Thie is an important introduction to even more important series of articles that is aimed to explain to you that Laurence Krauss and basically all celebrity "scientists" are giant assholes.

I have been promisisng this series of articles for some months already, promising to our Dear Banana and to myself mostly, and also to yourself as you are represented in my head. It is time to fullfil my promises, but nothing comes easy -- our topic belongs to the purest phylosophy ever concieved by man yet, challenging deCartes in abstraction and Molyneux in depth and social impact -- we are about to question our presuppositions and implicit "truths" and destroy the language you used to speak, the languange that is being abused by "scientists" and priests to your detriment, therefore it requires some very thorough understanding of simple math, nothing complex (only the simplest math pertains the strongest to our notion of truth) but you must really understand every bit of it, this is going to be boring, brace yourself.

Here is a problem: you took an unfair coin and have it tossed 10 times, it landed tails 6 times. What is a probability of it landing tails on the next toss?

Some of you with implicit understanding of "prbabilities" immediately divide 6 by 10 and wonder: how is it even a problem at all? For you this whole problem feels as a solid piece of data not a question but an answer in its own right. Afterall, this is the only meaningful number you can produce from the given data. But what exactly compels you to tke this number as the probability of tails for this coin? The PHYSICAL meaning of this number is the PROPORTION of tails in the given experiment. Translating this proportion into probability is YOUR ASSUMPTION. This assumption is not stupid, but it IS AN ASSUMPTION.

So what?

So we can justify this assumption! We can demostrate how much sense it makes within the mathematical framework of probability theory.

We have an experiment done, which gave us a certain fixed result. We can step back before the experiment and ask a question: what was the probability of this particular outcome? (before we started tossing this coin). For tackling this question we must introduce some symbols: let's call the coin's probability to land tails on any random toss "p" -- this is the number in question of the original problem; let's call the probability of this coin to land tails m times out of N -- "P(m/N)"; and without saying "m" is the amount of tails in an experiment consisting of "N" throws (to get rid of the original "6" and "10").

Thus we wish to represent the probability P(m/N) for any GIVEN m and N as a FUNCTION OF p.

In other words it is somewhat reverse to our original question, but it is a question specifically about the PROBABILITY p that belongs to our coin. We now investigate how this probability AFFECTS the proportion of tails coming out of an experiment that repeats the expirement of our original problem, of which the outcome is already known.

It is very easy to find out the general formula for P(m/N). Let's find a probability of one particular outcome: TTTTTTHHHH (10 throws 6 tails in a row).
It is p*p*p*p*p*p * (1-p)*(1-p)*(1-p)*(1-p), because we demand a certain known outcome for each of a trow in the sequence.
Now imagine we are to find the probability for a sequence of 6 tails, but not in one row, say: TTTTTHHHTH, we can do the same trick, simply multiply all the probability in sequence... and end up having the same number, because multiplication commutes. Thus the probability of any outcome having m tails of N throws is: p^m * (1-p)^(N-m). Let's call it p1.
But our question is about all of them, not each one separately, what is the probability of all such outcomes combined as opposed to all the outcomes containing not m tails? Simple! It is a plain sum of these individual probabilities, because all our individual outcomes are mutually exclusive -- we can sum their probabilities up due to mutual exclusivity. And all these probabilities are the same p1, so the sum becomes a multiplication: p1 by the amount of all such outcomes. Lets call this amount C(m/N). How exactly many are they? This question has a well known answer: C(m/N) = N! / m! / (N-m)! -- we are safe to believe wikipedia on this particular number, as it is not going to affect our reasoning in any manner.

Thus, P(m/N) = p^m * (1-p)^(N-m) * C(m/N)

This is a function of p for any given m and N. Problem solved. But we needed this solution for some justification of some assumptions? Rememeber?

Feel free to plug numbers in. Returning to our original problem m=6 N=10 p=0.6, it will give you 0.250822656, deviate ever so slightly from p=0.6, (plug 0.6-epsilon and 0.6+epsilon in) and see the P(6/10) drops below 0.250822656. Try it for any numbers m N, 0<m<N. It always gives you P(m/N) value at a point p=m/N is higher than its value at points m/N-epsilon and m/N+epsilon.

According to a well known theorem [wikilink] it suggests a local extremum IN BETWEEN m/N-epsilon and m/N+epsilon, but not necessarily at the point p=m/N.
A local extremum always means (or rather "IS" as opposed to "means") a zero of the first derivative over the free variable, in our case p.
It is very easy to find this derivative dP(m/N)/dp, and then we must find its zeroes: for what values of p it is true dP(m/N)/dp = 0?
As we equate the whole thing to 0 we can at once forget about the constant C(m/N) which is non-zero, and proceed to: d(p^m * (1-p)^(N-m))/dp = 0.
using the rules for compounded derivatives [wikilink]:
d(p^m)/dp * (1-p)^(N-m) + d((1-p)^(N-m))/d(1-p) * d(1-p)/dp * p^m = 0.
m * p^(m-1) * (1-p)^(N-m) - (N-m) * (1-p)^(N-m-1) * p^m =0.
since 0<p<1, we can divide by p^(m-1)*(1-p)^(N-m-1)
m * (1-p) - (N-m) * p = 0.
m * (1-p) = (N-m) * p.
(1-p) / p = (N-m) / m.
1/p = N/m.
p = m/N.

Wow! the only zero of this derivative is at the point p=m/N.

The p=m/N deliveres the Maximum to the probability function P(m/N), which is the probability of the given experimental outcome in the space of all identical experiments with the same coin (we are still talking about this sigle coin that was given to us in the original problem, remember?).

Tangent 1: probability only makes sense in some previously defined SPACE, so called probability space, which is on one hand ARBITRARY, on the other hand it HAS TO MAKE SENSE. I planned to write about this problem even before the present article, and I will eventually write it as a part of the present series. This problem of selecting/defining a relevant probability space for your probabilistic description of a problem is another aspect of PURE-ART that defies the sacred scientific method, this can not be regulated, prescribed, or even verified or proven for relevance -- you either do it right, or screw it, and there is no "scientific method" to convey your motivation for any particular choice of the probability space to a person who refuses to understand it. For now it is suffice to say that our space of all identical experiments with the same coin is highly relevant beyond resonable doubt, it just does not include anything besides tossin the same coin and counting outcomes in some singular consistent manner throughout experiments without overlaps and omissions. At this point you must simply SEE & FEEL that this is relevant, definite, and no shit involved. This is another occurence of "implicit knowledge" as our Dear Teacher Ryan Faulk calls it. And for the single coin toss experiment we implied (ah! how often we imply the most consequential things!) the probability space to be a set of two elements: either Heads or Tails -- we FEEL there is no natural alternative to this probability space in a coin toss experiment. To contrast the illustration of relevance it worth mentioning that the famous saying "how improbable is the origin of life!" conveys no meaning AT ALL, because it does not refer to any valid notion of probability, because it lacks the definition of the probability space.

So we have found that our preferred answer "0.6" if it is true, makes the condition of the problem MOST PROBABLE among all conditions of the same structure. This is THE justification for using the factual proportion m/N as an answer to the question of probability. It is imperative for you to understand this statment, and feel how the local extremum found above justifies our answer to the original problem in the complete absense of any formal rules that prescribe to use this particular method and its result. We take a number with some certain phisical meaning and make it designate a probability -- a leap of faith, but very well justified faith. You can better feel the justification of declaring m/N to be the correct answer if you imagine a slight deviation from it. Although the "0.6" does not mean "absolute truth" and does not make the probability of the problem's condition to be 1, the actual number is 0.250822656, you can say it is the probability of your GUESS "0.6" to be true, and it is the absolute BEST GUESS you can make, for if you deviate from 0.6 any amount any direction, the probability your guess is right is even less than 0.250822656. The Principal Unspoken Rule of Probability Theory forces you to make your best possible guess and avoid funny stuff. It is apparent that the probabilistic description of this coin derived from the solution above is not going to be perfect, but it is THE LEAST IMPERFECT among all alternative descriptions within the framework of the theory. As long as we are going to continue using the apparatus of this theory we must proceed with the answer "0.6", or "m/N" in the general case.

Problem solved? Right? What else?!

Let us now toss the same coin another 90 times. Imagine it landed in total 70 tails from 100 throws. What is the probability of this coin to land tails on the next toss?

Using the same method as explained and throughly justified above... We divide m by N and get the answer "0.7". And I am not even trying to make you feel surprised... This is THE CORRECT ANSWER, for the sake of accuracy, we must proceed with the new probability describing our coin in all subsequent experiments. This is not some revelation, but a routine...

Tangent 2: at this point some might say: wait! are we simply going to discard the previously used data, and forget about the "6 of 10" as it never happened? but it did! just before we observed "70 of 100" we observed "6 of 10" in two equally valid experiments. isn't it more likely that the truth is somewhere BETWEEN 0.7 and 0.6?
No, we do not forget it, we INCLUDE it into the "100". But I understand it still feels as if we ignored "6 of 10" by including it into some bigger domain... Let's run two experiments of 50 tosses each: first gives us "30 of 50", the total gives us "70 of 100" (you can even see how our "6 of 10" fits smoothly inside the first 50). Given the "70 of 100" means the latter 50 tosses came "40 of 50". And if you insist on the "truth between 0.6 and 0.7", then you must EQUALLY insist on the "truth between 0.7 and 0.8". Obviously the only shared point in these two intervals is "0.7", with which we proudly proceed. Please note, i haven't used any nothin from the probability theory -- this rebuttal is effing universal for very many false claims that are based on jerrymeandering where some integration is required by common sense. You can not extract any MEANINGFUL information from ARBITRARY division of the domain (and the mere sequence of the identical experiments occuring in time (absent the physical world) is arbitrary (and again you can either feel it, "implicitly knowing" how is it arbitrary, or I can not help you)) -- the integral characteristic holds more truth unless you specify some physical meaning for the exclusion of a sub-domain. I hope, I just helped you to rebut a thousand false proposals from your political enemies.

Ok, we accept the new value of p. What's the point?

The point is: THE COIN HAS NOT CHANGED. The probabilistic description of the coin (which is in our case a single probability value p) has changed while the coin itself remained the same. The probability turned out not to be a property of the coin!

At this point I must defend the relevance of the entire reasoning scheme layed down above. You could blame my particular method of giving an answer to the original question for being that "flimsy" under the influence of new data added to the input. You can find a more refined approach in the video lectures of 3Blue1Brown on the topic of "probability of probabilities" [tubelink], he uses a more practical example of quality assurance in industrial production (seriously, watch 3Blue1Brown, you will see that our coin was just a simplified version of this extremely prominent and routinely practical problem of assessing the quality of a batch of produce without biting each apple) and he is way more careful with non-discreet values -- according to his approach instead of folowing my path you could have asked a different question: what is a probability of the p<0.6, and similarly what is the probability of p>0.6, but in the end (I leave this task for the reader) you would come to the conclusion that these probabilities are both 1/2 (upon further reflection it reveals that an interval around 0.6 is most probable among all intervals of same width). It is my arbitrary decision to omit the "probability of probabilities" approach as more obscure to my taste.

Nevertheless, the method itself is not that much important, feel free to invent your own, better than mine, but you can not avoid the dependency on m and N because it is YOUR ONLY INPUT DATA! Even if all my results are wrong, then your results will also CHANGE as the new data arrives to the input! And it is the change that is our point of interest not the specific values! But wait, if it works with any method resembling common sense, why have I bored you with all these math? Firstly, for the sake of completeness, so you can not accuse me of not giving you any method, and deriving my conclusions from void -- you see, it is easier to defend a method than its absense. Secondly, it is a method that gives you the correct answer and facilitates further understanding of the entire framework.

So the probabilistic description of the coin happened to be wrong at least once, regarding the same coin, so it does not reflect the state of the coin!
Then what DOES it reflect, for fuck's sake?! IT REFLECTS OUR KNOWLEDGE ABOUT THE COIN.

The probability value p (the answer to our problem) has changed because the additional data altered our knowledge about the coin.
And nothing else has been altered!!!

<bold as fuck>
PROBABILITY IS NOT A STATEMENT OF FACT
PROBABILITY IS A STATEMENT OF KNOWLEDGE 
</bold as fuck>


Read it again. And then again. It is THE MOST CONSEQUENTIAL CLAIM you have ever heard. Half of the scientific narratives, and half of the religious narratives REQUIRE THIS CLAIM TO BE FALSE. But it is not false! It renders half of all popular narratives, both scientific and religious devoid of any meaning. It reveals the absolute assholeness of all celebrity scientists, you name them. It puts Laurence Krauss in the same category with Deepak Chopra right next to each other surrounded by quantum computers. It puts an end to all phylosophic attempts at "quantum" free will, but more importntly, it destroys all seemingly "scientific" modern interpretations of quantum mechanics that do have place in mainstream "scientific" publications, beginnning with the quantum computing, which is a billion dollar industry of promises. This claim if known to the general public is gonna cost billions to very many snake-oil industries and individual assholes. It does not stop at quantum computing, you can use it to undermine the Holy Global Warming too. Read it carefully again, and reflect on it EVERY TIME YOU HEAR "probability". Invent your own applications. If we sue every fraudulent salesman on the grounds of this claim, we could pay off the US debt.