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I hope to have convinced with my previous answer that the mathematical definition of probability is key to resolving your paradox here. Mathematically (using th

I hope to have convinced with my previous answer that the mathematical definition of probability is key to resolving your paradox here. Mathematically (using the concept of infinity) and also physically (using the concept of large numbers) there is a simple and commonly accepted answer to your question. Since that definition is very basic, it is hard to find a direct answer to your question. You might find it interesting that in physics when we deal with large numbers that are not infinity it is usually good enough to know that a certain event is pretty damn unlikely (and it is essential to corroborate this with numbers as in "this happens once in 10^100 years"). In physics, we rarely have 100 % certainty. Logical consistency and a certainty that is large enough are usually key to success (i.e. growth of knowledge) in natural sciences.

prmph 10y ago
I guess another way of stating the problem is this: say you toss a coin and get heads the first 1000 times. Now you are likely to believe that the coin is biased. But that cannot be proved. It may be that you have simply not tossed the coin enough times to perceive its fairness. Maybe after the 1000th toss you start to get enough tails such that after the millionth toss, it is not at all clear that there is any bias. So practically we can make a judgement as to whether the coin is biased on not based on how many tosses we think is sufficient, but theoretically it is impossible to distinguish a biased coin from a fair one if we toss to infinity
- danbruc 10y ago
  Yes, that is exactly what I had in mind. Probabilities are weired in the way that they say what will happen but then still leave open the possibility that this will not happen at all. You toss a coin a billion times, you will get about 500 million heads. Well, or you don't and there are exactly zero heads.
- srean 10y ago
  > theoretically it is impossible to distinguish a biased coin from a fair one if we toss to infinity correction: if you replace infinity by any large finite number your statement is correct.

prmph 10y ago

I guess another way of stating the problem is this: say you toss a coin and get heads the first 1000 times. Now you are likely to believe that the coin is biased. But that cannot be proved. It may be that you have simply not tossed the coin enough times to perceive its fairness. Maybe after the 1000th toss you start to get enough tails such that after the millionth toss, it is not at all clear that there is any bias. So practically we can make a judgement as to whether the coin is biased on not based on how many tosses we think is sufficient, but theoretically it is impossible to distinguish a biased coin from a fair one if we toss to infinity

danbruc 10y ago
Yes, that is exactly what I had in mind. Probabilities are weired in the way that they say what will happen but then still leave open the possibility that this will not happen at all. You toss a coin a billion times, you will get about 500 million heads. Well, or you don't and there are exactly zero heads.
srean 10y ago
> theoretically it is impossible to distinguish a biased coin from a fair one if we toss to infinity correction: if you replace infinity by any large finite number your statement is correct.