In this lesson we will learn about such a concept as coefficient. We will also look at several problems, using examples of which we can easily find the coefficients of various expressions.
This is the product: the number 2 is multiplied by the letter.
In such a work we agreed to name the number coefficient.
A coefficient is a numerical factor in a product where there is a letter.
For example:
Therefore the coefficient is 4.
Therefore the coefficient is 1.
Therefore the coefficient is -1.
Therefore the coefficient is 5.
In mathematics, we agreed to write the coefficient at the beginning, therefore:
There may be several letters, but this does not affect the coefficient. For example:
Coefficient -17.
Factor 46.
If the product has several numerical factors, then this expression can be simplified:
The coefficient in this expression is 100.
A numerical factor in a product that contains at least one letter is called a coefficient.
If there are several numbers, you need to multiply them, simplify the expression, and thus obtain a coefficient.
There is only one coefficient in one product.
If there is a sum, for example, this:
Then each term has coefficients: and .
If there is no number, then you can put one. This is the coefficient.
, coefficient 1.
Find the coefficient: a) ; b) 
.
a) , coefficient -50.
b) coefficient.
So, coefficient is a number that stands in a product with one or more variables. It can be integer or fractional, positive or negative.
When planting potatoes, the yield is 10 times greater than the number of potatoes planted. What will the harvest be if you planted 65 kg?
Solution
What if 90 kg of potatoes are planted?
What if we don’t know how much has been planted? How then to decide in this case?
If you planted kg, then the harvest will be kg.
So, 10 is a coefficient here (let’s call it yield), and is a variable. can take any value, and the formula will calculate the amount of harvest.
If the yield is different, for example 9, then the formula looks like this: .
The coefficient in the formula has changed.
If we consider different yields, the formula will remain the same in appearance, only the coefficient will change.
So we can write general form all such formulas.
Where is the coefficient; - variable.
This is the yield, it can be equal to, for example, 10 or 9, as before, or another number.
So, how to answer the question “what is the coefficient in the entry?”?
If nothing is known about this record, then they are just letters, variables. Coefficient one.
If it is known that this is part of the formula for calculating the potato yield, then this is the coefficient.
In other words, the coefficient can often be denoted by a letter.
In mathematics, physics, and other sciences there are many formulas where one of the letters is a coefficient.
Example
The density of matter in physics is denoted by the letter.
The higher the density, the more the same volume of a substance weighs.
If you know the volume of a substance and its density, then you can easily find the mass using the formula:
Any person who is familiar with this formula, when asked “what is the coefficient here?” will answer "".
A coefficient is a number in a product where there are one or more variables.
There is an agreement to write the coefficient before the variables.
If there is no number in the product, then you can put a factor of 1, which will be the coefficient.
If we have a formula in front of us, then one of the letters may well be a coefficient.
Bibliography
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.
- Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course grades 5-6 - ZSh MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at the MEPhI correspondence school. - ZSh MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for grades 5-6 high school. Math teacher's library. - Enlightenment, 1989.
- Internet portal "Uchportal.ru" ()
- Internet portal “Festival of Pedagogical Ideas” ()
- Internet portal “School-assistant.ru” ()
Homework
One of the main statistical indicators of a sequence of numbers is the coefficient of variation. Quite a lot of work is done to find it. complex calculations. Microsoft Excel tools make it much easier for the user.
This indicator is the ratio of the standard deviation to the arithmetic mean. The result obtained is expressed as a percentage.
In Excel there is no separate function for calculating this indicator, but there are formulas for calculating standard deviation and average arithmetic series numbers, namely they are used to find the coefficient of variation.
Step 1: Calculate Standard Deviation
Standard deviation, or, as it is otherwise called, root mean square deviation, is Square root from . To calculate the standard deviation, use the function STANDARD DEVIATION. Starting with Excel 2010, it is divided depending on population the calculation takes place either by sample, into two separate options: STDEV.G And STDEV.V.
The syntax for these functions looks like this:
STANDARDEVAL(Number1,Number2,…)
= STANDARD DEVIATION.G(Number1;Number2;…)
= STANDARDEV.B(Number1;Number2;…)
Step 2: Calculate the arithmetic mean
The arithmetic mean is the ratio of the total sum of all values in a number series to their number. There is also a separate function to calculate this indicator - AVERAGE. Let's calculate its value using a specific example.
Step 3: Finding the Coefficient of Variation
Now we have all the necessary data to directly calculate the coefficient of variation itself.
Thus, we calculated the coefficient of variation, referring to cells in which the standard deviation and arithmetic mean had already been calculated. But you can do it a little differently, without calculating these values separately.
There is a conditional distinction. It is believed that if the coefficient of variation is less than 33%, then the set of numbers is homogeneous. Otherwise, it is usually characterized as heterogeneous.
As you can see, the Excel program allows you to significantly simplify the calculation of such a complex statistical calculation as finding the coefficient of variation. Unfortunately, the application does not yet have a function that would calculate this indicator in one action, but using the operators STANDARD DEVIATION And AVERAGE this task is greatly simplified. Thus, even a person who does not have high level knowledge related to statistical laws.
Beginners face problems where there are no obstacles for experienced and successful bettors. Beginners cannot regularly find adequate bets with odds around two. In this article we will analyze betting options with quotes from 1.80 to 2.20.
- A coefficient of 2.0 is quite high. To make money when playing on such quotes, it is enough to show 53-55% passability.
- A coefficient of 2.0 is not too high if the quotes in a particular game reflect the real probability of the outcome. This is 50%, excluding the bookmaker's margin. Finding adequate events with a 50/50 probability is not as difficult as it seems. It is much more difficult to take a coefficient of 2.5.
- Many betting strategies are designed to play with odds of 2.0. First of all, these are the “martingale” and “dogon” financial systems. That is why beginners often look for information about what betting options they can play with this odds.
First, open the bookmaker's line and look at the types of bets. There are many markets with coefficients around 2.0, but which of them are adequate?
Below are optimal options bets with odds of 2.0. Each transaction must be justified and based on the analysis performed, and not made blindly, based on quote values.
Clear victory
Standard net gain. When they offer to bet on a team’s success at 2.0, then it is a favorite, but a hidden one. The triumph of the expressed favorite has less value. If the analysis indicates a confident victory for one of your opponents, feel free to flirt with this outcome.
Handicap (-1)
When the favorite is clear (odds 1.3-1.7), and the analysis speaks of a defeat, and not just a win, take the negative handicap for two.
Handicap (0)
If the opponents have equal chances, a zero handicap for each team is valued at the same prices. Usually at 1.85-1.95, excluding margin. If you think that the team will probably not lose, but rather even win, then a zero handicap with a coefficient of about two is an excellent option in terms of profitability and risks.
Handicap (+1), (+1.5) and (+2)
There are fights in which the outsider has good chances for a draw or minimal defeat. It is advisable to take a positive handicap. In the list you can rarely find decent options with a positive handicap on the underdog.
Team goal
This is a “team to score” bet or ITB (0.5). Bookmakers often give odds close to two for an underdog goal. There are fights when such a deal is justified. Bet if the underdog has attacking potential, but the bookmaker overestimates the reliability of the favorite's defensive line.
Individual total over (1)
Bet on ITB (1) with odds. 2.0 is possible in confrontations between equal opponents and matches where the favorite is not clearly defined. If a weaker team plays in front of its home fans, it is capable of scoring even against the championship leaders. The main thing is to back up your choice with facts.
You can also play ITB (1) in games when many goals are predicted. The advantage of the bet is that it is not tied to the result, because even if the team loses 3:2, the deal will still be successful. Determine the team's potential in a duel with a specific opponent.
Individual total over (1.5) and (2.0)
Bigger total. Naturally, this is a bet on the clear favorite when predicting a scoring extravaganza. It is important to consider the risks here. Calculate whether the football players are motivated to score two or more goals. What if they are satisfied with a minimal victory or the opponent closes down so much that they miss the maximum time?
Total over/under (2.5)
Standard total value. In most fights, both totals are given quotes close to two. If the analysis points in favor of a certain side, then the bet is quite good. The main thing is to justify the choice.
Remember that the overall match total is a more dangerous outcome than the ones we looked at earlier.
Total under/over (2.0)
When an ineffective meeting is expected at the office, the main total drops to two. If you agree with the opinion of the bookmaker’s analysts and do not look at more than one goal, play with TM (2).
TB (2) in the main list is usually found in non-scoring championships, for example, the RFPL and FNL, where bookmakers sometimes even offer TB (1.5). I often find underestimated totals and make money by underestimating bookmakers.
Total over/under (3)
The main total (3) is set where many goals are expected to be scored. Limit yourself to 3 goals. Flirting with TB (3.5) or more is risky. In some events, depending on the analysis performed, you can take TB (3) and TM (3). On the one hand, you will increase the coefficient, and on the other, you will reduce the risks. TB (3) is the same TB (2.5), just with the possibility of a return.
Both will score
A bet with a 50% probability, regardless of bookmaker quotes. Play if your HP is estimated at a high coefficient, minimum 1.85. But consider other, less risky outcomes.
Health + TB (2.5)
This is a double bet consisting of both goals and total. It is logical to flirt with the outcome when there is confidence in HP and the top total. However, individually these rates are estimated at 1.7-1.8, or even less. And for combined option given already 1.9-2.1.
Of course, there are many more outcomes in the line with odds of 2.0, but most often these are unjustified and risky bets. It is not recommended to take large handicaps, totals, combined bets, etc.
Summary
A coefficient of about two allows you to make a profit, even if the throughput is slightly above 50%. With meager quotes, the traffic level should increase by 2-3 times. It is often easier to show 55% passability with quotes of 1.8-2.2 than 80% with quotes of 1.25.
Now you know the options on how to take a coefficient of about two. There is nothing complicated about this. The main thing is to analyze events and justify every bet.
Where x·y, x, y are the average values of the samples; σ(x), σ(y) - standard deviations.
In addition, the linear pair correlation coefficient can be determined through the regression coefficient b: , where σ(x)=S(x), σ(y)=S(y) - standard deviations, b - coefficient before x in the regression equation y= a+bx .
Other formula options:
or 
K xy - correlation moment (covariance coefficient) 
The linear correlation coefficient takes values from –1 to +1 (see Chaddock scale). For example, when analyzing the closeness of the linear correlation between two variables, a paired linear correlation coefficient equal to –1 was obtained. This means that there is an exact inverse linear relationship between the variables.
Geometric meaning of the correlation coefficient: r xy shows how different the slope of two regression lines: y(x) and x(y) is, and how much the results of minimizing deviations in x and y differ. The greater the angle between the lines, the greater r xy.The sign of the correlation coefficient coincides with the sign of the regression coefficient and determines the slope of the regression line, i.e. general direction of dependence (increasing or decreasing). The absolute value of the correlation coefficient is determined by the degree of proximity of the points to the regression line.
Properties of the correlation coefficient
- |r xy | ≤ 1;
- if X and Y are independent, then r xy =0, the converse is not always true;
- if |r xy |=1, then Y=aX+b, |r xy (X,aX+b)|=1, where a and b are constants, a ≠ 0;
- |r xy (X,Y)|=|r xy (a 1 X+b 1, a 2 X+b 2)|, where a 1, a 2, b 1, b 2 are constants.
Instructions. Specify the amount of input data. The resulting solution is saved in a Word file (see Example of finding a regression equation). A solution template is also automatically created in Excel. .
Hi all!
Having entered the sports betting community, I did not find any articles on betting theory, although I bet myself and know that theoretical material in betting no less than in poker. Therefore, I want to post here some posts about the mathematical and analytical foundations of sports betting. I hope it is useful to someone.
I would like to start where every player starts: with the bookmaker’s line. The first question that arose in my mind when I first picked up a printed line: How does a bookmaker determine all this mass of odds?
Bookmakers operate solely for the purpose of making a profit. And, contrary to popular belief, the bookmaker’s profit does not depend on the number of lost bets, but on the correctly set odds. What does "correct" mean? This means that in case of any, even the most unexpected, outcome of the event, the bookmaker must remain profitable.
Let's look at how the coefficients are formed. First, analysts determine the teams' chances. This is done in many ways, which can be divided into two groups: analytical and heuristic. Analytical ones are mainly statistics and mathematics (probability theory), heuristic ones are expert assessments. By combining the results obtained in one way or another, the probabilities of the outcome of the event are derived. Let’s assume that as a result of the activities of analysts and experts, the following probabilities of outcomes were obtained:
These are "pure odds", but these odds will never line up because the bookmaker will not make a profit in this case. The line odds for these events will look something like this:
That is, out of every one hundred thousand rubles bet by all players, 75,000 were bet on victory 1, 15,000 on a draw and 10,000 on victory 2. Most players most often bet on obvious favorites, making up most of the express bets based on such outcomes . What will the bookmaker get for each hundreds of thousands of dollars invested by players in the event of different outcomes?
It can be seen that if the favorite wins, which happens most often, the bookmaker will suffer losses. This is completely unacceptable for business, and the bookmaker is obliged to exclude even the theoretical possibility of such a situation arising.
To do this, he must artificially lower the odds on the favorite. The bookmaker does not know in advance exactly how the bets will be distributed, but he knows for sure that the players will “load” on the favorite, therefore, for insurance, he overestimates the probability of the favorite’s victory.
In reality, neither the real chances nor the distribution of funds by players can be accurately calculated; there is always some error. Therefore, bookmakers try to initially lower the odds for the favorite in order to guarantee their profit, i.e. determine the teams' chances and add 10-20% to the calculated probability of victory for the favorite. And as bets are received, depending on their actual current distribution, the odds are varied so that the profit is greatest.
Conclusion: the main principle that guides the bookmaker is the distribution of finances between two or more groups of players in such a way as to pay winnings from the funds of the losers, leaving a certain percentage for themselves. Very often, the coefficients obtained in this way have nothing to do with the probabilities of certain events. Therefore, you need to have your own system for evaluating sporting events.
Thank you for your attention!
