TITLE: Flip a coin until a head comes up. Why is "infinitely many tails" an event we need to consider?
QUESTION [7 upvotes]: Suppose we're considering the problem of flipping a coin until "heads" comes up.  The sequences H, TH, TTH, TTTH, ... are all part of the sample space we need to consider.
But what about the sequence TTT... (infinitely many tails)?  I would think that this "event" should be excluded by definition, because our experiment doesn't end until "heads" comes up.  Nonetheless, one could imagine a scenario where "heads" never comes up.
So when should we include TTT... in our sample space, and in doing so, what does that "buy" us?

REPLY [7 votes]: I had previously written an answer which got a fair number of votes.  In retrospect, this was a nice answer, but not to this question!  (Added: I changed my mind again after reading Qiaochu Yuan's comment.  Apologies for leaving two somewhat lengthy answers to the same question.  Maybe it will be "educational"?)
The point here is that the sample space here is much simpler than the "Bernoulli space" of countably infinite $\{H,T\}$-valued sequences.  This sample space consists of elements $H,TH,TTH,\ldots,T^n H,\ldots$ -- i.e., one element for each $n \in \mathbb{N}$, together with -- perhaps -- the one infinite sequence of all tails.  
In particular this sample space is countably infinite.  If one is interested only in countably infinite sample spaces then there is no need to bring out measure-theoretic ideas: it is enough to enumerate the elements of the space and assign each element a non-negative real numbered probability in such a way so that the sum of all of these numbers is $1$.  Here the element $T^n H$ -- i.e., first $n$ tails, then heads -- should be assigned the probability $\frac{1}{2^{n+1}}$.  If we sum over all these singleton probabilities we get $\sum_{n=0}^{\infty} \frac{1}{2^{n+1}} = 1$.  
In particular, if we want to put the last point $TTTTT\ldots$ (infinitely many tails) into our space, then we need to assign it probability zero.  If you have a discrete probability space then it's true that you don't lose anything by omitting singleton elements of probability zero, any more than you lose anything by omitting zero terms from an infinite series.  On the other hand you don't have to omit it: it's legal to put it in and assign it probability zero.  Although it won't make any mathematical difference, I would favor doing this from a modelling perspective, because after all flipping a coin infinitely many times and having it come up tails every time is not logically impossible: it just happens that the probability we want to assign to it is zero.