How a quota is created

How is a quota created?

—- Still to be completely revised from here —

Basically, there are two aspects that are responsible for the emergence of a quota. One is the philosophical aspect, the other is the mathematical aspect.

The philosophical aspect is that a quota only comes into being because there is, so to speak, a need for a quota. This need is recruited from the fact that there is some form of interest in predicting the occurrence of events that lie in the future. The extent of this need also depends partly on the significance of the event (see below).

In any case, the need to predict the occurrence of an event results in the need to support this opinion financially. This would result in a kind of bet in the sense we want to deal with it.

Please understand that it is not possible to derive such a concept and ultimately grasp its meaning entirely without mathematical considerations. Of course, this concept is just like many others: One has a conception of the term. Apart from being “very vague” or “wrong” or “right”, it can also be “incomplete”. If this is the case with you, I would like to try to gradually fill the term with meaning.

Now it is also the case that a prior understanding is necessary. This can be gained through other terms that are useful or indispensable for the construction of the term “ratio”, especially in mathematical terms.

—- Still to be completely revised up to here —

1. definitions of terms
   a. Events
   Defining always sounds so terribly mathematical, and it is. In the chapter “Philosophy of Happiness” I already talked a bit about how terms come into being, gain meaning, change their meaning, are used and subsequently (can) be so delightfully misunderstood. There is often no fixed definition. Rather, there are conceptions. And especially abstract terms, such as the term “event”, leave room for interpretation. Occasionally, however, it becomes necessary to reach an agreement on how one would like to have that term understood in such a context.

What do you immediately imagine when you hear the term “event”? George Clooney was in your home town?

First of all, there are at least two understandings of an event. One is the taking place of an event, the other is the occurrence of an event (synonymous with occurrence the word occurrence is also often used in the following).
Then it is important to distinguish between future and past events. The above example is representative of the concept: An event takes place. In this case, past, i.e. it took place. This view is also worth mentioning insofar as the major events that are so often cited, for example a football World Cup, are perfectly suited to provide questions for the other view, the occurrence of events. Representative of many others is the usual question that occupies us all and can usually be answered with “yes”: Will Germany be World Champion/European Champion again?
The main reason for this is that a major event is called such because it interests, fascinates, excites and captivates many people, so that they deal with what might happen. People discuss chances, possible outcomes and for the first time they even encounter the term “suspense”, which is somewhere partly responsible for all the gambling and betting.

So without defining, I’ll talk briefly about events in the past. And events that have taken place or those that have occurred. Events that have taken place essentially serve as reports of experiences (“I was live in the stadium when Maradonna…”). In this sense, they are irrelevant for bets, games and odds.
The only thing that happens now and then is that people get caught up in discussions about events, no matter what their opinion was, and in the course of the discussion they come across different memories. This occasionally even causes hotheads to place bets on it, trusting in their superior memory. Now this kind of bet is also irrelevant for us here, at least in the mathematical sense. We are betting on a fact. The truth content is certain (exceptions here: The exact formulation of a bet on a fact can lead to discussions. Likewise, there are such bets that are made on a fact that cannot be tested exactly. So you can bang your head even on bets on facts. However, there is only a very limited form of “truth” in this). The bet can be made and it can be paid out. One could even pay an odds (“I am quite sure that Hertha will be promoted in 1968. I’ll pay you ten times the money.”). But these kinds of bets (and odds) are also irrelevant for us here.

What is interesting for us here are events in the future whose occurrence (occurrence) has an unknown truth content. That is, their occurrence is still open. It can happen or not happen. At this moment, the term “probability” comes into play. The truth content is open, undetermined. After the event has occurred or not, we have the truth content at 1 or at 0. 1 means “it has occurred”, 0 “it has not occurred”. Before it was a probability between 0 and 1, then it ended up at 0 or at 1.
A probability perhaps also triggers an association: Percent, yes? So at this point it might be useful to mention that the numbers between 0 and 1 are excellent and perfect for reformulating into percentages. But here, too, we are in luck: it is synonymous. The number 0.5 is identical with the number 50%. Because the % sign only stands for division by 100. And 50/100 = 0.5. By the way, it is also interesting that the mathematician speaks of a “certain event” for an event with probability 1 and of an “impossible event” for one with probability 0.
The “certain event” applied to the past simply means: “It was so”. The impossible event, on the other hand, means “it was not so”.
b. Statements
Then there is the concept of a statement, at least mathematically, logically speaking. The mathematician has devoted all his love to statements. He constantly makes statements and tries to assign a truth content to them. A statement then becomes “true” if it has been proven on the basis of the existing axioms and previously proven statements. “False” is a statement that could be refuted (for statements without logic or with unclear truth content, please study the chapter “Paradoxes”).
In this sense, by the way, every event can also be automatically reformulated into a statement. However, the truth of such statements can only be checked with regard to past events.

c. Random experiment
Unfortunately, this is still not enough in terms of concepts to arrive at a quota. We still need the so-called “random experiment”. Unfortunately, the mathematician has made things far too easy for himself here too. First of all, however, it is good for understanding. A random experiment is carried out. To do this, one takes a certain experimental set-up. Here it is sufficient if we visualise it with a few examples: “A coin is tossed once”, “A die is rolled once” or “A roulette ball is thrown once”, just as “The lottery numbers are drawn”… Of course, you can also take “A die is thrown ten times” or “Ten dice are thrown at the same time”. These are the random experiments

d. The event space
The possibilities of investigating a large variety of possible events (in the sense of occurrence, as always from now on) in a random experiment perhaps become clear in this way: The dice can not only show a 1,2,3,4,5 or 6. It can also show an even number or an odd number. It can show a prime number or a number less than 4 or even a number greater than 4. All are possible events. You would always have to formulate them like this: “The dice shows an even number.” “The dice shows an odd number.” “The die shows a 5.” “The die shows a prime number.” Or even, quite curiously, “The points on the upper side of the cube are arranged point-symmetrically” (Incidentally, point-symmetrical on a conventional cube are the 1, the 4 and the 5. Please look at a cube and turn the side where a 5 is shown, while the side on top remains. The picture remains unchanged. It is different with the 3).

The roulette ball is similar, of course. It can show a number from 0 to 36, but also an even or odd number, a number that is divisible by 3, a number that has a black background in the roulette wheel or a red one (just not zero!). But it can just as well be a prime number or no prime number. One could also examine the event “prime number, greater than 12 and red”. I don’t know how many of these there are. I’m not looking now either. You can’t bet or wager on it either (at most all the numbers in question individually). It becomes impossible, however, if one should say prime number and divisible by 7. Because a prime number, of all things, has no divisors except 1 and itself.
You could even take the numbers with the checksum 8, all numbers that contain a “b” or all numbers with two “N “s (e.g. six-twenty, two-twenty). The variety here is almost infinite. And there is always a probability between 0 and 1 for the occurrence. If one should say “Any number between 0 and 36 will occur” then in this sense it is the “certain event”, if one should take “prime number and divisible by 7” then one has the impossible event. So the probabilities 0 and 1 are also represented.
In the case of the lottery, I don’t want to start another special investigation into what events can be teased out with 6 numbers plus an additional number (example?). All right: “At least four of the seven balls drawn have a closed rounding in the notation.” By the way, that would be the numbers 0, 6, 8 and 9. Please don’t argue! What is round? And what is a notation? It’s called “notation” etc…), but I would like to have at least two points mentioned: Combining the 6 first numbers alone, without the additional number, offers a sufficiently large variety of 13,983,816 possibilities. And secondly: Before the draw, one is always assured that the machine is in proper working order. I will examine the relevance of this well-known fact a little later.
Now I have perhaps provided a few more or less illustrative examples of “events”. However, I still owe you a definition, both of event space and of the event itself.

The event space comprises all possible outcomes of a random experiment. The mathematician likes to write it in the form, and let’s stay with the dice for simplicity’s sake:{1,2,3,4,5,6}. This is a quantity notation. The numbers 1,2,3,4,5,6 are the set of all possible outcomes in the random experiment “Roll a die once.” The resulting possible subsets are the events. So each subset of the set {1,2,3,4,5,6} is an event. That is how it is defined.

Above I have verbally paraphrased the ways in which subsets could be formed. All prime numbers from the set {1,2,3,4,5,6} is the set {2,3,5}. All numbers greater than 4 is the set {5,6} and so on. Certainly, one can also write down a subset just like that, without any recognisable commonality. I’ll take the subset {1,2,6}. You could either simply say “the numbers 1,2 or 6 interest me” or the numbers “less than 3 and greater than 5”. Of course, I chose them differently: They are the numbers between 1and 6 that contain a sibilant. Checked? Good. But even without commonality, they form a subset and thus an event.

But the fact remains that the mathematician has done a good job so far. All subsets that can be formed from the possible outcomes of a random experiment represent the possible events. If you include the empty set {} and the total set {1,2,3,4,5,6}, i.e. the certain and the impossible event, then you can also calculate quite well how many possible events there are. There are always 2^n. So when rolling a die, 2^6 = 64 possible events. (Please read the calculation of this elsewhere, in the chapter “”; in any case, it is the sum of the entries in the 6th row of Pascal’s triangle. And these are 1, 6, 15, 20, 15, 6, 1. Added up? Great, keep it up. Formulated, these numbers represent all zero-element subsets, all one-element subsets, all two-, three-, four- , five- and six-element subsets; zero-element subsets there is one, one-element subsets there is six, and so on).

The formulations one can come up with, as partly done above, form an almost endless variety. The resulting subsets, however, are bounded by n in the form 2^n. This is because one is guaranteed to describe identical subsets at some point with different formulations.

Please note here that the individual outcomes of the random experiment also represent a subset, i.e. they are also events at the same time. They are all one-element subsets. This should be taken into account insofar as the terms are unfortunately sometimes mistakenly used synonymously in everyday life.

e. Probabilities

I do not want to make any calculations here. I merely want to mention that the concept of probability will also be necessary sooner or later and make a few small preliminary remarks about it.
The individual outcomes of a random experiment each have a probability of occurrence. So do all the other subsets that we can put together at will from the possible outcomes.
These individual probabilities add up to 1. This was the content of the statement “there will be a number between 0 and 36” in roulette or the statement “a number 1 to 6” in dice rolling. For these are all possible outcomes for this event space. A number will come. The sum of the probabilities must be 1.
However, if we examine such events as defined above, i.e. other subsets, then we have to be a little more careful with the sums of the probabilities. We then have a great variety of events whose sum is by no means 1, but of course much greater. For the events “the die shows a 4” and “the number is even” and “the number is greater than 3” (also any number of others) can occur simultaneously (in this case obviously, and precisely when the die shows a 4). Prudence, however, is very simple to maintain. In the simplest case, one takes the event and its counter-event. So “the die shows a 4” and “the die does not show a 4”. These two events then add up again, mathematically cleanly and correctly, to 1. In other words, I am sure you will agree: The statement “The die shows a 4 or it does not show a 4” is guaranteed to be true. The probability of occurrence is 1 (it’s like my dog, Waldi, who obeys me to the letter. If I say: “Come here or don’t” then he comes here or not).

Here, too, there is the possibility of making the whole thing a bit more complex and still keeping to the mathematical laws. To be on the safe side, I’ll stick with the cube, because we can just imagine it so well. You could also take events like: “The cube shows a number smaller than 3”, “The cube shows a number larger than 2 but at the same time smaller than 5” and “The cube shows a number larger than 4”. With all three events together, we would obviously also have covered all numbers and no overlap. So the sum of these three individual probabilities will also add up to 1, and we have not argued with the counter probability. The first event here would be the numbers 1 and 2, the second would be the numbers 3 and 4, and the third would be the numbers 5 and 6. So we can see that for each outcome of the random experiment, one of the three events can subsequently be attested to the occurrence of the same as “true”, having occurred, and the other two as “untrue”, not having occurred.

The possible outcomes of a random experiment can therefore be broken down into individual events as desired. What remains to be considered, especially for later correct calculation, is that the sum of the individual possible outcomes, in the sense of this event, must add up to 1. The simplest way to achieve this is always to say: “The event occurs” and to contrast this with “the event does not occur”. The sum of these two is guaranteed to be 1.

In mathematical definitions it looks like this: Take a subset A. Then there is the counter-set A. So if we look for the counter-set of {1,2,3}, then we find all the elements that are not in it, but are still a possible outcome. So A is then {4,5,6}.
When you use the word “and” you mean, mathematically, the intersection. So if you intersect the sets A = {1,2,3} and B = {2,4,6}, then you get the set {2} as the intersection. So A intersects(I am missing the sign, an inverted U) B is {2}. The fact that a spoken “and” results in a reduction of the set is logical and plausible insofar as it is pronouncedly required that the event and the other also occur. And that obviously complicates the matter.
However, if one says “or”, then it is the union set. So with the sets A = {1,2,3} and B = {2,4,6} the union results in A U B = {1,2,3,4,6}. That a spoken “or” results in an increase of the set is equally logical, because one allows for more. So this or that can come. This increases the quantity of the individual outputs that provide for the occurrence of the event A U B.
This only deserves mention insofar as the words and and or in set theory provide for otherwise unfamiliar effects: the word and, which is otherwise also sometimes used synonymously with “plus”, provides for subtraction, reduction. The or may not have an automatic association in this sense, but one would not necessarily suspect an increase, would one?

2. the evaluation of a random experiment

The events that we have described above can be examined for their occurrence on the basis of the outcome that occurred in the random experiment.

Example: The dice showed a 4. That was the outcome of the random experiment. Now we assign a truth content to the events, in the sense of “occurred” and “did not occur”. We had the event “The die shows an even number of points.” This event has occurred. In other words, the statement “the die shows an even number of points” is true. We check the event “The die shows a 5.” The event did not occur or the statement, “The die shows a 5” is untrue. Likewise, 4 is not a prime number. The statement “The die shows a prime number” is therefore untrue. In the same way, the event “The points on the upper side of the cube are arranged point-symmetrically” has occurred. In each case, the counter-event then occurred or did not occur accordingly.

Summarise these two points again here:

1. should one want to formulate, i.e. verbally describe, what kind of events one is looking for, then there really is an almost endless variety. However, many of these formulations would result in the same subsets. The number of possible subsets, i.e. the number of events, is limited by the number of possible outcomes, i.e. by 2^n.

2 The possible outcomes of a random experiment ensure that defined events can be assigned a probability before the experiment is carried out and a truth content in the sense of right or wrong, i.e. 1 or 0, after the experiment has been carried out.

The concrete calculation of probabilities will be carried out a little later.

3 The LaPlace Experiment
This experiment represents an idealisation. In theory, there are n outcomes for every random experiment (throwing a die: n=6, throwing a roulette ball: n=37, flipping a coin n=2). In the LaPlace experiment, these n outcomes are all considered equally likely. This is what is called the “idealised probability space.”

It is not unsuitable for illustration. With the help of the LaPlace space, as it is also called, one can wonderfully and easily make numerous calculations that are also applicable to it. Reality always looks a little different, but it can be helpful to know the basic considerations.

4. my first small objections
   a. The actual event space
   I would at least like to have it mentioned that even the assumption that there are these n possible outcomes is itself already a form of idealisation.
   There are a few more possible outcomes, depending on the setup of the random experiment.
   When tossing a coin, for example, my friend Micha always liked to say, partly jokingly, partly to illustrate his lack of chances, “Come on, we’ll play heads or tails. I’ll take edge.” So theoretically, the coin really can land on the edge. That is simply not covered in the event space with the possible outcomes. It should actually read: In the coin toss, there are the following possible outcomes: heads; tails; edge; the coin disintegrates when it lands; the coin is lost because it rolls away; the coin “burns”. Can you think of any others?
   All of these can easily be applied to other random experiments. What do you do in practice? You repeat the experiment, of course. But you have no idea how often this leads to discussions. The cases don’t bother you until they happen. When did the cube burn? “But it doesn’t burn!” Oh dear, and just when money is at stake! An event has occurred that was not foreseen in the event space.

Albert Einstein:” If at first an idea is not absurd, then there is no hope for it.” If at first an idea is not absurd, then there is no hope for it.

I describe the role these considerations play in practice with some examples in the chapter “Practice of the Event Space”.

b. Past and future

This is also a purely practical aspect that bothers me about mathematics. It is indeed the case that random experiments of some kind are carried out ubiquitously. And something comes out of it. The inanimate definitions, on the other hand, succinctly state: an event is to be considered as having occurred if the outcome of the experiment is contained in the set.

Yes, has it occurred or not? It is if, but at some point it has to be. It somehow lacks practical relevance. Did this paragraph bring anything enlightening? I’m not quite sure.

It’s just that during my entire time at university, I either did practice or missed it. And this allergy to a lack of practice in mathematics has always put me off, despite all my admitted affection for it. Or, let’s put it this way: I have only done practical and applied mathematics. There is at least one side to mathematics that I love and a few others that I willy-nilly have to accept.

c. Between perfect calculability and absolute chaos

LaPlace himself, by the way, was well aware that this was only an idealisation. On the contrary, he even postulated the absolutely opposite assumption:
I quote: “A demon who knew for a given moment all the forces at work in nature as well as the mutual position of all the atoms, and who, moreover, would be astute enough to subject the given quantities to mathematics, would be able to comprehend in a single formula the movements of the largest world bodies and the lightest atom. Nothing would be uncertain to him, and future and past would lie open before his eyes.”

Well, reality, as usual, moves somewhere in between. There is some truth in both considerations. One aspect is discussed in the following section (who has what intention?). Part of it is also considered in the chapter “My Chaos Theory”.

Here I would just like to briefly mention that the philosophical considerations dominate. Absolute chaos, i.e. this part of chaos theory, describes complete non-predictability in mathematical terms. Everything is equally probable. Anything can happen.

The opposite perspective is that if all the parameters were known, everything could be calculated exactly. So in a given random experiment, there are very specifically certain conditions under which the experiment is conducted, at the very moment it is conducted, that make it predictable, and even exact. A random experiment is pure theory. It is the execution that makes it so. So it is predictable after all, at least in theory?

The physicists immediately jump out at me. And rightly so. Since Heisenberg, we have known that it is not possible to determine the location and speed of certain small particles at the same time (when Heisenberg once drove too fast in his car, he was stopped. The police officer: “Do you know how fast you were going?” “No, but I know where I am”). So it would not be possible, even theoretically, to calculate exactly the outcome of a practical random experiment, even if all the parameters were known. As I said, the transitions to philosophy or even religion are fluid in these areas. Was Piet Klokke possibly right after all when he sang: “This is fate, everything is predetermined.” Or is it simply a world view?

In any case, it seems clear to me that truth is somewhere between exact predictability and absolute chaos. I’ll provide you with another example here: If you take a cube in your hand and intend to throw it now, then I could possibly take a look at it. Then I might see along which axis you throw it, roll it. It is conceivable that the numbers that lie on this axis are slightly more likely to fall than those that lie on the outer positions. The probabilities could be slightly shifted. Is it possible to imagine something like that? Anyway, this leads on to the next section, the question namely….

d. Who has what intention?

It always remains with all these questions that philosophical considerations somehow always play a role. I would like to bring this closer to you again in the form of mentioning that the problem has existed since time immemorial. Something happens. One looks for explanations and reasons afterwards. And all you find is the inevitability, the unavoidability. It has happened. It cannot be undone. Was it fate, destiny, God’s will, chaos, simply coincidence? The question always remains the same: Could it have happened differently? The answer so far: no. In this context, I remind you once again of Biff Tannen, Back to the Future II.

You can never try the alternative. Of course, this also applies in this form to real, intended or even influenced chance experiments. “I knew that 32 was coming up at that table back there.”

Speaking of roulette: Rudolf Taschner writes in his book “Zahl, Zeit, Zufall” (Number, Time, Chance) that it is precisely in roulette (also in the construction of a dice) that the intention is to make it a pure and, as far as possible, equally distributed random experiment. I quote: “…the precision of the game, the subtle effort that the casino makes to provide for chance, and only chance.”
The organiser has a pronounced interest in ensuring that the numbers really do fall purely by chance. In this case, it most certainly guarantees the casino’s long-term profits. The game should be unbeatable for the player, nothing predictable, nothing calculable. That is why the profit margin is calculated very small, because it works so reliably. That makes it attractive for both players and organisers.
The lottery is a little different. Even if the people in charge are supposedly convinced of the “proper condition of the machine”, in the end they don’t care. Only part of the money is paid out anyway. The payout quotas are calculated according to the stakes received and the number of winners.

Nevertheless, I would like to remind you that there are people who manage to spot the mistakes in roulette and make a living out of it by betting, wagering on it. So even if the device is at least intended to be exact, the organiser does not necessarily succeed here in guaranteeing this equal distribution. The person watching has the chance to estimate the speed of the ball, to predict its course, even if only the rough direction, the area of the cauldron where it is more likely to hit, because he is there live.

Another case is when you play backgammon, for example. Then it may be that the equal distribution is reasonably assured (I remind you that in big tournaments they always play with precision dice), but I am not interested in that. That is a very essential point. If I need a 1, I would like to roll it. My opponent may either rely on mathematics or on his influence by jinxing my dice. But he certainly does not want me to roll a higher chance than I am entitled to. Either, at best, this one or a smaller one. So neither of us is interested in equal distribution.

Whether these intentions can in any way have the slightest influence on this may remain an unsolved mystery for some time to come. But I can assure you that practically every player, amateur or professional, has his thoughts on the matter. Sure, most are convinced they have bad luck (it’s different with me: unlike those who only believe it, I know it).

I have explored thoughts about luck and bad luck in some detail elsewhere (for example, in the chapter “On Luck and Bad Luck”). But a few extra thoughts can’t necessarily hurt: Every player is wringing his hands, looking for some way to influence events. It would be so nice, so tempting, if one could discover something that works reliably: Reading coffee grounds, palm lines, horoscopes; whenever I put on this jacket, I’m lucky; my lucky jumper (Udo Lattek); the pen I used to write down the chess game: I won with that, I’ll use it again. That is endless. It is the so-called superstition. And the danger is not in being superstitious. The only danger is that one actually makes relevant decisions dependent on it. I don’t know a general solution to this problem either. Only this much: my worldview dictates that I do not include these considerations, however much they may intrude from time to time, in decision-making. If it’s a coin toss decision – and that’s certainly not uncommon – then a tiny thing may well tip the balance. Above all, there is an argument for superstition: if you think you are so sure that it doesn’t help, then logically it can’t hurt in any case. That would be a real paradox!

But in any case, I accept if someone has a different view of things and just and always makes his decisions in this way. First of all, I cannot judge whether he is successful in doing so. Because that is conceivable. So why should I criticise him for his decisions? Secondly, it is possible that it makes him happy, even though he is not successful. It is his path in life. Then let him follow it. I am taking a different path and trusting in mathematics. Although in so many places I also despise it (and it me). But isn’t arguing and reconciliation part of a good relationship?

A friend and former partner had studied psychology. He had a plan at the time to do a study to see if you could have any kind of influence on the exits. He wanted to carry out a gigantic experiment in which a group of people would try, let’s say for example, to roll a certain number, of course under the most correct conditions possible (i.e. precision dice, dice cup etc.). If one succeeds, one gets a reward or overall winners and daily winners are determined or something like that. Then he wanted to hand in the results as a thesis. Of course, it would also be a result if it turned out that this is nonsense: you have no influence.
But ask any player. Everyone will be able to tell you a story, a theory, observations on the subject that at least keep open the chance that they do believe that there are other things between heaven and earth that have an influence on the outcome of a random experiment than just plain mathematics.

Elementary Combinatorics

What is your impression so far of mathematics in general and probability in particular? My guess: A lot of blah-blah and a lot of gaaaaaay. But surely not in the way you expected?

As they say, let’s “put our money where our mouth is”. So a few arithmetic operations, I’m already all fidgety.

No, it still doesn’t work. You first have to formulate the elementary law: The probability of occurrence in the idealised probability space is calculated by dividing the number of outcomes that are “favourable” for the event by the number of outcomes that are “possible”. So, in short, favourable/possible.
The event, roll an even number of dice in the random experiment “roll a die once” is the set {2,4,6}. That is a total of 3 outcomes. The possible outcomes are (ideally; see above) 6. So the calculation rule is favourable/possible, we have an arithmetic operation: 3/6. Now 3/6 after shortening is 1/2 or also 0.5 or also 50%.
Somehow I have the impression that this was not directly new to you?
Throwing a very specific number is an outcome, so you then divide 1/6 and so on. In roulette, the calculation is not much more complicated, but the resulting numbers are more unwieldy. So the event of getting “a number divisible by 3”, between 0 and 36, is 12 possibilities. However, stupidly, you now have to divide this 12 by 37 and get 12/37. No shortening or anything.

6. just one last definition
   We have defined all possible terms, but not yet the term “quota”. Maybe you already had an idea about it? Are you sure you know exactly? Or only vaguely?
   In any case, I am striving for further enlightenment, also for myself (I always maintain that you can tell if you have understood something if you can also explain it).
   In principle, the odds we are talking about here refer to bets and payouts. One could also generally always call such an odds “payout odds” (which may also occur here and there).
   Many people I have met do not have such a concrete idea of an odds. They usually bet for “even money”, often not even knowing that. “I bet 10 euros that I…” then actually automatically means “against 10 euros of someone else.”
   The odds I want to talk about here and henceforth are payout odds in the form:
   You bet on an event, preferably in the future. You give money to the provider of the odds. Then the “random experiment” is carried out. The event on which you have bet occurs, in the best case. The provider pays you an amount. This amount is calculated by multiplying the stake and the odds on which you have bet.
   This type of odds is also called “European odds”. The assumption is that the provider of the bet is running a business. He guarantees the payout. So the bettor, the bettor, has to put in the money beforehand. The stake must be placed. The provider then pays back the stake plus the winnings.
   In the concrete example: Today you bet Bayern to win against Florence, away (today is the game, 5.11.2008). Odds 3.80. Bet 10 euros. Bayern actually wins. You get back 10 * 3.80 = 38 euros. Net profit, however, “only” 28 euros, please always bear this in mind (I often asked my mother what my final financial result was on a day/weekend. I answered e.g. “I won 3000 DM.” Then she asked back, “Aha, and how much did you lose?” 28 Euros is the net result. For my mother to understand, they would have to answer her “I won 38 euros and lost 10 euros”).

Note: There are other types of odds in England. There the one unit for the stake is not deducted. So an odds of 11/10 would mean that you risk 10 units, the provider in turn 11. However, even there you usually have to deposit the money. So you pay the 10 units in advance and get back 10 + 11, i.e. 21 units, in case of success. In European terms, this would correspond to odds of 2.10. 10 euros stake, won, 10 * 2.1 = 21 euros payout.
There are even crazier odds in America, typical America. But you can see from this an advanced understanding of betting. But I won’t explain to you how that works now.

But now let’s get on with it: How are the odds really created?

Well, hopefully you are now “warmed up” enough for me to present this calculation to you easily and without you running away: The odds are primarily the reciprocal of the probability. You get a reciprocal value very simply by writing a 1, then making a fraction line under it and then writing the number of which you want to have the reciprocal value under it (excerpt from the book “Mathematics made easy”, published in 2036 {self-published}, author Dirk Paulsen; total edition 2, yes, right, 2, my children took one; even Otto could have contributed to this with the book “Mathematics for Runaways”, mathematics for advanced learners, before they run away).

So the odds on the event “throwing a certain number”, shame on me, the event “rolling a 2” (first of all I’m not a mathematician and also modest, therefore not always just the 6) with the probability 1/6 should give a payout odds of 6/1. 6/1 can also be written as 6.0.
Now two things might have irritated you here. The phrases “the odds are primarily…” and this one “…. would have to be a payout ratio of…”.
I wrote both because, once again, it is a question of who has what interest. I introduced a term for this original formulation “the reciprocal of the probability is the odds”, which I myself have been using ever since. I then speak of the “fair odds”. The fair odds are thus the — for the ideal probability space — correct payoff odds at which neither side would profit in the long run.
You can check this relatively easily: So if you throw the much-quoted die, say, randomly 120 times, you would roll 2 about 20 times. If someone pays you the odds of 6.0 on it, then the following calculation results: 100 times you lose the 10 euros. This makes a loss of 1000 euros. You win the remaining 20 times. You get back 60 euros each time, but the net profit is only 50 euros each time. 20 * 50 = 1000 euros. So you have lost 1000 Euros and won 1000 Euros. The game ended in a draw. And this is how it should be in the long run, with the above-mentioned restrictions: you will play at about par, a draw. Neither side had an advantage. 6.0 was the correct payout odds, in that sense therefore the “fair odds”. It was fair, wasn’t it?
But the perspective here is that one usually depends on a provider to be able to pursue one’s passion, betting and gambling. This operator has to earn something, that is market economy. Playing among friends would relatively quickly break friendships, become boring or, most likely, dry up. (“You want to play Bavaria tonight? Not with me. I’m playing them too!”)
So in my terminology there is a “fair quota” and a “pay quota”. Since I have also tried my hand at betting, without much success (see the chapter on “my betting shop”), you calculate the probability of an event on which you want to offer odds, take the reciprocal value of that, so you have the fair odds, and finally deduct a profit from that.
In the concrete case of the dice, the offerer would just as nimbly turn the 1/6 into a 6.0, then, trusting the nature of the game and its balance, subtract a little from it and then write down odds of 5.50 or also, more cautiously or greedily, a 5.0.
That would, at least in theory, give him a long-term advantage.
Have I reached my goal now? Yes, I think you are. I just made a quota, didn’t I? And before that it came into being, right?
But still, I can’t do without it, on

8. this cruel reality
   to this cruel reality. Because in reality things look different from the beautiful, ideal world of the mathematician, and quite different.
   Further above I have already drawn attention to the case where the event space does not keep its promise. Events occur that one simply did not expect, could not even expect. Obviously, however, it was a possible outcome of the experiment. It just hadn’t been considered.
   The consequences of this are not entirely insignificant in the world of gaming. And I refer you once again to the chapter “Practice of the Event Space”. There are really quite a few absolutely curious cases.
   And even our cube, which is so handy, is already having problems. It burns. The case is still okay. It rolls down somewhere where you can’t see it. But at least one side is up. Does that count or not? They bring it up, but who? Whoever picks it up twists the side. Well, problems upon problems.
   But the crucial effects are on the probabilities. There are no LaPlace cubes. There is not even anyone who would have the intention to roll balanced dice. Maybe, chance sets it up that way, world view. But the perfect cube, the ideal random experiment does not exist in practice.
   There is always a room temperature, a cube base, a throwing axis, a rolling friction and a throwing speed. And even with perfect intention to throw the dice “correctly”, there would be a few parameters that would be guaranteed to have an influence, even a calculable one. There are even centre of gravity dice, please don’t forget!
   And what would you have in addition in terms of results, if carried out many times? Nothing more than relative frequencies. Probabilities no longer count, they practically don’t exist. Imagine you are doing a “real random experiment”. You take the dice. You throw it (I’ll do it in a moment). It shows a 4. You stop the experiment. Once thrown. No statistics, nothing. No prior knowledge. I don’t even look at your dice. And I dare to make a statement, no other would make sense, think about it for a moment. My statement: “The most likely number is four.”
   I don’t even know if there are other numbers on it. And whether perhaps this side is heavier, that is, a centre of gravity cube. What else am I supposed to conclude?
   But, well, I’m ready, I’ll now, without a notary, throw my cube 100 times here. I note it down quite correctly. I’m not trying to control anything either. But I’m only doing that because of you. Because secretly I’d like to have a six, it’s from my childhood, the six in Parcheesi and you were allowed to roll again. Did that help? So here are my numbers: Number of dice/frequency

		Number of dice/frequency
1	2	3	4	5	6
16	19	17	16	13	24
	Total:	105
		Relative frequencies
1	2	3	4	5	6
15.24%	18.10%	16.19%	15.24%	12.38%	22.86%
		Expected frequencies
1	2	3	4	5	6
16.67%	16.67%	16.67%	16.67%	16.67%	16.67%
		Deviations
1	2	3	4	5	6
-1.43%	1.43%	-0.48%	-1.43%	-4.29%	6.19%

This is the practice. First of all, I had a purely practical problem: I miscounted when rolling the dice. I couldn’t reconstruct the last 5 throws, so I increased (left) the attempt count to 105.
And my training in youth seems to have helped: I still do 6s best. And I just couldn’t get my mind off the 6, it’s programmed in.
There’s just some result. If you want to draw any conclusion at all, it’s that this is the correct distribution appropriate to the experiment. It’s the best we could get, so far, mind you. If I were to repeat, there would be a new result, possibly totally different. But who knows, maybe I changed the room or the base. Or I threw higher, harder. Or shorter, in order to finish faster. Or I always turned the cube when I picked it up, why not? What one doesn’t do to live up to the statistician’s expectations. But it is also possible that the “relative frequencies” that occurred here would be confirmed once again. Why not, actually?
But the statistician would be speechless on the whole. He would only have to make absolutely terse statements, none of which would serve any purpose. He would say that certain deviations are normal. One simply tolerates them. And he would also tolerate greater deviations. But then he would sometimes start to wonder a little. Or calculate the probability of such a large deviation occurring.
But I would really provoke him and ask: “Ok, you tolerate this deviation. But what did the numbers deviate from?”
Answer, but already intimidated, I know these people: “Well, deviation from the expected value.” Good, now it’s my turn again: “How do you know the expected value?” Well, he is a conscientious mathematician. He even takes the cube in his hand. He examines it. He says, “6 sides, all look similar, only each side has a different number of dots. These range from 1 to 6, each represented once.”
“Aha, that’s how you do it. And what did you do afterwards?” “Well divided, 6 possible outcomes, each 1/6, makes a total of 1.” “Very clever and wise. But who threw when, how, where and with what intention, what ulterior motive and how at all? Why then, if different numbers of points are already represented, should the sides have equal weight…?”
All the mathematics takes place in theory. That’s not so bad, as long as you always and constantly know where you’re pushing the envelope.
I also know that the assumed 1/6 is a pretty good approximation for estimating the probability of the die falling on one of the 6 sides. But it is certainly not exactly 1/6. There are many influencing variables. Many different experimental setups. Many different people who can carry out the experiment in different ways and with different intentions. And in practice there is only ever the one throw that counts right now.

But now we are still in random experiments, which have a reasonably clear order.

Look at the reality of real and actual betting. The event spaces are not “ideal” in any single respect. There is a football match in which 11 players of one team play against 11 players of the other team. Then there is a referee with his assistants, an audience, a pitch, a weather (yes, it’s raining today), a ball that has a certain hardness and moves on the ground or even in the air, controlled by an absolutely confusing number of parameters that no human being will ever know. And then there are the intentions of the individual participants in this game. The round has to go into the square. And of course the counter-intention: the zero must stand. And there are abilities that vary individually and also as a team (I quote a famous German coach and philosopher: “a team is more than the sum of its individual players”).
And then someone is supposed to calculate probabilities and odds? Absurd and out of the question? Well, at least I found a way. And there are bookmakers, too. Only: we are all playing a gigantic guessing game. Still, one can be better or worse. No, I caught myself: one can be more successful or less successful. Whether one was “good”, better than another, is again only measured “statistically”. This “statistic” works like this: counting money.

This can be shocking but also reassuring. No one knows the right estimates. Everyone guesses. Feel free to participate. Just form opinions and play along. It’s really not that difficult. I’ll be happy to give you a few tips, but elsewhere.

You can also use the reciprocal value in the other direction. You take an odds, let’s stay with the Bavarians for the 5.11. The odds are 3.80, they actually exist (again, just to back up “guessing game”: different providers pay different odds). Then we take the reciprocal. 1/3.80 = 26.32%. If you think Bayern have more than 26.32& to win, you would have a good bet. If you think Bayern has less, you can of course still play it. For fun. Or out of conviction. Because what do Bayern care about these percentages? “Mir san mir” and win. So up with the dough!

Personally, I’ve read this chapter for the tenth time and still haven’t learned anything. But that doesn’t matter. It reminds me of Eckart von Hirschhausen, who once gave some clever advice on improving the quality of life: “If you look in the mirror tomorrow morning and someone smiles at you, just smile back. You will see. It helps.”

Exactly.