#### Lucky (mushrooms) and unlucky (birds)

1) Philosophy of a point

A much quoted piece of wisdom goes like this: luck and bad luck balance each other out. This is a sentence worth thinking about. Even more than that, my reflections have unearthed a result: I doubt this wisdom. To put it bluntly, it is even impossible for it to balance out. If one were to go in search of an impossible event, one would definitely have an example of it here. I realise that the frown that has been caused demands an explanation for the sake of relaxation. I am ready if you are.

First of all, you have to open your mind to a certain statement, which unfortunately has a mathematical basis. I would also like to explain it with a practical example. I will take the example in which it became clear to me myself. The statement is: It is impossible to divide a loaf of bread in the middle, to cut it in half. I was also surprised at first when I heard that. But it is so. The impossible event, the probability of it is = 0. By the way, I learned this in the basic course of probability theory.

The naïve ideas we have about a point are just unfortunately, mathematically speaking, mistaken. A misconception. Perhaps the best way to understand this is to remember what a point actually is. Or rather, how big it is. “A point has no extension.” Unfortunately, that is the truth. It may be possible to define it, but finding it, quite unlike looking for a needle in a haystack, where in truth there is a chance of success (slight, I might add), is impossible. You can’t find the point, you can’t hit it.

I always make this clear to myself by imagining when I could be sure of having found it. And if we presuppose or assume infinite measuring accuracy, which is absurd in itself, then it would still be impossible.

So in the concrete case, I cut the bread at a certain point. Afterwards I measure whether it was the middle, i.e. exactly the point I was trying to hit. We can measure infinitely accurately for our example, but that doesn’t necessarily simplify matters. But it would still be necessary. So we measure. We have engaged two absolutely neutral people to do this. One is proficient in measurement, the other in writing. The task is also clearly formulated: “Please check whether the bread has been divided in the middle.”

So the person measuring always gives new places. He says in our concrete case, completely surprisingly, the following sentence: “The intersection of the bread is at…”, the notator listens attentively, ready to write, “… 0.50000000000000000000000000000000000000000000000000…”, , the notator notes, equally attentively. I also don’t want to insist that he is beginning to get restless. He will certainly ask himself at some point when the result of the measurement will be known, but he is still neutral and committed to his task. The measurer, what else can he do, continues to measure. He finds another 1085 zeros, the note taker notes them down. He even gradually gets into the routine of writing zeros. Then the surprise. He has finally come across a 6. Both seem irritated at first, but the second measurement confirms the result. But then they are rather reassured. If they hadn’t come across the 6 now, they might still be measuring today…. Of course, this is only a fairy tale, because how should one be able to measure so precisely?

With this, the exact intersection, the “point of intersection”, is still not determined, but at least the two can now stop measuring and noting. Because, it was finally confirmed what was certain anyway: the bread was not divided exactly in the middle. It was impossible, now it was also confirmed in this experiment.

Of course, there is always an accuracy of measurement. As long as you take this as a basis, it is of course possible, even calculable. But I’ll try to explain this aspect again with another example: You take a pencil in your hand, a very fine one, perfectly sharpened to boot. I draw you a straight line, but a really perfect straight line, of length 1 cm. Then I ask you to pierce the pencil exactly in the middle. You don’t even touch it correctly, you have also hit the middle, everything is perfect. Only, unfortunately, the disappointment: you have hit an infinite number of points. The one single point I was looking for actually had no extension at all. So: you hit the point, but you didn’t fulfil the condition. You simply hit an infinite number of points, of course you had a measurable probability, that’s clear. The centre point was one of these, that’s no wonder at least.

With the accuracy of measurement, of course, you have a chance anyway. If you measure to 1000ths of a millimetre and the pencil tip has an extension of 1000ths of a millimetre, the calculation is simple: one centimetre consists of 10 millimetres, so the total distance is 10000 units. The chance of hitting then if you happen to stab is 1/10000. Of course, if you have a very good eye and it becomes a game of skill where the intention is to hit the centre point, the probability can be much higher.

2) Offside

For me, understanding and internalising this idea has helped in other areas as well. For example, you look at a discussion in football about “same height – is that offside?” or “was that the same height?” with completely different eyes. Just for fun, let me try to contribute to the discussion:

We have a foot, a ball, two bodies and a clock running along. We want to measure whether the two bodies are at exactly the same distance from the goal line when the ball leaves the foot (for those not familiar with the rules: for offside, the moment of play counts).

I first ask for an exact definition of which part of the body is involved. Ok, let’s say we fix that in the rules. The centre of gravity is what counts. Would you prefer the foremost point? But, no matter how precisely you define it, the probability that these two points are at the same height is 0, guaranteed and incontrovertible. And if, when measuring, the difference is still 0 to this day, then a difference will be found at some point, unless life ends before then.

In contrast to the first example, however, there is at least one more factor in football: the time of measurement. So we would first have to define the time when we measure. Cleverly, the rules experts have written into the rulebook that the time of ball release is decisive. Unfortunately, this moment does not exist. In principle, the ball leaves the foot over a longer period of time. There is a phase in which it is guaranteed to still touch the foot. All well and good. Later, there is also a phase where it is guaranteed not to touch the foot. Also fine. But in between, apart from the fact that we can only ever determine the point in time, as above, within the limits of measurement accuracy, there is at least a phase where it is undefined.

However, if the measurement accuracy only works to a hundredth of a second anyway, the discussion is superfluous. Because the course of a hundredth of a second is already a phase, a period of time. But to even attempt to define here when the ball has “finally” left the foot is already unsuitable. Now we are combining two impossible events. But there we are lucky, it’s like infinity + infinity or infinity * infinity, it just doesn’t get any more. So also here. Impossible * impossible is nothing more than still just impossible.

Funny though: a few years ago the rule was changed. That was an initiative, a very daring one, by the rules commission. Since then, the rule has been: “Equal height is not offside.” After all, the experts have managed to hold heated discussions about absolutely impossible events for several months.

This is comparable to a bill that provides for baking a cake for everyone who has a birthday at Christmas and Easter at the same time. To then pass the law and then have a new discussion about changing the law so that in future he should get a cake with candles.

But why have I had this whole discussion here in the first place? “With that, ladies and gentlemen, I now come to our topic for today.”

3) Everyone is lucky — or unlucky.

Lucky and unlucky people really do exist. To prove this statement, however, I have to set a few preconditions: Every bet, every participation in a game, a lottery, a stock purchase, everything that is connected with money stakes and that can later produce a profit or loss, is, from a purely theoretical point of view, convertible into an equity. So everyone who stakes their money to gamble has a certain equivalent value based on the probabilities of occurrence, maybe a few other parameters (payment morale?). One could therefore convert every stake into equity. This would mean that each player would have expected a total payout as the sum of all these equities.

So if someone who regularly goes to the casino once a month or takes part in the South German class lottery or has already thrown 427 euros into a slot machine in his life, has played the lottery regularly for two years, has wagered a total of 1613.44 euros in the lottery, has wagered a total of 145 euros on his visits to the racetrack (cautiously but successfully), has held a few stable stocks for a few years and so on, then one could theoretically calculate his equity, the monetary value to which he would be entitled, exactly. And if then, as sorry as I am for this person, it comes out as equity that he should have got back only 65723.143725009871 euros of the total 86906 euros invested, because he made most of his bets at a (great) disadvantage, then at least one thing will be guaranteed: He did not get this amount back.

Maybe he even got more than he was entitled to, let’s say 69721 euros. That is absolutely not unrealistic. Then this person would be a lucky man, at least up to this point.

Sure, you may now object that this is nonsense, because recently his wife ran away from him and he is extremely unhappy. The luck or misfortune I am talking about here refers only to the financial aspects. It’s an interesting synonym anyway that equates “luck” in a playful sense with “good fortune” in life. One is simply lucky. Or just unlucky. It is and remains philosophy. “You just have to recognise your luck.” There’s some truth to that.

But let’s stay with the gambling and thus financial aspects. Everyone who uses money in his life in the way described above is either lucky or unlucky. That is still true today. Of course, the tide can turn today. Perhaps you will play the lottery this weekend?

To illustrate this with a concrete example, I once again carried out a small simulation. And I really enjoyed it. How do you feel about it? Well, I simulated a typical small gambler’s career. Also to make the fairy tale of “everything evens out” ad absurdum.

For my example, I chose a really good player who makes absolutely sensible decisions. He places his money in many and very different games. The probability of occurrence varies greatly. His betting amount also varies. He gets 250 units as a budget. He bets a quantity of 1 – 10 units per bet, more or less at random. Sometimes he has an advantage and sometimes a disadvantage. This is all randomly controlled, but he himself does not know this (of course, if he knew, he would only play the “good” bets). Nevertheless, he plays with a very slight long-term advantage. This is that he can estimate the quality of the bets a little bit. So if a bet is very good, he tends to bet a higher amount.

First of all, here is a small diagram showing a career over 1000 such bets:

There you can see what can happen to you. The purple line represents the equity. Due to the minimal advantage that the man has only because of his feeling for the really good bets, this equity line moves slowly but continuously upwards. And what happens to his budget? Look at that! He didn’t really come into danger of going completely broke, but nevertheless he was still in the red after almost 1000 bets. Then, however, there was suddenly a boost, a lucky streak, and he even exceeded his expected value at the very end.

If you now want to say, quite subtly, that he was exactly equal to his expectation at least 12 times, namely at all intersections of the two lines, then unfortunately I have to contradict you there too: The intersections are, of course, always between two bets. So before the bet he was still in the minus, after the bet in the plus, or vice versa. Apart from that, we may again have a measurement accuracy. And if this were accurate to a cent, then it would of course again be possible to reach his exact (to the cent) equity. Sure, even then it would be extremely unlikely.

Before I show you a few more examples of such careers, I will now show you the diagram of his hit expectancy. It’s a little less exciting, but still worth looking at:

So we see that the curves are quite parallel (is there something wrong with my random number generator?). Nevertheless, the person’s hit expectation is rather in the minus (specifically: even at the end, after his final lucky streak, he still remains in the deficit: 502.69 hits expected, 497 achieved). This shows you how many different ways you can be lucky: He did not reach his hit expectation (which in this case is objective, since the probabilities were fixed; always remember, in reality even that looks different), and yet exceeded his equity, his money expectation value.

The reason for this is simple: he was lucky in his choice of betting levels. And this was indeed luck and not intuition. After all, his equtiy is also objectively calculated, based on the stakes. To exceed these (equity) is luck and will always remain luck.

It’s easy for me to run another simulation and copy the diagrams here. The simulation itself doesn’t take a second. So here is another player’s career, the player plays exactly the same (well) as the previous one:

Here first the curves for equity and budget.

Here is the curve for the expected hits and the hits themselves.

The interpretation of the curves: In this run, the player was rather unlucky. In any case, it is always bad luck when you don’t get what you are entitled to (although in practice it is not known what you are really entitled to). However, the person has been “lucky” in terms of his hit yield. He has therefore exceeded his hit expectation, even if only by 7 hits. So the effect here is reversed: poor choice of engagement levels. That ultimately worked against him.

But one can also look a little closer here. In the initial phase, he was lucky, both in terms of hit yield and win. Logically, these values will always go a little hand in hand. If you significantly exceed your hit expectation, then you will almost always be financially in the black. In order for this not to happen, you would have to play the losing bets at a high price and the winning bets at a lower price on a regular basis.

So we have now seen that you can be lucky or unlucky in many ways. In addition, we have seen that it is not true that everything evens out. And thirdly, that everyone who takes part in the game, that is, who bets money, in any of the games presented or in another game not presented, is either lucky or unlucky. Everyone either has more than they are entitled to or they have less.

Whether this is relevant, however, is questionable. I had already given other examples where I showed that it is difficult to call it bad luck if you play on the wrong side and lose too much. Based on the above definition, when luck or bad luck is only measured in terms of the financial aspect, it would of course be relevant. But then, for me, the other consideration prevails: Whoever loses too much and then stops playing has saved time. He would have lost the money sooner or later.

The idea that everything evens out is simply nonsense. Nevertheless, it naturally contains a certain amount of truth. As a mathematician, how could I see it differently? These are the laws of large numbers. The curves always look like this, like the hit expectation and the hit yield. They nestle up against each other. But the following facts get in the way:

What are these much-cited long periods of time? How many attempts do I have to make before I achieve my hit yield? Mathematics simply says: “The relative deviation goes below each epsilon when n (the number of attempts) approaches infinity.” Even this can be discussed, but I prefer to call it philosophising. But I will come back to that afterwards. First, let’s analyse this statement, with its possible consequences.

One would have to divide the expected hits by the actual hits. And this number, as you can see from the diagrams, is always somewhere near 1. I have copied this out again for you in the diagram of the second example. Here, only the quotient of these two values is shown.

The line with the 1 is not highlighted, but that is the one that our curve should follow according to mathematical laws. Thankfully, it does here. But how close is close? It could also suddenly deviate again in another 5000 attempts. And more so than can be seen here. It would be possible, even if very unlikely. The mathematician would then quickly say that 6000 attempts were not enough, try more, you’ll see. And he would probably not be entirely wrong. The statement “below every epsilon” means that if you make enough attempts, the deviation is arbitrarily small. So you say epsilon should be 1/10000, then we make so many attempts until the difference of the curve to 1 is smaller than 1/10000. The mathematician would again rub his hands in satisfaction. And I would once again live up to my reputation as a nag: Why do you stop when it is below this value 1/10000? Who guarantees that it will stay below that now? Do another 100000 attempts, maybe it will deviate more than the 1/10000 again? And strangely enough, I would also be right.

And now there are all the curious developments in life itself. Maybe, despite playing at one’s best, one has already been broke for a long time, has done another job? Or a player plays at a disadvantage and doesn’t notice it for the rest of his life because he still wins. Then, according to mathematicians, the period was not long enough. I just call him a lucky newt. And I know newts like that.