a person cannot always accurately determine the likelihood of certain events.
For example, without knowing the statistics, most people tend to think that the chance of dying in a plane crash is greater than in a car accident. Now we know, having studied the facts (which, I think, many have heard about), that this is not at all the case. The fact is that our life “eye” sometimes fails, because air transport seems much more frightening to people who are accustomed to walking firmly on the ground. And most people do not use this type of transport very often. Even if we can estimate the probability of an event correctly, it is most likely extremely inaccurate, which will not make any sense, say, in space engineering, where parts per million decide a lot. And when we need accuracy, who do we turn to? Of course, to mathematics.
There are many examples of the real use of probability theory in life. Almost all modern economy is based on it. When releasing a certain product to the market, a competent entrepreneur will certainly take into account the risks, as well as the likelihood of purchase in a particular market, country, etc. Brokers on world markets practically cannot imagine their life without probability theory. Predicting the money exchange rate (which definitely cannot be done without the theory of probability) on money options or the famous Forex market makes it possible to earn serious money from this theory.
The theory of probability is important at the beginning of almost any activity, as well as its regulation. By assessing the chances of a particular problem (for example, spaceship), we know what efforts we need to make, what exactly to check, what to generally expect thousands of kilometers from Earth. Possibility of a terrorist attack in the subway, economic crisis or nuclear war- all this can be expressed as a percentage. And most importantly, take appropriate counteractions based on the data received. Any activity in any sphere can be analyzed using statistics, calculated using probability theory and significantly improved.
Chapter 2 Practical part 2.1 Coin in probability theory.
From the point of view of probability theory, a coin has only two sides, one of which is called “heads”, and the other is called “tails”. A coin is tossed and lands with one side up. No other properties are inherent in the mathematical coin.
Let's conduct an experiment. To begin with, let's take a coin in our hands, throw it and write down the result sequentially. In our case, tossing a coin is a test, and getting heads or tails is an event, that is, a possible outcome of our test (see Appendix 2).
|
Test No. |
Event: heads or tails |
Test No. |
Event: heads or tails |
Test No. |
Event: heads or tails |
After carrying out 100 tests, heads fell out - 55, tails - 45. The probability of heads falling out in this case is 0.55; tails - 0.45. Thus, we have shown that the theory of probability is valid in this case.
2.2 Solving problems in probability theory in OGE
The very first application of probability theory that came to mind was solving problems on this topic included in the upcoming 9th grade mathematics exam. It is most appropriate to consider the key problems in probability theory, which are number 9 in the OGE.
Formulas used to solve problems:
P = , where m is the number of favorable outcomes, n - total number outcomes.
Task No. 1. The coin is tossed twice. What is the probability of getting one “head” and one “tail”?
Solution: When throwing one coin, two outcomes are possible - “heads” or “tails”. When tossing two coins, there are 4 outcomes (2*2=4): “heads” - “tails” “tails” - “tails” “tails” - “heads” “heads” - “heads” One “head” and one “ “tails” will appear in two out of four cases. P(A)=2:4=0.5. Answer: 0,5.
Task No. 2. The coin is tossed three times. What is the probability of getting two heads and one tail?
Solution: When throwing three coins, 8 outcomes are possible (2*2*2=8): “heads” - “tails” - “tails” “tails” - “tails” - “tails” “tails” - “heads” - “tails” "heads" - "heads" - "tails" "tails" - "tails" - "heads" "tails" - "heads" - "heads" "heads" - "tails" - "heads" "heads" - "heads" - “Heads” Two “heads” and one “tails” will appear in three cases out of eight. P(A)=3:8=0.375. Answer: 0,375.
Task No. 3. In a random experiment, a symmetrical coin is tossed four times. Find the probability that you will get no heads at all.
Solution: When throwing four coins, 16 outcomes are possible: (2*2*2*2=16): Favorable outcomes - 1 (four heads will appear). P(A)=1:16=0.0625. Answer: 0,0625.
Task No. 4. Determine the probability that when throwing a die you get more than three points.
Solution: The total possible outcomes are 6. Large numbers 3 - 4, 5, 6. P(A)= 3:6=0.5. Answer: 0,5.
Task No. 5. A die is thrown. Find the probability of getting an even number of points.
Solution: The total possible outcomes are 6. 1, 3, 5 are odd numbers; 2, 4, 6 are even numbers. The probability of getting an even number of points is 3:6=0.5. Answer: 0,5.
Task No. 6. In a random experiment, two dice are rolled. Find the probability that the total will be 8 points. Round the result to hundredths.
Solution: This action - throwing two dice - has a total of 36 possible outcomes, since 6² = 36. Favorable outcomes: 2 6 3 5 4 4 5 3 6 2 The probability of getting eight points is 5:36 ≈ 0.14. Answer: 0,14.
Task No. 7. Throw it twice dice. A total of 6 points were rolled. Find the probability that one of the rolls results in a 5.
Solution: Total outcomes of 6 points - 5: 2 and 4; 4 and 2; 3 and 3; 1 and 5; 5 and 1. Favorable outcomes - 2. P(A)=2:5=0.4. Answer: 0,4.
Task No. 8. There are 50 tickets in the exam, Timofey did not learn 5 of them. Find the probability that he will come across the learned ticket.
Solution: Timofey learned 45 tickets. P(A)=45:50=0.9. Answer: 0,9.
Task No. 9. 20 athletes are participating in the gymnastics championship: 8 from Russia, 7 from the USA, the rest from China. The order of performance is determined by lot. Find the probability that the athlete competing first is from China.
Solution: There are 20 outcomes in total. Favorable outcomes are 20-(8+7)=5. P(A)=5:20=0.25. Answer: 0,25.
Task No. 10. 4 athletes from France, 5 from England and 3 from Italy came to the shot throw competition. The order of performances is determined by drawing lots. Find the probability that the athlete competing fifth is from Italy.
Solution: The number of all possible outcomes is 12 (4 + 5 + 3 = 12). The number of favorable outcomes is 3. P(A)=3:12=0.25. Answer: 0,25 .
2.3 Practical application of probability theory. Determination of air temperature.
We can say for sure that each of us is interested in the weather forecast at least once a day. However, not everyone knows that behind the modest numbers of temperature and wind speed there are complex mathematical calculations. Meteorology in general and predictive meteorology in particular are a kind of ideal area of uncertainty.
Experiment No. 1.
For 20 days we measured the air temperature outside. To calculate the probability that on September 21 the outside air temperature will be above +15 0 C (see Appendix 1).
|
Date and month |
Day of the week |
Air temperature |
|
|
Sunday |
|||
|
Monday |
|||
|
Sunday |
|||
|
Monday |
|||
|
Sunday |
|||
|
Monday |
|||
|
TOTAL: m = 20, n = 9, P = 9 / 20 = 0.45 |
|||
Conclusion: Having carried out the calculations, we conclude that since the probability is less than 0.5, then most likely on September 21 the air temperature outside will be below 15 0. Which is practically confirmed. Air temperature on September 21 +13 0.
Experiment No. 2.
For 15 days we measured the air temperature outside. To calculate the probability that on October 7 the outside air temperature will be below +10 0 C (see Appendix 3).
|
Date and month |
Day of the week |
Air temperature |
|
|
Sunday |
|||
|
Monday |
|||
|
Sunday |
|||
|
Monday |
|||
|
Sunday |
|||
|
TOTAL: m = 15, n = 12, P = 12 / 15 = 0.8 |
|||
Conclusion: Having carried out the calculations, we conclude that since the probability is greater than 0.8, then most likely on October 7 the air temperature outside will be below +10 0. Which is practically confirmed. Air temperature on October 7 +7 0 .
Conclusion
In the course of the work, basic information about the application of probability theory in life was studied. The ability to solve problems in probability theory is necessary for every person, since the ability to predict this or that event allows us to succeed in many areas of our activity.
As a result of the work, it was revealed:
Probability theory is a huge branch of the science of mathematics and the scope of its application is very diverse. Having gone through many facts from life and conducting experiments, using probability theory you can predict events occurring in various spheres of life;
The theory of probability is a whole science, which, it would seem, has no place for mathematics - what kind of laws are there in the kingdom of Chance? But here, too, science has discovered interesting patterns. If you toss a coin, you cannot say for sure which side it will face up - the coat of arms or the number. But after testing, it turns out that when the experiment is repeated many times, the frequency of the event takes values close to 0.5.
The theory of probability has wide application: for weather forecasting, for purchasing serviceable cars, also for purchasing serviceable light bulbs, and various other things. We conducted two experiments to predict the weather at a certain date and time. The theory of probability is indeed used not only for textbooks, but can also find application in everyday life.
Based on the example of this work, more general conclusions can be drawn: stay away from all lotteries, casinos, cards, and gambling in general. You always need to think, assess the degree of risk, choose the best possible option - this will be useful in later life. Thus, the goal set in the work was fulfilled, the assigned tasks were solved and the appropriate conclusions were drawn.
List of used literature
1. Borodin A.L. Elementary course in probability theory and mathematical statistics / A.L. Borodin. - St. Petersburg: Lan, 2004.
2. Klentak L.S. Elements of probability theory and mathematical statistics / L.S. Klentak. - Samara: SSAU Publishing House, 2013.
3. Mordovich A.G. Events. Probabilities. Statistical data processing / A.G. Mordovich, P.V. Semenov. - M.: Mnemosyne, 2004.
4. Open bank assignments for mathematics OGE [Electronic resource] // URL:
http://oge.fipi.ru/os/xmodules/qprint/index.php?theme_guid=5277E3049BBFA50A46567B64CE413F29&proj_guid=DE0E276E497AB3784C3FC4CC20248DC0 (accessed 09/10/2018).
5. Fadeeva L.N. Probability theory and mathematical statistics/ L.N. Fadeeva, A.V. Lebedev; edited by Fadeeva. - 2nd ed. - M.: Eksmo, 2010. - 496 p.
Applications Appendix 1 Appendix 2 Appendix 3
Introduction………………………………………………………..……………………………..… 2Theoretical part
Chapter I. Probability theory - what is it?………………..………………........................... .........…3
History of the emergence and development of probability theory …………………………..…..3
Basic concepts of probability theory…………………………………………….…….3
Theory of probability in life……………………………………………………………....6 Practical part
Chapter II. Unified State Exam as an example of using the theory of life probabilities……….…......... 7
2.1. Single state exam ………………. 7
Experimental part………………………………………………………...……………………….………..9
Questionnaire………………………………………………………………………………..…9
Experiment………………………………………..………………………………………………………9
Conclusion………………………………………………………..………………………………………… 10
Literature………………………………………………………………………………………....………11
Appendix…………………………………………………………..……………… 12
The highest purpose of mathematics...is to
to find hidden order in the chaos that surrounds us.
N. Viner
Introduction
We have heard or said more than once “this is possible”, “this is not possible”, this will definitely happen”, “this is unlikely”. Such expressions are usually used when talking about the possibility of an event occurring, which under the same conditions may or may not occur.
Target my research: identify the probability successful completion exam for 11th grade studentsby guessing the correct answer using probability theory.
To achieve my goals, I set myselftasks :
1) collect, study and systematize material about probability theory,Vusing various sources of information;
2) pconsider the use of probability theory in various spheres of life;
3) pconduct a study to determine the probability of receiving a positive assessment when passing the Unified State Exam by guessing the correct answer.
I nominatedhypothesis: Using probability theory, we can predict with a high degree of confidence the events occurring in our lives.
Object of study – probability theory.
Subject of research: practical application of probability theory.
Research methods : 1) analysis, 2) synthesis, 3) collection of information, 4) work with printed materials, 5) survey, 6) experiment.
I believe that the question explored in my work isrelevantfor a number of reasons:
Chance, chance – we encounter them every day.It seems, how can one “foresee” the occurrence of a random event? After all, it may happen, or it may not come true!But mathematics has found ways to estimate the probability of random events occurring. They allow a person to feel confident when encountering random events.
A serious step in the life of every graduate is the Unified State Exam. I also have to take exams next year. Is its successful completion a matter of chance or not?
Chapter 1. Probability theory.
Story
The roots of probability theory go back centuries. It is known that in ancient states China, India, Egypt, Greece have already used some elements of probabilistic reasoning for the population census, and even to determine the size of the enemy army.
The first works on probability theory, belonging to the French scientists B. Pascal and P. Fermat, the Dutch scientist X. Huygens, appeared in connection with the calculationdifferent probabilities in gambling. Largethe success of probability theory is associated with the nameSwiss mathematician J. Bernoulli(1654-1705). He discovered the famous law of large numbers: he made it possible to establish a connection between the probability of any random event and the frequency of its occurrence, observed directly from experience. WITHthe next period in the history of probability theory (XVIIIV. and the beginningXIXc.) is associated with the names of A. Moivre, P. Laplace, C. Gauss and S. Poisson. During this period, probability theory finds a number of applications in natural science and technology..
The third period in the history of probability theory, ( secondhalfXIXc.) is associated mainly with the names of Russian mathematicians P. L. Chebyshev and A. M. Lyapunov.The currently most common logical scheme for constructing the foundations of probability theory was developed in 1933 by mathematician A. N. Kolmogorov.
Definition and basic formulas
So how useful is this theory in forecasting and how accurate is it? What are its main theses? What useful observations can be drawn from current probability theory?
The basic concept of probability theory isprobability . This word is used quite often in everyday life. I think everyone is familiar with the phrases: “It will probably snow tomorrow,” or “I’ll probably go outdoors this weekend.”In S.I. Ozhegov’s dictionary the word probability is interpreted as “the possibility of something happening.” And here the concept of probability theory is defined as “a branch of mathematics that studies patterns based on the interaction of a large number of random phenomena.”
In the textbook “Algebra and the beginnings of analysis” for grades 10-11, edited by Sh.A. Alimov, the following definition is given: tprobability theory - a branch of mathematics that “engages in the study of patterns in mass phenomena.”
When studying phenomena, we conduct experiments during which various events occur, among which we distinguish: reliable, random, impossible, equally probable.
Event U called reliable Uwill definitely happen. For example, the occurrence of one of six numbers 1,2,3,4,5,6 with one throw of a die will be reliable.The event is called random in relation to some test, if during the course of this test it may or may not occur. For example, when throwing a dice once, the number 1 may or may not appear, i.e. an event is random because it may or may not happen. Event V called impossible in relation to some test, if during this test the eventVwon't happen. For example, it is impossible to get the number 7 when throwing a die.Equally probable events - these are events that, under given conditions, have the same chance of occurring.
How to calculate the probability of a random event? After all, if it’s random, it means it doesn’t obey laws or algorithms. It turns out that in the world of randomness certain laws apply that allow one to calculate probabilities.
Accepted probability of eventA designateletter P(A), then the formula for calculating the probability is written as follows:
P(A)=, wherem ≤ n(1)
Probability P(A) of event A in a test with equally possible elementary outcomes, the ratio of the number of outcomes is calledm, favorable to event A, to the number of outcomesnall test outcomes. From formula (1) it follows that
0≤ P(A)≤ 1.
This definition usually calledclassical definition of probability . It is used when it is theoretically possible to identify all equally possible outcomes of a test and determine outcomes favorable to the test under study. However, in practice there are often tests in which the number of possible outcomes is very large. For example, without repeatedly tossing a button, it is difficult to determine whether it is equally likely to fall “on the plane” or on the “edge.” Therefore, the statistical definition of probability is also used.Statistical probability name the number around which the relative frequency of an event fluctuates (W ( A ) – the ratio of the number of trials M in which this event occurred to the number of all trials performedN) at large number tests.
I also became acquainted with Bernoulli's formula- this is the formula in , allowing one to find the probability of occurrence of event A during independent trials. Named after the outstanding Swiss mathematician , who derived the formula:
P(m)=
To find the chances of event A occurring in a given situation, it is necessary:
find the total number of outcomes of this situation;
find the number of possible outcomes in which event A occurs;
find what proportion of the possible outcomes are from the total number of outcomes.
The theory of probability in life.
In the development of probability theory, problems associated with gambling, primarily with dice, played a very important role.
Dice games
The tools for the game are cubes (dice) in the amount of one to five, depending on the type of game. The essence of the game is to throw out dice and then count points, the number of which determines the winner. The basic principle of dice is that each player takes turns throwing a number of dice (from one to five), after which the result of the roll (the sum of the points rolled; in some versions, the points of each dice are used separately) is used to determine the winner or loser.
Lottery
A lottery is an organized game in which the distribution of gains and losses depends on the random drawing of a particular ticket or number (lot, lot).
Card games
A card game is a game using playing cards, characterized by a random initial state, to determine which a set (deck) is used.
An important principle In almost all card games, the order of the cards in the deck is random.
Slot machines
It is known that in slot machines the speed of rotation of the reels depends on the operation of the microprocessor, which cannot be influenced. But you can calculate the probability of winning on a slot machine, depending on the number of symbols on it, the number of reels and other conditions. However, this knowledge is unlikely to help you win. Nowadays, the science of chance is very important. It is used in selection when breeding valuable plant varieties, when accepting industrial products, when calculating the schedule for unloading cars, etc.
Chapter II. Unified State Exam as an example of using the theory of life probabilities
2.1. Unified State Exam
I am in 10th grade and next year I have to take exams.
Among careless students, a question arose: “Is it possible to choose an answer at random and still get a positive mark for the exam?” I conducted a survey among students: is it possible to practically guess 7 tasks, i.e. pass the Unified State Exam in mathematics without preparation. The results are as follows: 50% of students believe that they can pass the exam using the above method.
I decided to check if they were right? This question can be answered by using elements of probability theory. I want to check this on the example of subjects required to pass exams: mathematics and Russian language and on the example of the most preferred subjects in 11th grade. According to 2016 data, 75% of graduates of the Kruzhilinskaya Secondary School chose social studies.
A) Russian language. For this subject, the test includes 24 tasks, of which 19 are multiple-choice tasks. In order to pass the threshold for the exam in 2016, it is enough to correctly complete 16 tasks. Each task has several answer options, one of which is correct. You can determine the probability of receiving a positive grade on an exam using Bernoulli’s formula:
Bernoulli's scheme describes experiments with a random outcome, which are as follows. N consecutive independent identical experiments are carried out, in each of which the same event A is identified, which may or may not occur during the experiment. Since the tests are identical, then in any of them event A occurs with the same probability. Let us denote it p = P(A). We denote the probability of an additional event by q. Then q = P(Ā) = 1-p
Let event A be the correctly chosen answer out of four proposed in one task of the first part. The probability of event A is defined as the ratio of the number of cases favorable to this event (i.e., a correctly guessed answer, and there are 1 such cases) to the number of all cases (4 such cases). Thenp=P(A)= and q=P(Ā)=1-p=.
119759850
0,00163*100%0,163%
Thus, the probability of a successful outcome is approximately 0.163%!
Using the demo example as an example Unified State Exam test In 2016, I invited 11th grade students to choose answers by guessing. And this is what I got. The average score for the class was 7. Yana Sofina scored the most points - 15, and Danil Zykov scored the least (3 points). 1 student scored 16 points, which is 12.5%. (Appendix I)
Social science
First part of the demo version of the Unified State Exam The 2016 social studies class contains 20 multiple-choice questions, of which only one is correct. Let's determine the probability of receiving a positive assessment. Rosobrnadzor has established a minimum primary score in social studies – 19.
Probability of receiving a positive rating:
15504
0,000003*100%=0,0003%
Thus, the probability of a successful outcome is approximately 0.0003%!
I asked 11th grade students to guess the answers in social studies. The average score was 4.2 points. Most high score-7, the lowest - 1. Thus, not a single student was able to score the required number of points in social studies. (Appendix I)
Mathematics
In 2016, the demo version of the KIM Unified State Exam in MATHEMATICS contains 20 tasks. To successfully pass the exam, it was necessary to solve at least 7 tasks. Let's apply Bernoulli's formula.
(8)=* *; ==9; (8)=9**=0,000102996;
0,0001*100%=0,01%
Conclusion: the probability of receiving a positive rating is 0.01%.
An experiment conducted among my classmates showed that the largest number of matches was 3, GPA was 1.7 points.
Experimental part
Questionnaire
The survey was conducted among students in grades 9-11. They were asked to answer the following question:
1.Is it possible to pass exams without preparation by guessing the answer in the tasks?
The results of the survey are reflected in the diagrams. (Appendix II)
Experiment
1. Among 11th grade students, using the example of a demonstration version of the testing and measuring materials of the Unified State Exam-2016, I conducted an experiment with guessing the answer in the Russian language and social studies. The results are shown in Table 1 (Appendix I).
2. She invited her classmates to guess the answer in demo version in mathematics for 2016, the results are also presented in Appendix I.
As a result of the experiment and using Bernoulli's formula, I proved that it is impossible to pass exams by guessing the answer. Only systematic, thoughtful and conscientious study at school will allow the graduate to be well prepared to participate in the Unified State Exam, and to successfully solve the fateful problem when moving to a higher level of study at a university.
Conclusion
As a result of the work I did, I achieved the implementation of the tasks I set for myself:
Firstly , I realized that probability theory is a huge branch of the science of mathematics and it is impossible to study it in one go;
secondly , Having sorted through many facts from life and conducted experiments, I realized that with the help of probability theory it is really possible to predict events occurring in various spheres of life;
thirdly , having examined the probability of students successfully passing 11 Unified State Examination class in mathematics, I'm atcame to a conclusion, what tOnly systematic, thoughtful and conscientious study at school will allow the graduate to be well prepared to participate in the Unified State Exam. Thus, the hypothesis I put forward was confirmed; with the help of probability theory, I proved that you need to prepare for exams, and not just rely on chance.
Using the example of my work, more general conclusions can be drawn: stay away from all lotteries, casinos, cards, and gambling in general. You always need to think, assess the degree of risk, choose the best possible option - this, I think, will be useful to me in later life.
Literature
Alimov Sh.A. Algebra and the beginnings of mathematical analysis. Grades 10-11: textbook for general education institutions: basic level. M.: Education, 2010.
Brodsky Ya.S. "Statistics. Probability. Combinatorics" -M.: Onyx; Peace and Education,2008
Bunimovich E.A., Suvorova S.B. Guidelines for the topic “Statistical Research” // Mathematics at school. - 2003. - No. 3.
Gusev V.A. Extracurricular work in mathematics in grades 6-8. - M.: Education, 1984.
Lyutikas V.S. Optional course in mathematics: Theory of probability.-M.: Education 1990.
Makarychev Yu.N. Algebra: elements of statistics and probability theory: textbook. manual for students 7-9 grades. general education institutions - M.: Education, 2007.
Ozhegov S.I. Dictionary of the Russian language: M.: Russian language, 1989.
Fedoseev V.N. Elements of probability theory for grades VII-IX of secondary school. // Mathematics at school. - 2002. - No. 4,5.
What's happened. Who is this: In 3 vols. T.1 – 4th ed. revised and additional - M.: Pedagogy-Press, 1997.
Resources:
