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, 00.6, 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!!!
PROBABILITY IS NOT A STATEMENT OF FACT
PROBABILITY IS A STATEMENT OF KNOWLEDGE
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.