Best random variable for infinite trials of a true/false event? if you were to toss a fair coin a finite amount of times, and a success = heads, then the best random variable to represent it would be a binomial random variable. 
However, if you were to toss it an infinite amount of times, what would be the best random variable to represent it now? 
Would it be a Poisson random variable? 
The thing I don't understand is with an infinite amount of trials, you will also get an infinite amount of successes, so does any random variable represent this situation?
 A: The problem posed in the question may be a bit unclear, but I suspect that it exposes the problems in moving from a countable to an uncountable sample space. Notice that it is the sample space that is uncountable - not a specific infinite sequence of tosses. Or, differently stated, $\Omega =\{0,1\}^{\mathbb{N}}$ has cardinality $\aleph_1$. To see this, it is probably just good enough to see that irrational numbers are uncountable, and their binary expressions would correspond to infinite sequences of coin tosses.
Indeed, in a finite number of tosses, there is no need to explicitly formulate a sigma algebra: implicitly, the sigma algebra is $2^\Omega$, i.e. the collection of all subsets of the sample space, $\Omega$. In the scenario of an infinite coin toss, the sample space $\Omega = \{0,1\}^\infty$ is uncountable, and there is a need to define a probability space $(\Omega,\mathcal F,\mathbb P)$ before worrying about a random variable, or function $X: \Omega \rightarrow \mathbb R$, and its distribution.
Otherwise, the questions at the root of the OP remain undefined, as the last paragraph tentatively insinuates. What is the probability of an infinite coin toss of all heads? Or the probability of any specific infinite coin toss in an definite ordered sequence of heads and tails?
In defining a sigma algebra, compromises will need to be adopted to be able to impose a probability measure.
