Another chapter that serves to introduce a world view based on thinking in probabilities. Everyone surely has a certain idea of the “Murphy’s law” that is so often quoted. Here we look a little more closely at where this actually comes from. The assertion stands that the view expressed here so far still differs from that of the reader. This may confidently be called audacious.
A problem is known from physics in which one wants to measure radioactive decay in time. Somewhere radioactivity has escaped (this is supposed to have happened in large quantities on 26.4.1986?). Is that when it was called the SuperGau, which is so often associated with the name Murphy?) Afterwards, one wonders how long it will continue to radiate. Unfortunately, this question cannot be answered exactly. Or rather, the only halfway acceptable, correct answer would be: “It keeps on radiating. It never stops.”
Therefore, physicists have found a solution by rephrasing the question. At some point they asked more cleverly, “When will the radiation be half as strong?” or “How long will it take for the radiation currently present to be halved?” So what one should be interested in is the question of what is called the half-life. It keeps halving and at some point it’s so small that it becomes irrelevant. The time it takes to halve is always the same. So from 20 to 10 takes just as long as from 10 to 5 etc. Of course, this regularity had to be observed (i.e. measured) first. The natural constant that was found for such processes and has been responsible for them ever since is e, Euler’s number.
From the player’s practice, a similar problem has arisen quite frequently. Unfortunately, the very formulation of this problem often causes enough confusion. However, the story of Pasch-Jürgen (chapter: “I roll a double in five throws”) gave a small, practical, introduction to this problem. There it was that a person was allowed to roll the dice 5 times to roll a double. He had five chances to make an event of 1/6 occur. The probability calculated for this was well over 50%, so, because of the identical payout ratio in the story (DM 100 against DM 100, as it was played there, so-called equal money), the person rolling the dice had a clear advantage.
Three tries would not have been enough to get to 50%, let alone above. But four attempts would have been enough. It would not have been safe even with 100 attempts, since there is a chance, even with absolutely correct dice rolls, that 100 times in a row no double will occur. Can you see the relationship with the physics problem mentioned above?
Generally speaking, one would like to know when an event is “certain” or “guaranteed to occur at least once”. But, analogous to the above, it is never certain. Just as radioactivity has never completely decayed.
The analogous reformulation to this practical problem, i.e. after the half-life, which here should better be called the “half chance”, is thus:
“How many trials on an event with probability of occurrence 1/n are required before the chance that the event has occurred at least once is above 50%.”
Here a variable called n has been used, indicating that it is a natural number. For most game-like problems, this is even adequate. Otherwise, of course, the question, and indeed the solution, is no different in any problem in which an event with a probability of occurrence between 0 and 1 is repeatedly carried out (just as mathematicians sometimes have the finger of warning that SuperGau , whose probability, though unknown, is very small and not determinable by an n or 1/n, will eventually occur).
So in the Pasch problem, n=6. If n=6, i.e. the probability of occurrence = 1/6, then it quickly becomes obvious that the probability “The event occurs at least once” exceeds the 50% hurdle with 4 attempts. With 3 attempts it cannot possibly be above 50%. Because even with the most naive (but wrong) calculation method of adding up the 1/6, you just reach 50% (1/6 + 1/6 + 1/6 = 3/6 = 50%) with 3 attempts. But that it exceeds 50% with four attempts? Yes, why not? Of course, it can also be calculated. But it is not done here.
The answer can be used relatively often. Here, however, it is not to be deduced, but only mentioned once: If n becomes larger and larger — one then says that n goes towards infinity — the value sought approaches ln2 * n more and more. The ln2 is about 0.69. For ever decreasing probabilities, this means — which is intuitively obvious — that one must perform an ever increasing number of trials to increase the chance to over 50% that the event will occur at least once. Incidentally, the logarithm naturalis is the inverse function of the e-function. So the relationship exists here too.
If you have an event with a probability of occurrence of 1/100, then you have to make about 0.69*100 = 69 attempts to be the favourite, i.e. to have over 50% that the event occurs once.
By the way, if you are looking for a random experiment in which any such probability can be mapped, it is advisable to think of the experiment “drawing a ball”. If there are 100 balls in a drum, one of which is red and the others white, one would have ( quite exactly?!; see other chapters on doubts in this regard) the probability of 1/100 in an attempt to draw the red one.
To construct a probability of 1/100, you can also take 10 balls, one of which is red and the other nine are white, and draw 2 times in a row, but don’t forget to put them back! Then the probability is 1/10 in each case. Since the events “draw red ball” in the first and second attempt are equally probable and independent, multiply and you get 1/10 * 1/10 = 1/100. The chance of drawing the only red ball out of 10 balls twice in a row would then also be 1/100.
The 69 attempts or, generalised, 69% of n or ln2*n, are here, however, only the value where it is guaranteed to be over 50%. It is possible that 68 attempts are also sufficient. Because in the Pasch example, only four attempts were needed. And 4/6 is 66.67%. And this number is smaller than ln 2, i.e. smaller than 69%.
Here is a diagram to illustrate the whole thing:
The diagram shows the development for an event with a probability of occurrence of 1/100 (one of 100 balls is red. Will it be drawn?) for up to 500 attempts. The purple line describes the chance that it has already occurred at least once by the number of attempts, the blue line is the counter probability (it has not yet occurred). The shape of such a curve is also called a “hyperbola”. A hyperbola nestles at its limit value. The limit value for purple is 1, that for blue 0. These values are never reached. For each number of attempts, you can calculate how likely it is that it has not yet occurred up to this number (and read off the diagram if you want to know beyond 500). Nevertheless, at 500 attempts it already seems pretty certain that it will come once (the value at 500 attempts is 99.34% that it has come at least once, by the way).
So what does this have to do with Murphy’s law? A brief search on the internet revealed that the intuitive usage “everything goes wrong” is actually correct. However, there is an actual law, if you rephrase it only slightly: “Everything goes wrong. If you try it often enough.” Or, more briefly: “Everything goes wrong at some point.”
The real formulation of the mathematical law and – since Murphy was an English speaker, as one might already suspect from the name – actually used in English is thus: “What ever can go wrong, will go wrong.” This statement is absolutely confidence inspiring. The law of large numbers – there are actually two, but they are very related – is expressed in this way. The statement of this is made vividly in the diagram above: every event with a probability of occurrence >0 will occur at some point if you just try it long enough. The hyperbola nestles up against the 1. 1 is the certain event. It will happen at some point. The mathematician then formulates it so strangely (but correct in his world): “the relative deviation goes below each epsilon”. Just as you can see clearly in the diagram: The curve inexorably approaches 1. But it will never touch 1. There is always a space in between. It will always be smaller, even smaller than epsilo, and that can be 10^-100, despite all arbitrariness.
The basic assumption was that Murphy had expressed this law in a simple sentence that everyone could understand. And from this, according to the further assumption, the legend would have arisen that he was the one who painted the devil on the wall: “What ever can go wrong, will go wrong.” Everything that can go wrong will go wrong at some point, including such bad things as superGAUs. All these things are true, but they simply do not go back to Murphy and his insight. He said it in a different context and even meant it. So that’s how it was probably known until now. Something about the greatest possible catastrophe, which is sure to happen. The vernacular also has its own laws…
So if you want to invoke Murphy in the future because something has gone wrong again, just stick to it. Nevertheless, there is a mathematical law behind it – and it is valid.
Let’s go back to the example of the “half-entry time”. First of all, the point was sought at which the one-time occurrence of an event, even a very improbable one, reaches 50%. This made sense insofar as one cannot guarantee a point in time from which it is certain that it will occur. This actually refers to any event with a probability >0. Events with very small and with very large probabilities of occurrence are no different: the question: “When will the 50% hurdle be crossed?” is a fixed and determinable number of trials. When it is 100%, i.e. certain, is not determinable. Only that it will happen is certain. Don’t forget: Try long enough. For many, and very small, chances there is probably not enough time in life.
It applies to large and small probabilities as well as to pleasant and most unpleasant events. So in this sense: don’t forget to hand in your lottery ticket! It will come eventually.
Here is another diagram with a much less likely event. You can see:
Here the probability is plotted for the repeated occurrence of an event with a probability of occurrence of 1/20000. The hyperbola is not yet quite moving. Because with 20000 attempts (Excel is overstretched at some point. Keep in mind: 1/20000 is not yet an extremely improbable event.) the chance rises to just over 60% that at least one event will have occurred by the 20000th attempt (shown here only up to 17034; the value is read off). This value should not be confused with the chance of exceeding 50%. This happened with attempt number 13863, which is 69.31% (out of 20000). The logarithm naturalis (short: ln) of 2 is also 0.6931.
The hyperbolic movement would of course continue in this way and likewise nestle up against the 1 (and the 0). Even if one considers the SuperGau to be much more improbable: The curve would be similar. If you try often enough (here: long enough), it will happen at some point. Even winning the lottery. So stay tuned.
