The role of probability theory in everyday life. Probability theory and math
Gataullina Lilia
In my research work, I will try to check whether the theory of probability really works and how it can be applied in life.
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X Republican Scientific and Practical Conference
"Christmas Readings"
Section: mathematics
Research
Coincidence or pattern?
or
The theory of probability in life
Gataullina Lilia,
School No. 66, 8 B grade
Moskovsky district, Kazan city
Scientific supervisor: mathematics teacher 1st quarter. cat Magsumova E.N.
Kazan 2011
Introduction........................................................ ........................................................ ............3
Chapter 1. Probability theory - what is it?………………...................................5
Chapter 2. Experiments………………………………………………………7
Chapter 3. Is it possible to win the lottery or roulette? …………………........9
Conclusion................................................. ........................................................ ......eleven
Bibliography................................................ .............................................12
Application
Introduction
People have always been interested in the future. Humanity has always been looking for a way to predict or plan it. At different times in different ways. In the modern world there is a theory that science recognizes and uses to plan and predict the future. We're talking about probability theory.
In life we often encounter random phenomena. What is the reason for their randomness - our ignorance of the true reasons for what is happening or is randomness the basis of many phenomena? Disputes on this topic do not subside in various fields of science. Do mutations occur randomly, how much does historical development depend on an individual, can the Universe be considered a random deviation from the laws of conservation? Poincaré, calling for a distinction between the contingency associated with instability and the contingency associated with our ignorance, asked the following question: “Why do people find it completely natural to pray for rain, while they would consider it ridiculous to ask in prayer for an eclipse?”
Every “random” event has a clear probability of its occurrence. For example, look at the official statistics on fires in Russia. (see Appendix No. 1) Does anything surprise you? The data is stable from year to year.
Over 7 years, the range is 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).
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.
A reasonable person should 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. The probability that a passenger boarding an airplane will die inplane crashis approximately
1/8,000,000. If a passenger boards a random flight every day, it will take him 21,000 years to die. (See Appendix No. 2)
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. Out of 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 are scared on television: bird and swine flu, terrorism..., but the likelihood of these events is negligible compared to real threats. It is more dangerous to cross the road at a zebra crossing than to fly on 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.It is estimated that the chance of a person being attacked by a shark is 1 in 11.5 million, and the chance of dying from such an attack is 1 in 264.1 million. The average annual number of drownings in the United States is 3,306 people, and deaths from sharks are 1. Probability rules the world and it is necessary remember this. They will help you see the world from a chance perspective. (see Appendix No. 3)
In my research work, I will try to check whether the theory of probability really works and how it can be applied in life.
The probability of an event in life is not often calculated using formulas, but rather intuitively. But checking whether the “empirical analysis” coincides with the mathematical one is sometimes very useful.
Chapter 1. Probability theory - what is it?
Probability theory or probability theory is one of the branches of Higher Mathematics. This is the most interestingScience section Higher MathematicsProbability theory, which is a complex discipline, has applications in real life. Probability theory is of undoubted value for general education. This science allows not only to obtain knowledge that helps to understand the patterns of the world around us, but also to find practical application of the theory of probability in everyday life. So, each of us every day has to make many decisions under conditions of uncertainty. However, this uncertainty can be “transformed” into some certainty. And then this knowledge can provide significant assistance in making a decision. Learning probability theory requires a lot of effort and patience.
Now let's move on to the theory itself and the history of its origin. The main concept of probability theory is probability. This word is “probability”, a synonym for which is, for example, the word “chance”, which is often used in everyday life. I think everyone is familiar with the phrases: “Tomorrow it will probably snow,” or “I’ll probably go outdoors this weekend,” or “this is simply incredible,” or “there’s a chance to get an automatic test.” These kinds of phrases intuitively assess the likelihood that some random event will occur. In turn, mathatic probability gives some numerical estimate of the probability that some random event will occur.
Probability theory took shape as an independent science relatively recently, although the history of probability theory began in antiquity. Thus, Lucretius, Democritus, Carus and some other scientists of ancient Greece in their reasoning spoke about equally probable outcomes of such an event, such as the possibility that all matter consists of molecules. Thus, the concept of probability was used on an intuitive level, but it was not separated into a new category. However, ancient scientists laid an excellent foundation for the emergence of this scientific concept. In the Middle Ages, one might say, the theory of probability was born, when the first attempts at mathematical analysis and such gambling games as dice, toss, and roulette were made.
The first scientific works on probability theory appeared in the 17th century. When scientists such as Blaise Pascal and Pierre Fermat discovered certain patterns that occur when throwing dice. At the same time, another scientist, Christian Huygens, showed interest in this issue. In 1657, in his work, he introduced the following concepts of probability theory: the concept of probability as the value of chance or possibility; mathematical expectation for discrete cases, in the form of the price of chance, as well as theorems of addition and multiplication of probabilities, which, however, were not formulated explicitly. At the same time, probability theory began to find areas of application - demography, insurance, and assessment of observation errors.
Further development of probability theory led to the need to axiomatize probability theory and the main concept - probability. Thus, the formation of the axiomatics of probability theory occurred in the 30s of the 20th century. The most significant contribution to laying the foundations of the theory was made by A.N. Kosmogorov.
Today, probability theory is an independent science with a huge scope of application. In this section of the site you will find cheat sheets on probability theory, lectures and problems on probability theory, literature, as well as many interesting articles on the application of probability theory in life.
Chapter 2. Experiments
I decided to check the classic definition of probability.
Definition: Let the set of outcomes of an experiment consist of n equally probable outcomes. If m of them favor event A, then the probability of event A is called the number P(A) = m/n.
Take the coin game for example. When tossing, there can be two equally probable outcomes: the coin can land up head or tail. When you toss a coin once, you cannot predict which side will end up on top. However, after tossing a coin 100 times, you can draw conclusions. You can say in advance that the coat of arms will appear not 1 or 2 times, but more, but not 99 or 98 times, but less. The number of drops of the coat of arms will be close to 50. In fact, and from experience one can be convinced of this that this number will be between 40 and 60. Who and when first performed the experiment with the coin is unknown.
The French naturalist Buffon (1707-1788) in the eighteenth century tossed a coin 4040 times; the coat of arms landed 2048 times. The mathematician K. Pearson at the beginning of this century tossed it 24,000 times - the coat of arms fell out 12,012 times. About 20 years ago, American experimenters repeated the experiment. In 10,000 tosses, the coat of arms came up 4,979 times. This means that the results of coin tosses, although each of them is a random event, are subject to an objective law when repeated several times.
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 the form of a line: O, P, P, O, O, R. Here the letters O and P indicate heads or tails. 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. The results of the experiment are presented in Appendix No. 4. After 100 tests, heads fell - 55, tails - 45. The probability of heads falling in this case is 0.55; tails – 0.45. Thus, I have shown that the theory of probability has its place in this case.
Consider a problem with three doors and prizes behind it: “Car or goats”? or "Monty Hall Paradox". The conditions of the problem are:
You are in the game. The presenter offers to choose one of three doors and tells that behind one of the doors there is a prize - a car, and goats are hidden behind the other two doors. After you have chosen one of the doors, the presenter, who knows what is behind each door, opens one of the remaining two doors and demonstrates that there is a goat behind it (a goat, the sex of the animal in this case is not so important) And then the presenter slyly asks: “Do you want to change your choice of door?” Will changing your selection increase your chances of winning?
If you think about it: here are two closed doors, you have already chosen one, and the probability that there is a car/goat behind the chosen door is 50%, just like with a coin toss. But this is not true at all. If you change your mind and choose another door, your chances of winning will increase by 2 times! Experience has confirmed this statement (see Appendix No. 5). Those. By leaving his choice, the player will receive a car in one of three cases, and by changing two out of three. Statistics from the TV show confirm that those who changed their choices were twice as likely to win.
This is all probability theory and it is true over “many options.” I hope that this example will make you think about how to quickly pick up a book about probability theory, and also start applying it in your work. Believe me, it is interesting and exciting, and there is a practical sense.
Chapter 3. Is it possible to win the lottery or roulette?
Each of us has bought a lottery or gambled at least once in our lives, but not everyone used a pre-planned strategy. Smart players have long stopped hoping for luck and turned on rational thinking.
The fact is that each event has a certain mathematical expectation, as higher mathematics and probability theory say, and if you correctly assess the situation, you can bypass the unsatisfactory outcome of the event.
For example, in any game, such as roulette, it is possible to play with a 50% chance of winning by betting on an even number, or a red cell. This is exactly the game we will consider.
To ensure profit, we will draw up a simple game strategy. For example, we have the opportunity to calculate the probability with which an even number will appear 10 times in a row - 0.5 * 0.5 and so on 10 times. Multiply by 100% and we get only 0.097%, or approximately 1 chance in 1,000.
You probably won’t be able to play so many games in your entire life, which means that the probability of getting 10 even numbers in a row is practically equal to “0”. Let's use this game tactic in practice.
But that's not all, even 1 time in 1,000 is a lot for us, so let's reduce this number to 1 in 10,000. You ask, how can this be done without increasing the expected number of even numbers in a row? The answer is simple - time.
We approach the roulette wheel and wait until an even number appears 2 times in a row. This will be each time out of four calculated cases. Now we place the minimum bet on an even number, for example 5p, and win 5p for each occurrence of an even number, the probability of which is 50%.
If the result is odd, then we increase the next bet by 2 times, that is, we already bet 10 rubles. In this case, the probability of losing will be 6%. But don't panic if you lose even this time! Make the increase twice as much each time. Each time the mathematical expectation of winning increases, and in any case you will remain in profit.
It is important to take into account the fact that this strategy is only suitable for small bets, since if you initially bet a lot of money, you risk losing everything due to bet restrictions in the future. If you have any doubts about this tactic, play a game of guessing the side of a coin with fictitious money with a friend, betting twice as much if you lose.
After a while you will see that this technique is simple to practice and very effective! We can conclude that by playing with this strategy, you will not earn millions, but will only win for small expenses.
Conclusion
While studying the topic of “probability theory in life,” I realized that this is a huge section of the science of mathematics. And it is impossible to study it in one go.
After going through many facts from life and conducting experiments at home, I realized that the theory of probability really has a place in life. The probability of an event in life is not often calculated using formulas, but rather intuitively. But checking whether the “empirical analysis” coincides with the mathematical one is sometimes very useful.
Can we predict with the help of this theory what will happen to us in a day, two, a thousand? Of course not. There are a lot of events related to us at any given time. A lifetime alone would not be enough to typify these events. And combining them is a completely disastrous business. With the help of this theory, only events of the same type can be predicted. For example, something like tossing a coin is an event of 2 probabilistic outcomes. In general, the applied application of probability theory is associated with a considerable number of conditions and restrictions. For complex processes it involves calculations that only a computer can do.
But we should remember that in life there is also such a thing as luck, luck. This is what we say - lucky, when, for example, some person never studied, did not strive for anything, lay on the couch, played on the computer, and after 5 years we see him being interviewed on MTV. He had a 0.001 chance of becoming a musician, it happened, he was lucky, such a convergence of circumstances. What we call is being in the right place and at the right time, when those same 0.001 are triggered.
Thus, we work on ourselves, make decisions that can increase the likelihood of fulfilling our desires and aspirations, each case can add those cherished 0.00001 that will play a decisive role in the end.
Bibliography
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,
theory of observation 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 substantiation 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 the real meaning of the concept of event, different types of events can be defined.
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 that form a complete group are pairwise inconsistent, then one and only one of these events will appear as a result of the trial. This particular case is of greatest interest, as it will be 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.
X Republican Scientific and Practical Conference
"Christmas Readings"
Section: mathematics
Research
Coincidence or pattern?
The theory of probability in life
Gataullina Lilia,
School No. 66, 8 B grade
Moskovsky district, Kazan city
Scientific supervisor: mathematics teacher 1st quarter. cat Magsumova E.N.
Kazan 2011
Introduction…………………………………………………………………………………………………………3
Chapter 1. Probability theory – what is it?………………………………………………………….5
Chapter 2. Experiments………………………………………………………7
Chapter 3. Is it possible to win the lottery or roulette? ………………………..9
Conclusion……………………………………………………………………………………………………………11
References…………………………………………………………………………………12
Application
Introduction
People have always been interested in the future. Humanity has always been looking for a way to predict or plan it. At different times in different ways. In the modern world there is a theory that science recognizes and uses to plan and predict the future. We're talking about probability theory.
In life we often encounter random phenomena. What is the reason for their randomness - our ignorance of the true reasons for what is happening or is randomness the basis of many phenomena? Disputes on this topic do not subside in various fields of science. Do mutations occur randomly, how much does historical development depend on an individual, can the Universe be considered a random deviation from the laws of conservation? Poincaré, calling for a distinction between the contingency associated with instability and the contingency associated with our ignorance, asked the following question: “Why do people find it completely natural to pray for rain, while they would consider it ridiculous to ask in prayer for an eclipse?”
Every ‘random’ event has a clear probability of its occurrence. For example, look at the official statistics on fires in Russia. (see Appendix No. 1) Does anything surprise you? The data is stable from year to year. Over 7 years, the range is 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).
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.
A reasonable person should 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. The probability that a passenger boarding airplane death toll in a plane crash is approximately
1/8,000,000. If a passenger boards a random flight every day, it will take him 21,000 years to die. (See Appendix No. 2)
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. Out of 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 are scared on television: bird and swine flu, terrorism..., but the likelihood of these events is negligible compared to real threats. It is more dangerous to cross the road at a zebra crossing than to fly on 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. It is estimated that the chance of a person being attacked by a shark is 1 in 11.5 million, and the chance of dying from such an attack is 1 in 264.1 million. The average annual number of drownings in the United States is 3,306 people, and deaths from sharks are 1. Probability rules the world and it is necessary remember this. They will help you see the world from a chance perspective. (see Appendix No. 3)
In my research work, I will try to check whether the theory of probability really works and how it can be applied in life.
The probability of an event in life is not often calculated using formulas, but rather intuitively. But checking whether the “empirical analysis” coincides with the mathematical one is sometimes very useful.
GlAva1 . Probability theory - what is it?
Probability theory or probability theory is one of the branches of Higher Mathematics. This is the most interesting Science section Higher Mathematics Probability theory, which is a complex discipline, has applications in real life. Probability theory is of undoubted value for general education. This science allows not only to obtain knowledge that helps to understand the patterns of the world around us, but also to find practical application of the theory of probability in everyday life. So, each of us every day has to make many decisions under conditions of uncertainty. However, this uncertainty can be “transformed” into some certainty. And then this knowledge can provide significant assistance in making a decision. Learning probability theory requires a lot of effort and patience.
Now let's move on to the theory itself and the history of its origin. The main concept of probability theory is probability. This word is “probability”, a synonym for which is, for example, the word “chance”, which is often used in everyday life. I think everyone is familiar with the phrases: “Tomorrow it will probably snow,” or “I’ll probably go outdoors this weekend,” or “this is simply incredible,” or “there’s a chance to get an automatic test.” These kinds of phrases intuitively assess the likelihood that some random event will occur. In turn, mathematical probability gives some numerical estimate of the probability that some random event will occur.
Probability theory took shape as an independent science relatively recently, although the history of probability theory began in antiquity. Thus, Lucretius, Democritus, Carus and some other scientists of ancient Greece in their reasoning spoke about equally probable outcomes of such an event, such as the possibility that all matter consists of molecules. Thus, the concept of probability was used on an intuitive level, but it was not separated into a new category. However, ancient scientists laid an excellent foundation for the emergence of this scientific concept. In the Middle Ages, one might say, the theory of probability was born, when the first attempts at mathematical analysis and such gambling games as dice, toss, and roulette were made.
The first scientific works on probability theory appeared in the 17th century. When scientists such as Blaise Pascal and Pierre Fermat discovered certain patterns that occur when throwing dice. At the same time, another scientist, Christian Huygens, showed interest in this issue. In 1657, in his work, he introduced the following concepts of probability theory: the concept of probability as the value of chance or possibility; mathematical expectation for discrete cases, in the form of the price of chance, as well as theorems of addition and multiplication of probabilities, which, however, were not formulated explicitly. At the same time, probability theory began to find areas of application - demography, insurance, and assessment of observation errors.
Further development of probability theory led to the need to axiomatize probability theory and the main concept - probability. Thus, the formation of the axiomatics of probability theory occurred in the 30s of the 20th century. The most significant contribution to laying the foundations of the theory was made by A.N. Kosmogorov.
Today, probability theory is an independent science with a huge scope of application. In this section of the site you will find cheat sheets on probability theory, lectures and problems on probability theory, literature, as well as many interesting articles on the application of probability theory in life.
Chapter 2 . Experiments
I decided to check the classic definition of probability.
Definition: Let the set of outcomes of an experiment consist of n equally probable outcomes. If m of them favor event A, then the probability of event A is called the number P(A) = m/n.
Take the coin game for example. When tossing, there can be two equally probable outcomes: the coin can land up head or tail. When you toss a coin once, you cannot predict which side will end up on top. However, after tossing a coin 100 times, you can draw conclusions. You can say in advance that the coat of arms will appear not 1 or 2 times, but more, but not 99 or 98 times, but less. The number of drops of the coat of arms will be close to 50. In fact, and from experience one can be convinced of this that this number will be between 40 and 60. Who and when first performed the experiment with the coin is unknown.
The French naturalist Buffon (1707-1788) in the eighteenth century tossed a coin 4040 times; the coat of arms landed 2048 times. The mathematician K. Pearson at the beginning of this century tossed it 24,000 times - the coat of arms fell out 12,012 times. About 20 years ago, American experimenters repeated the experiment. In 10,000 tosses, the coat of arms came up 4,979 times. This means that the results of coin tosses, although each of them is a random event, are subject to an objective law when repeated several times.
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 the form of a line: O, P, P, O, O, R. Here the letters O and P indicate heads or tails. 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. The results of the experiment are presented in Appendix No. 4. After 100 tests, heads fell out - 55, tails - 45. The probability of heads falling out in this case is 0.55; tails – 0.45. Thus, I have shown that the theory of probability has its place in this case.
Consider a problem with three doors and prizes behind it: “Car or goats”? or "Monty Hall Paradox". The conditions of the problem are:
You are in the game. The presenter offers to choose one of three doors and tells that behind one of the doors there is a prize - a car, and goats are hidden behind the other two doors. After you have chosen one of the doors, the presenter, who knows what is behind each door, opens one of the remaining two doors and demonstrates that there is a goat behind it (a goat, the sex of the animal in this case is not so important) And then the presenter slyly asks: “Do you want to change your choice of door?” Will changing your selection increase your chances of winning?
If you think about it: here are two closed doors, you have already chosen one, and the probability that there is a car/goat behind the chosen door is 50%, just like with a coin toss. But this is not true at all. If you change your mind and choose another door, your chances of winning will increase by 2 times! Experience has confirmed this statement (see Appendix No. 5). Those. By leaving his choice, the player will receive a car in one of three cases, and by changing two out of three. Statistics from the TV show confirm that those who changed their choices were twice as likely to win.
This is all probability theory and it is true over “many options.” I hope that this example will make you think about how to quickly pick up a book about probability theory, and also start applying it in your work. Believe me, it is interesting and exciting, and there is a practical sense.
Chapter 3 . Is it possible to win the lottery or roulette?
Each of us has bought a lottery or gambled at least once in our lives, but not everyone used a pre-planned strategy. Smart players have long stopped hoping for luck and turned on rational thinking. The fact is that each event has a certain mathematical expectation, as higher mathematics and probability theory say, and if you correctly assess the situation, you can bypass the unsatisfactory outcome of the event.
For example, in any game, such as roulette, it is possible to play with a 50% chance of winning by betting on an even number, or a red cell. This is exactly the game we will consider.
To ensure profit, we will draw up a simple game strategy. For example, we have the opportunity to calculate the probability with which an even number will appear 10 times in a row - 0.5 * 0.5 and so on 10 times. We multiply by 100% and we get only 0.097%, or approximately 1 chance in 1,000. You probably won’t be able to play that many games in your entire life, which means that the probability of getting 10 even numbers in a row is practically equal to “0”. Let's use this game tactic in practice. But that's not all, even 1 time in 1,000 is a lot for us, so let's reduce this number to 1 in 10,000. You ask, how can this be done without increasing the expected number of even numbers in a row? The answer is simple - time.
We approach the roulette wheel and wait until an even number appears 2 times in a row. This will be each time out of four calculated cases. Now we place the minimum bet on an even number, for example 5p, and win 5p for each occurrence of an even number, the probability of which is 50%. If the result is odd, then we increase the next bet by 2 times, that is, we already bet 10 rubles. In this case, the probability of losing will be 6%. But don't panic if you lose even this time! Make the increase twice as much each time. Each time the mathematical expectation of winning increases, and in any case you will remain in profit.
It is important to take into account the fact that this strategy is only suitable for small bets, since if you initially bet a lot of money, you risk losing everything due to bet restrictions in the future. If you have any doubts about this tactic, play a game of guessing the side of a coin with fictitious money with a friend, betting twice as much if you lose. After a while you will see that this technique is simple to practice and very effective! We can conclude that by playing with this strategy, you will not earn millions, but will only win for small expenses.
Conclusion
While studying the topic of “probability theory in life,” I realized that this is a huge section of the science of mathematics. And it is impossible to study it in one go.
After going through many facts from life and conducting experiments at home, I realized that the theory of probability really has a place in life. The probability of an event in life is not often calculated using formulas, but rather intuitively. But checking whether the “empirical analysis” coincides with the mathematical one is sometimes very useful.
Can we predict with the help of this theory what will happen to us in a day, two, a thousand? Of course not. There are a lot of events related to us at any given time. A lifetime alone would not be enough to typify these events. And combining them is a completely disastrous business. With the help of this theory, only events of the same type can be predicted. For example, something like tossing a coin is an event of 2 probabilistic outcomes. In general, the applied application of probability theory is associated with a considerable number of conditions and restrictions. For complex processes it involves calculations that only a computer can do.
But we should remember that in life there is also such a thing as luck, luck. This is what we say - lucky, when, for example, some person never studied, did not strive for anything, lay on the couch, played on the computer, and after 5 years we see him being interviewed on MTV. He had a 0.001 chance of becoming a musician, it happened, he was lucky, such a convergence of circumstances. What we call is being in the right place and at the right time, when those same 0.001 are triggered.
Thus, we work on ourselves, make decisions that can increase the likelihood of fulfilling our desires and aspirations, each case can add those cherished 0.00001 that will play a decisive role in the end.
Bibliography
Mathematics, the queen of all sciences, is often put on trial by young people. We put forward the thesis “Mathematics is useless.” And we refute it using the example of one of the most interesting mysterious and interesting theories. How probability theory helps in life, saves the world, what technologies and achievements are based on these seemingly intangible and far from life formulas and complex calculations.
History of probability theory
Probability theory- a field of mathematics that studies random events and, naturally, their probability. This kind of mathematics originated not in boring gray offices, but... in gambling halls. The first approaches to assessing the probability of a particular event were popular back in the Middle Ages among the “Hamlers” of that time. However, then they had only empirical research (that is, evaluation in practice, by experiment). It is impossible to attribute the authorship of the theory of probability to a specific person, since many famous people worked on it, each of whom contributed their own share.
The first of these people were Pascal and Fermat. They studied probability theory using dice statistics. She discovered the first laws. H. Huygens had done similar work 20 years earlier, but the theorems were not formulated precisely. Important contributions to probability theory were made by Jacob Bernoulli, Laplace, Poisson and many others.
Pierre Fermat
The theory of probability in life
I will surprise you: we all, to one degree or another, use the theory of probability, based on the analysis of events that have happened 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 the entire 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 malfunction (for example, a spacecraft), we know what efforts we need to make, what exactly to check, what to expect in general thousands of kilometers from Earth. The possibilities of a terrorist attack in the metro, an economic crisis or a nuclear war - all this can be expressed as a percentage. And most importantly, take appropriate counteractions based on the data received.
I was lucky enough to attend a mathematical scientific conference in my city, where one of the winning papers spoke about the practical significance theories of probability in life. You probably, like all people, don’t like standing in lines for a long time. This work proved how the purchasing process can be accelerated if you use the probability theory of calculating people in line and regulating activities (opening cash registers, increasing the number of salespeople, etc.). Unfortunately, now the majority of even large networks ignore this fact and rely only on their own visual calculations.
Any activity in any sphere can be analyzed using statistics, calculated using probability theory and significantly improved.
Many people ask what is theory of probability, cognition and everything, what it affects and what its functions are. As you know, there are many theories and few of them work in practice. Of course, the theory of probability, knowledge and everything has long been proven by scientists, so we will consider it in this article in order to use it to our advantage.
In the article you will learn what the theory of probability, knowledge and everything is, what its functions are, how it manifests itself and how to use it to your advantage. After all, probability and knowledge are very important in our lives and therefore we need to use what has already been tested by scientists and proven by science.
Certainly Probability theory is a mathematical and physical science that studies this or that phenomenon and what is the probability that everything will happen exactly the way you want. For example, how likely is it that the end of the world will happen in 27 years, and so on.
Also, the theory of probability is applicable in our lives, when we strive for our goals and do not know how to calculate the probability of whether we will achieve our goal or not. Of course, this will be based on your hard work, a clear plan and real actions, which can be calculated for many years.
Theory of knowledge
The theory of knowledge is also important in life, as it determines our subconscious and consciousness. Because we are learning about this world and developing every day. The best way to learn something new is by reading interesting books written by successful authors who have achieved something in life. Knowledge also allows us to feel God within ourselves and create reality for ourselves the way we want, or trust God and become a puppet in his hands.
Theory of everything
But here theory of everything tells us that the world came into existence precisely because of the big bang, which separated energy into several cells in a matter of seconds and as we see large populations, this is actually the division of energy. When there are fewer people, this will mean that the World is returning to its original point again, and when the world is restored, there is a high probability of another explosion.
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