this may sound easy or hard or whatever but i cant seem to find anything after searching around for a similar question/answer
The question is this:
What can we say (probabilistically) about the next throw of a (fair) coin after having observed n (the exact value of n is left unspecified, use as you wish) previous throws??
There are 2 possible answers:
Since the coin is fair (although irrelevant to the main question) the probability of a next throw (lets say tails) equals 1/2
Since by entropy (or ergodic) considerations a sequence of (fair) coin throws will have to manifest (in the sequence) the relative probabilities of the various outomes as expected (meaning HEADS=1/2, TAILS=1/2). So in the course of the sequence (the value of n here might be relevant) there should be some point where the next throw should have different probabilities for head and tails (considering the above rationale)
A possible objection is that entropy and ergodicity (related) may be manifested in the limit towards infinity (so for a practical n, it could have no observational difference)
However the opposite argument can also be made, from 2 angles
a. it seems awkward (it can be put in more mathematical form) that abruptly at some point n the probabilities will "change" (quotes intended, since thet are not the original probabilities of heads and tails)
b. "typical sequences" (in the context of entropy) have a structure where the outcomes are interspersed in the sequence more densely (for clarifying this point consider the digits of various prime numbers and their distribution)
Last another point of the question is how much should one observe a sequnence before deciding (with a probability threshold) that a coin is NOT fair?
sorry for delaying to comment on the given answers (so far), it seems no notification was sent
regarding first 2 answers, i tried to enhance the meaning of the question in the comments and explain why they dont point to the main question (the wikipedia article posted was close though)
a related question if that helps to enhance understanding is trying to figure out how random a (black box) random genarator is, this is very similar instead relying on basic probability and entropy (and their "interplay" ) although it is phrased as a gambler's fallacy
saying one option is correct but without trying to explain why the other is not correct or not applicable, is not what the question is about
the question actually "doubts" this, either one is correct and the other is in some way not correct or not applicable or there is some synthesis of the two or maybe sth else?