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"whether it is possible to always get heads, an infinite number of times" No if we actually make the experiment, for we cannot toss coin infinitely many times.
by effie 10y ago
"whether it is possible to always get heads, an infinite number of times" No if we actually make the experiment, for we cannot toss coin infinitely many times. Yes, if we are just considering and measuring all possible infinite sequences. Measure of such mathematically possible sequence is zero, like measure of a point somewhere in a circle is zero. The point exists and is a possible result of choosing a point, but its probability is zero. Thus zero probability does not mean thing is impossible; it is only very unlikely.
- danbruc 10y agoIf an infinite sequence of all heads is a possible outcome, then it is not true that the relative frequency converges to 0.5 in the limit. It is obviously true for many infinite sequences but if all heads is a possibility then it is not true in general. Just to clarify, I am talking about the frequentist's claim that in the long run the relative frequency becomes the probability, i.e. P(x) = lim[n -> ∞] nx / n where n is the number of trails and nx is the number of trails that yielded x. So again, if an infinite sequence of heads is a possible outcome, then that would mean P(heads) = 1 while the frequentist asserts that it necessarily must become 0.5.
- effie 10y agoDoes the frequentist view really claim that infinite sequence nx/n always converges, for all possible runs? That is obviously false. The laws of large numbers only state that probability that the limit of nx/n is equal to 1/2 is 1. This is because all sequences are equally probable and there are overwhelmingly many more sequences that limit to 1/2 than sequences that do not. That is different from saying that the limit is always equal to 1/2, because there are these exceptional cases of low measure like HHHHH..., HTTHTTHTT...