So - I am no math genius but I do have shower thoughts. And there is one thought about normal distribution that I just couldn't let go. I converted it into a little story to visualize it a little better. Let's see if it makes sense and if it really is a paradox I came up with. Here is the story:
A man is in court. He is said to have murdered someone. There is evidence that stats that he did it - but chances are it is all a coincidence. The judge comes up with a simple solution: "Tomorrow at 8am on market square - you are to toss a coin. Head and your head comes off - tails and you go home a free man. Let the gods decide whether you are to die or not."
The man gladly accepts this offer. You must know - even though it is the middle ages, he is a mathematican - not one to believe in gods. And he also knows probabilites and thinks he has a way of how to manipulate those.
The man takes his fate deciding coin home with him and begins tossing it all night.
The morning comes and everyone is waiting on market square. It is 8 am sharp and the man, as promised shows up with his coin in his hand. He is very confident, because he knows - his chances of dying are at about 0.1 %. In front of everybody, he tosses the coin and: tails. Then man is free to go. Not even the tiniest bit nervous about his fate.
How was that possible? He must have known that his chances where 50 - 50 (assuming the coin cannot land on its edge and will always be tossed and flipped randomly).
Well, here is the thing that I cannot explain:
Last night, the man was home - as I said - flipping his coin over and over again. Since this is a normal distribution, in within the first 10 tosses, the coin showed head 5 times, and tails 5 times. But, after many, many tosses - the coin finally showed head 9 times in a row. This happening comes with a likelihood of 0.2% (according to one of those tree-diagrams). Now - for the 10th time, the chances of head again would be only 0.1% percent if I am not mistaken. Now - in my eyes: All the man had to do was to NOT throw that coin again until his fate was about to be decided - because heads again? That would be insanly unlikely - wouldn't it be?
So, that is my paradox. A random coin toss cannot be manipulated only by waiting for it to be unlikely to show a certain outcome over and over again - or can it?
Thanks for reading my little story :) I hope you guys understand what I am trying to convey here :)