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If someone makes the argument that it is impossible to flip tails $n$ times on a two-sided coin, then we can argue there is a $1$ in $2^n$ chance. There is not a definable point at which it becomes impossible.

Is the following statement true or false?

It is possible to flip tails indefinitely on a two-sided coin.

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cyclochaotic
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1 Answers1

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Possible, yes, but extremely unlikely, in the sense that the probability is $0$. That does not mean that it is impossible, especially since "probability" in such circumstances is a human model of phenomena. That is, for example, there is not actually any "force of nature" that "prevents" flipping heads for an arbitrarily long time... a.k.a. "forever". But one should bet against it, as in the surely-googleable "gambler's ruin" phenomenon/scenario.

"Probability zero" and "impossible" are very different notions, though for day-to-day purposes they are similar in function. At extremes, the presumed equivalence is much less useful for reasoning about either the mundane world or mathematical things.

(One should certainly keep in mind that formulation of mathematical ideas does not directly lend enforcement power over physical phenomena...)

paul garrett
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  • +1, but also I wonder if there is some point to be made about the difference between "infinitely many times" and "indefinitely many times." If the latter means something different, it's not clear that we can assign a probability to it. – Trevor Wilson Nov 25 '13 at 02:59
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    @TrevorWilson, indeed, to my perception, the distinction between "indefinitely-many times" and "infinitely-many times" in the mundane world is potentially significant. In standard contemporary mathematics, it may be harder to distinguish the two, which may be a bad thing, depending, ... – paul garrett Nov 25 '13 at 04:21
  • What if the antagonist made the argument that the chance of flipping heads at least once is $(2^n-1)/2^n=1-2^{-n}$, therefore eventually we have a $100$% chance of flipping heads, so the answer is false? – cyclochaotic Nov 28 '13 at 00:23