The role of probability theory in everyday life. Probability theory as a tool for business success
1.2. Applications of Probability Theory
Probability theory methods are widely used in various branches of natural science and technology:
in the theory of reliability,
queuing theory,
theoretical physics,
geodesy,
astronomy,
shooting theory,
the theory of observational errors,
Theories of automatic control,
general theory of communication and in many other theoretical and applied sciences.
The theory of probability also serves to substantiate mathematical and applied statistics, which, in turn, is used in the planning and organization of production, in the analysis of technological processes, preventive and acceptance control of product quality, and for many other purposes.
In recent years, the 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 were born 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 the theory of probability is associated with the name of Jacob Bernoulli (1654 - 1705). The theorem he proved, later called the "Law of Large Numbers", was the first theoretical substantiation of the facts accumulated earlier.
Probability theory owes further success to Moivre, Laplace, Gauss, Poisson and others. . Lyapunov (1857 - 1918). During this period, the theory of probability becomes a coherent mathematical science. Its subsequent development is due primarily to Russian and Soviet mathematicians (S.N. Bernshtein, V.I. Romanovsky, A.N. Kolmogorov, A.Ya. Khinchin, B.V. Gnedenko, N.V. Smirnov, etc.). ).
1.4. Tests and events. Event types
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”, we will say briefly: “tested”. Thus, the event will be considered as the result of the test.
Definition. random event any fact that may or may not occur as a result of experience is called.
In this case, one or another result of the experiment can be obtained with varying degrees of possibility. That is, in some cases it can be said that one event will almost certainly happen, the other almost never.
Definition. Space of elementary outcomesΩ is a 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 denoted by the letter ω ("omega").
Then the subsets of the set Ω are called events. It is said that as a result of the experiment, the event A Ω occurred if one of the elementary outcomes included in the set A occurred in the experiment.
For simplicity, we 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 occur as a result of the test (three points on a die roll, a phone call at the moment, etc.).
Example 1 The shooter shoots at a target divided into four areas. A shot is a test. Hitting a certain area of the target is an event.
Example 2 There are colored balls in the urn. One ball is drawn at random from the urn. Removing 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 the initial one, which is not defined and which is characterized only by its own properties. Based on the real meaning of the concept of an event, different types of events can be defined.
Definition. A random event is called authentic, if it is known to occur (rolling one to six points on a roll of the die), and impossible, if it certainly cannot occur as a result of experience (seven points rolled when throwing a die). In this case, a certain event contains all points of the space of elementary events, and an impossible event does not contain any point of this space.
Definition. Two random events are called incompatible if they cannot occur at the same time for the same test outcome. And in general, any number of events are called incompatible if the occurrence of one of them excludes the occurrence of the others.
A classic example of disjoint events is the result of a coin toss - the fall of the front side of the coin excludes the fall of the reverse side (in the same experiment).
Another example is a part taken at random from a box of parts. The appearance of a standard part excludes 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 certain event. In particular, if the events that form a complete group are pairwise incompatible, then one and only one of these events will appear as a result of the test. This special case is of the greatest interest, since it is used below.
Example. Purchased two tickets of the money and clothing lottery. One and only one of the following events will necessarily occur: “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”, “the winnings did not win 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 is sure to occur: hit, miss. These two disjoint 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 ball 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 likely 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 a coin is tossed are equally likely events. Indeed, it is assumed that the coin is made of a homogeneous material, has a regular cylindrical shape, and the presence of a coinage does not affect the loss of one or another side of the coin.
Example. The appearance of one or another number of points on a thrown dice are equally probable 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 the box contains the same number of red and green balls. If there are more red balls in the box than green ones, then the appearance of a green ball is less probable than the appearance of a red one.
Many people use probability theory regularly. Especially often it is used by entrepreneurs in their business. But almost no one associates personal calculations and thoughtful actions with her. The theory of probability in life helps to avoid many troubles, including losses. Most businessmen own it at a practical level. On the other hand, often those to whom the theory of probability should seem to be very well understood are, in fact, completely ignorant in it. By the way, an Israeli scientist, Nobel Laureate Daniel Kahneman and his friend Amos Tversky proved experimentally: specialists with mathematical education do not really understand the theory of probability. They do not take it into account even in cases where it would be possible to avoid losses or gain benefits. And they act in exactly the same way as people who are completely unfamiliar with this theory.
For your business (in the sense of your business), the theory of probability is necessary. Its understanding and constant application is one of the foundations of success and efficiency in work.
Probability theory is simple, if you do not complicate it
Consider the theory of probability for a very simple examples. If we have 10 numbered balls in a box with numbers from 1 to 10, then the probability of drawing a ball with the number 10 is 10 percent. But it is more likely that we will draw any other number from 1 to 9, and not the largest (not 10), since such a probability is 90 percent. Drawing the ball with the highest number out of 10,000 numbered balls is already too unlikely. Most likely, we will draw any other number (not 10000). With 10 million balls, pulling out the largest number (10,000,000) is almost impossible. The logical result will be the pulling of any other number, but not the largest. The given examples with balls led us to the law big numbers. It says:
Phenomena that are probable with a small number of them become regular with a large number, and inevitable with a very large number.
In our examples, it is possible to draw a ten out of 10 balls, but it is more likely that we will draw any other number. But as the number of balls increases, the probability of drawing not the most a large number increases more and more and turns into a pattern when a large number of balls is reached, and when there are a huge number of them - into inevitability.
The law of large numbers includes several provisions (several theorems). To the formulation already known to you, you should add one more:
With an increase in the number of probable phenomena, their average values tend to become constant, and in the case of a large number they practically become so.
Consider this situation on the example of a coin. When a coin is tossed 10 times, its fall heads or tails up is likely in the ratio of 5 to 5, and 6 to 4, and 3 to 7 ... But as the number of tosses increases, this ratio will inexorably approach equality (to constant average values) , that is, to a ratio of 50% to 50%. With a million rolls, getting even a 60% to 40% ratio is almost impossible - it will be very close to a 50% to 50% ratio. Some people believe that the probability of getting one side of the coin 100 times in a row is 1 percent. And they are very mistaken, because such an event is too unlikely: like one chance in several billion.
I think you understand that probability theory is really simple. Since its publication (several centuries ago), its provisions have been checked in almost all states great amount once. The students were especially successful in this. As a rule, coins were used for verification. And everyone was convinced of the complete coincidence of theory with practice.
Application of the theory of probability in your business
When assessing the situation on the market (in your niche), when working with statistical data, you inevitably have to use probability theory - as a rule, at a practical level. But it is better if you apply this theory, understanding it. theoretical basis. After all, it is really simple. It is only important to understand the theory of probability and apply it consciously. And situations in which its use is necessary arise all the time, especially in business. Therefore, remember the two given formulations of probability theory. They are highlighted in red above. Try to understand their meaning! This is really important for you!
Probability theory, which immediately after its discovery became a separate branch of mathematics, helped people long before it. scientific justification.
As soon as they did not explain the development of an unpredictable event according to the desired scenario - some by the intervention of gods and spirits, some by the power of prayer, and some by mere chance. And only in the seventeenth century, by the works of the great physicist and mathematician Blaise Pascal, it was clearly proved that any "accidents" obey a certain pattern, which was called the theory of probability. It is she who claims that with a sufficiently large number of coin tosses, the number of heads and tails will be equal; if some player does not win for a long time, then in the next game he must definitely win and similar inevitable coincidences.
That is why the theory of probability has found one of its areas of application in gambling. Intuitive calculations in gambling were used in ancient times, and only in our time people were able to determine that these calculations obey mathematical laws! But, unfortunately, any win in gambling, as a rule, is random - and it is almost impossible to calculate the time of occurrence of a win, as well as create any effective winning combination, so players have to rely only on probability theory. True, she can let a person down very much - for example, throwing coins into slot machine and without winning a penny, the player can lose all hope and move away from the machine - and then the first newcomer who has just started the game wins stunning money, actually "earned" by the previous player! You can practice mathematical calculations of the probability of winning on any specialized gaming portal, for example,.
It is important to start analyzing the mechanisms of gambling without serious financial investments, and even better for free, since some sites today provide such an opportunity. However, it is important to understand that you can calculate the probability of winning as much as you like, starting from the theory of probability, but not a single theory, not a single most rigorous calculation will allow you to calculate the possibility of winning one hundred percent. But in a more responsible business, that is, in business, the theory of probability really works! Only by applying this theory, a businessman avoids possible losses and gains - after all, according to the law of large numbers, with a small number of expected events, the number of desired outcomes is probable, and with a very large number of events they become inevitable. And certain business moves in world history have been used countless times, so they can be used almost without error.
Consciously using the theory of probability, you will be able not to make mistakes in assessing the situation on the market, to work skillfully and benefit from statistical data. But even applying your knowledge of probability theory in practice, you must also understand its theory, especially the postulate that an increase in the number of probable phenomena entails the constancy of their average values. And the more events that happen, the more permanent their outcome will become.
In the section on the question Theory of Probability ... Where does the theory of probability occur in life? thanks in advance :) set by the author Adam Axmatov the best answer is The entire theorver is taken from life. Any more or less massive or frequently recurring phenomena.
- Probability of winning the lottery / roulette in the casino
- Probability of equipment breakdown
- Production - a forecast of the number of defects.
- Evaluation of the reliability of different systems. An example - at work you need an "uninterrupted" (99.9995% working capacity) Internet. Theorver helps.
- Probability that parents will give 3.14 zd for homework not done
Remember about MASS AND REPEATED
"If I now bet in roulette on 8, will it fall out or not", "now I will go outside, will an icicle fall on me?" - HZ.
But if you bet 100 times on 8 / then you will probably lose money, because the probability of winning is slightly less than losing, but by multiplying the probabilities, your chances fall more and more /
or 30 icicles fall down the street in a month, and 50,000 people pass by - then the theorver works great.
Answer from Man with paddle[guru]
Everywhere.
Please.
Answer from OchloPhob[guru]
Just not in Russian politics)
Answer from The enemy will not pass![guru]
A professor of physics is asked: What is the probability that a dinosaur will come here right now? The professor counted for two days, then says: The probability is 0.0 in minus 300 0000 00000000000000%
The saleswoman is also asked. She says: 50%
How is that? - And usually - Either he will come (50%), or he will not come (50%) ...
Answer from Murzik99rus[guru]
In a trolleybus. The controller will enter or not enter when YOU eat without a ticket.
Answer from Grumm[guru]
Falling coconuts kill ~150 people a year. This is ten times more than from a shark bite. But the movie "Coconut Killer" has not yet been filmed :))
Answer from Silver Shadow[guru]
Brick on the head will fall or not. . car crashes or not..
Webinar about how to understand probability theory and how to start using statistics in business. Knowing how to work with such information, you can make your own business.
Here is an example of a problem that you will solve without thinking. In May 2015, Russia launched spaceship"Progress" and lost control over it. This pile of metal, under the influence of the Earth's gravity, should have crashed onto our planet.
Attention, the question is: what was the probability that Progress would have fallen on land, and not in the ocean, and whether we should have been worried.
The answer is very simple - the chances of falling on land were 3 to 7.
My name is Alexander Skakunov, I am not a scientist or a professor. I just wondered why we need the theory of probability and statistics, why did we take them at the university? Therefore, in a year I read more than twenty books on this topic - from The Black Swan to The Pleasure of X. I even hired myself 2 tutors.
In this webinar, I will share my findings with you. For example, you'll learn how statistics helped create an economic miracle in Japan and how this is reflected in the script for the movie Back to the Future.
Now I'm going to show you some street magic. I don't know how many of you will sign up for this webinar, but only 45% will turn up.
It will be interesting. Sign up!
3 stages of understanding the theory of probability
There are 3 stages that anyone who gets acquainted with the theory of probability goes through.
Stage 1. “I will win at the casino!”. Man believes that he can predict the outcome of random events.
Stage 2. “I will never win at the casino!..” The person is disappointed and believes that nothing can be predicted.
And stage 3. “Let's try outside the casino!”. A person understands that in the seeming chaos of the world of chances one can find patterns that allow one to navigate well in the world around.
Our task is just to reach stage 3, so that you learn how to apply the basic provisions of the theory of probability and statistics to the benefit of yourself and your business.
So, you will learn the answer to the question "why the theory of probability is needed" in this webinar.