First of all, let me summarise again: The whole of life is a game of probabilities. Even God is sometimes said to play dice. So you see. No one can escape this game, everyone takes part. And everyone bets. You bet too. The state even tells us to bet (compulsory insurance). Here, the payout ratios are obscure, often incomprehensible. (the last three insurance agents who visited me always asked for a handkerchief after some time; and I had not overheated).
So, unfortunately, we have to spend a moment on what the correct connection is between the calculated, guessed, estimated or even known probability of occurrence and the payout ratio.
Quite obvious to me at first seems to be the connection: small probability – high odds or high probability – small odds. For example, “Greece becomes European Champion 2004” is a very unlikely event. So you would get high odds. It is possible that you would get 40.0, i.e. 40 times the money paid out. On the other hand, it is an absolute misconception that, as one likes to read in the newspaper, “the bookmakers really had to bleed” or some such nonsense. The “normal bettor” bets on the events that he thinks will happen. So certainly not Greece. The odds are secondary. Slightly advanced gamblers might play an outsider’s tip, because you can win a lot of money at low risk. Nevertheless, the stakes are usually kept within limits and remain small. Favourites are sometimes played at a high price. So for the bookmakers, betting providers, there is nothing better than such an underdog success.
Since the probability and the odds increase in opposite directions (one up – the other down and vice versa), mathematicians like to speak of an inverse or reciprocal relationship. This is expressed mathematically as a reciprocal value. So the reciprocal of the probability is the ratio (and the inverse also applies here). The reciprocal itself looks like this: 1/ratio = probability or 1/probability = ratio.
Now, of course, this is only correct to a certain extent. Because someone would like to profit from the bet. Most of the time, both would like to, but in a mathematical sense, there can only be one side that has the advantage. There is one side that wins in the end and one side that has the advantage. Whether these are congruent depends to a large extent on the much-quoted coincidence. And the attraction of gambling and betting lies to a large extent in the fact that a) you don’t know beforehand and b) that occasionally the wrong side wins. If, as in chess, the better, the “right” side were to win with nice reliability and regularity, the game itself, this particular one at least, would become boring very quickly. Apart from that, the problem of proving the advantage, analysed later, remains.
One can argue long, much, well and also cleverly: If you lose a bet, you have to pay it (see above, if reliable). Advantage or not. If you win it, you can let the arguments bounce off you with relish. “You talk as long as you pay. I won the bet. And you argue. Which is better?”
So it is that an event has to be estimated in terms of its probability of occurrence so that one can make any odds at all. In addition, all probabilities that are possible outcomes of a certain random experiment (usually sporting events) have to add up to 100%. In the simplest case, one could always make do with this: The event occurs: 1 per cent. The event does not occur: 1 percentage. (More on this topic in the chapter “
a) Betting for “equal money
The way you may have been betting among friends so far, you have been betting virtually without odds. This means that everyone risked the same stake. So when you said, “I bet 10 euros that…”, you basically meant who bets 10 euros against it. And the bets often come about when people bet on facts. Everyone is convinced that it will happen as they remember. One bets and then looks it up. The loser pays (if he has a good payment record). This bet is for equal money, so to speak. That’s how it’s said in gambling parlance. But “I bet equal money that…” already expresses a higher understanding. Because the same person could also say “I’m not betting equal money, but I’m so sure that I’ll even pay you a quota.”
Such conversations also occasionally take place in gambling circles. Nevertheless, it is urgent to distinguish between betting on established facts and betting on future, mostly sporting, events (but nowadays one can also bet on the outcome of the Grand Prix d’Eurovision or the Bundestag elections).
I actually like to use betting on facts whenever someone is talking nonsense. And of course I only imagine that it is nonsense. But I once heard the phrase “there are 400,000 Greeks living in Chicago.” Then I would like to say, “Please, let’s bet, we’ll look. I say it’s less, we formulate the bet and change the topic of conversation.” Or: “Hertha didn’t play against Alsenborn in the last game of the 1967/68 promotion season, nor did they play 1-1.” That was my last such bet, by the way.
Of course, betting without odds is still betting with odds. Because the actual payout ratio, expressed as a bookmaker’s odds, is a 2.0. In bookmaker language, 2.0 means, and the bookmaker is the one who “holds” a bet, that if you bet 10 euros, you get the stake multiplied by the odds paid out. So stake = 10, multiplied by odds = 2.0, gives 2 * 10 = 20 euros. So you have lost 10 euros if your prediction does not come true and he, the bookmaker, provider, has lost 10 euros if your prediction comes true.
So that was a bet for exactly the same money. He loses 10 or you lose 10, odds are 2.0. There are two reasons why it was worded this way: The first is that the bookmaker first takes the money, that is, collects it. He has his official licence and his (earned) reliability, which you would have to rely on at that moment. Among friends, however, it would be silly to say, we bet, but you have to deposit the money with me first, i.e. pay your possible loss beforehand. Neither of the two is the provider or bookmaker. You bet and hopefully you trust each other.
The second reason is that you can combine several predictions. And to calculate the profit in case all predictions (you can also call them tips) come true, you may multiply the odds together. The underlying law is explained in the chapter “independent events”. Nevertheless, it is mentioned here. It simplifies the calculation or, better still, makes the calculation possible.
Among friends, you occasionally hear phrases like “we bet 1:1” to express the phenomenon of “equal money”. And traditional English bookmakers still express their odds this way. Equal money for them is still 1/1 or even, more classically, evs. The evs. Stands for evens, meaning equal or even money. Just now, Wednesday 24.9.2008, I called an English bookmaker to find out if they still write down the prices in this form. And they do, in fact. And the only exception where it is not noted as a decimal fraction is at evs, same money. For combination bets and their calculation, 1 must still be added to each odds.
b) The calculation of an advantage bet
So if you go to a betting shop today (outside England), you get odds to bet on. These are what are known as fixed odds. The odds are fixed beforehand. You get paid a guaranteed amount when the event happens. So if the odds for the Bayern -Schalke match are 1.70 on Bayern winning and you bet 100 euros, you will receive 100*1.7 = 170 euros if the match is won, i.e. a net profit of 70 euros. What would the probability of occurrence have to be for the bet to be “worthwhile”?
Well, of course you can also deduce that. So let’s say the game takes place again and again, you can bet it 100 times in a row. Then you would have bet a total of 100 * 100 euros = 10000 euros. To get this back, you would have to be paid 170 euros so often that this sum exceeds the 10000 euros, i.e. 10000/170 times. That would be about 59 times. We check this: You get paid 59*170 euros, i.e. 10030 euros. The other 41 times you get nothing back. But still, you would have profited. 30 euros earned. So, if the probability of Bayern winning the match against Schalke is higher than 59% and a bookmaker offers you odds of 1.70, then you would have just had a profitable bet. The bet is “worthwhile” from 59%.
And you can believe me: nobody really knows this probability exactly.
The worrying aspect: you do not know the probability of occurrence. The pleasing aspect: the provider does not know it either.
The bookmaker’s odds are not per se a bad bet for you. The man is speculating just like you. He does not know. And after all, I have managed to live on such bets for 18 years. In doing so, I have always set my probability of occurrence against that of the provider.
Apart from the fact that on closer inspection you realise that the bookmakers themselves offer very different odds. This already shows the accuracy of the statement: They all guess. But they don’t guess what will happen, they guess roughly how high the chance is. So the relationship between the probability of occurrence and the payout rate is like this: One is the inverse of the other. Everyone who gambles and bets naturally tries to play on the “right” side, as such that the probability of occurrence is in favourable proportion to the payout ratio. “Favourable” is the one who ends up in the financial plus in the long run if the experiment is carried out repeatedly. So it is clear that the person who offers 1.70 as odds has the hope that the ratio is favourable for him too. So, in plain German, he estimates the chance of victory to be “certainly below 59%” (he does not do this consciously, by the way; so much only for the qualities of bookmakers. They always just write down some values as the sum of their experience; but be careful: intuition is usually a good advisor, isn’t it). The more often you make profitable bets, the more certain it is in the long run that you will actually win.
Unfortunately, you can’t prove anything, or anything at all, in this business. I always say you can only count your money every day. If there is some, you did well, if there is a lot, you did very well, if it is gone, all of them, if you are broke, you probably did something wrong. But every single bet on a sporting event is only ever made at one time and on that one event. Whether it was good or bad cannot be judged conclusively. One wins, one loses. But the event is not repeated. It is impossible to produce long-term statistics. You can say, look it up, Bayern won 8 of the last 10 games against Rostock, but they all had different conditions.
You could also reformulate every bet that comes to a conclusion. We take the odds at which the bet is placed. Now let’s take a different odds than before, let’s say 2.50. The reciprocal of 2.50 is 1/2.50 = 0.4 or 40% (the % sign is just a synonym for /100, 40/100 = 0.4). The party offering the bet, i.e. “holding” it, claims, so to speak, that the probability is at most 40%, the party placing the bet, i.e. betting itself, claims that the probability is at least 40%.
By the way, you would be superior to most gamblers by mere knowledge and understanding of these two assertions.
The quality of a single prediction, a single assessment, cannot in fact be definitively verified. If a bet was made, one side won and one side lost. But the “wrong” side may also have won, the one where the odds/probability ratio was unfavourable. Surely, the person will then claim that you can tell that he was right because he won. But this is where things get philosophical again relatively quickly. For example, one can discuss the question “is there an objectively correct probability assessment for any sporting event”, or is there not? If the good Lord were to take part in the game, could one then say that he knows the true probability? With God, one would actually have to assume that he knows much more than just the probability of occurrence. So already the winner or some other final result. Or could he just say, “Objectively, the estimate is this: 45% — 33% — 22% (1-X-2).” The rest I leave to your fate. It will be “diced”, so to speak.
Here’s a quote I found in a book called ‘Alles Zufall’ by Stefan Klein: “Zufall is the pseudonym of God, if he doesn’t want to sign himself.”
This shows once again how close religious, philosophical and mathematical questions and considerations actually are to each other. And often a problem can only be answered with the help of one or both of the others. You would have to put it like this: “According to my world view, it is like this…”.
Mathematics itself, just to show the parallel, always helps itself in such cases with the axioms it has set up. These are the smallest, unprovable and unalterable assumptions on which the whole (lying) edifice is built. That was just a joke, of course: mathematics relies on a few axioms. Without these, one simply could not work at all. They are not provable. They are, so to speak, the “world view of mathematics”. Every theorem, every statement in mathematics, everything that seems to be proven and considered irrefutable should always begin with the sentence: “Assuming that my basic assumptions (axioms) are correct…”.
So I would have to formulate the one question that accompanies me through life in this way, and the paradox is already revealed by the terms used: “Is there a true probability?