1) The LaPlace Experiment

The LaPlace experiment is a theoretical random experiment. Mathematics often uses idealised states to cope with the chaos around us. The real occurrence of such random experiments is in fact impossible. Nevertheless, mathematics can be worked out wonderfully using idealised examples.

In the so-called LaPlace experiment, it is assumed that there are n different outcomes and that all n outcomes are equally probable. From this, a large part of the common mathematics of probability calculation can be derived. So usually every mathematician can deal with it nimbly and effortlessly. This is also referred to as “idealised probability space”. And as a classic example, one always takes a dice and describes the throwing of this dice (also called rolling the dice) as a “random experiment”. Each of the six chances then has 1/6 and that’s it. Then you can derive any number of statements from it. And the mathematician feels perfectly at ease and believes that he has now dealt with this subject.

But now come my “ifs” and “buts”:

Take any cube in your hand. You actually intend to roll the dice honestly and fairly, purely by chance. How to do that? Close your eyes, put it in the dice cup, roll it out, look. All right, what’s the question?

Well, I’d like to explain: On what surface did you roll the dice? Does the dice have any punch-outs? Oh yes, by the eyes. But there are the so-called “precsion dice”, precision dice, well then…

Yes, so all sides have the same weight? Where is the centre of gravity? Is it homogeneous in itself? What is the air pressure and humidity? How hard do you throw the dice? Which side is on top when you throw the dice, along which axis do you throw? Room temperature? And so on.

Going even further, I could even claim that if you know all the parameters and include them correctly, you could even calculate the outcome exactly (this was even a claim made by LaPlace himself, by the way. 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”).

So is throwing dice now a true random experiment? When would it be?

While I am explaining to you that there is no such thing as a true random experiment, you have probably just thought of one. One that is guaranteed to work. What about a pack of cards? We take a pack of cards, shuffle them “well” and then draw a card. And ask, for example: How likely is it that this card is of the colour spades, that it is a 7 or that it is red? In any case, these seem to be fixed probabilities. Where is the problem? Here is the problem: When are the cards “well shuffled?” Which arrangement of cards would be called “well shuffled” and which would be called “badly shuffled”? What’s more, on closer inspection (looking through the card, for example? ) it would be clear which card was coming up anyway. Oh, you draw from the middle, do you?

All right, I’ll stick to the example of throwing the dice. So, you enjoy the game, you take a dice, you start recording. You make 1296 attempts (preferably divisible by 6, but take any other number, doesn’t matter). You only record the number of dice rolled. Now you notice that after 1296 attempts, the 1 has been thrown 229 times. Now the mathematician (who happens to be male, otherwise probability, mathematics and statistics are female, so one also speaks of statistics being the sister of probability), i.e. the statistician, comes and calculates for you: 1296/6 = 216. Actually, the 1 should come 216 times. But it came 229 times, that is within the simple standard deviation, I tolerate that, there is no reason to doubt the assumption that “the probability of rolling a 1 is 1/6”.

Except: who made the assumption in this case? I assert the following: You conducted an experiment with a very specific die under very specific conditions. Then the value 229/1296, i.e. the so-called “relative frequency” with which the 1 occurred, is a better approximation to the actual probability of occurrence of the event “roll a 1” than the value simply assumed by the mathematician. After all, you only have the one value. The mathematician has only one theory on his side, one basic consideration. He assumes 1/6 and simply does not doubt. The practice, the practical result, he “interprets”. “This deviation is within the expected.”

The mere fact that the mathematician sees no reason to doubt his intuitive assumption of equal distribution of probabilities does not justify this assumption. You have relative frequencies (cases occurred/total number of trials) that is your probability estimate. The best value you could get. And he, on the other hand, simply says 1/6? It’s a dice and it has six sides. Who is right now? Is the estimate 1/6 or the estimate 229/1296 better?

Here it would be a good idea to propose a bet to this person: He claims 1/6, you simply claim “>than 1/6”, based on your experiment. He pays you the “correct rate” or “the fair rate”, which is 6.0 (yes, pay attention, the reciprocal). And now I interfere: I’ll take your side and say that you have more of an advantage than he does. Daring? The surface is getting decidedly slippery now, was there an ice warning? Black ice? This one’s thin on top of that… This is more a warning to myself.

Of course, these considerations also apply to the so often quoted roulette. There, too, it is always said, not only among mathematicians, “no chance in the long run, the bank wins.” There are 37 numbers, each with 1/37, a typical, real and pure LaPlace experiment. The payoff is 36:1, the bank wins. But there are also good reasons to doubt the equal distribution (see chapter “Permanences and the consequences”).