Infinite heads from Infinite coin tosses? If I toss a coin an infinite amount of times, can I be sure to get an infinite amount of heads? Is it possible for it to be tails every flip meaning I get no heads at all?
 A: 
If I toss a coin an infinite amount of times, can I be sure to get an infinite amount of heads?

According to the Borel-Cantelli lemma, since each coin toss is an event of probability $\frac12$ and a sum of $\frac12$ diverges, the probability of $\limsup_{n\to\infty}\{\text{heads at $n$-th flip}\}$ is 1. But the $\limsup$ is precisely the event of infinite heads. So the probability is 1. And obviously the same goes for tails.
Addendum:
This does not mean it is impossible to only get a finite number of heads or tails. For example, if you throw a dart, the probability that it hits the exact center of a circular target is 0 (assuming a uniform probability on the circle), but that doesn't mean it is absolutely impossible to hit exactly the center. Example adapted from here.

Is it possible for it to be tails every flip meaning I get no heads at all?

The probability of getting no heads is the probability of getting tails always, which is an infinite product of $\frac12$ due to independence of the flips. Thus, it is 0.
Addendum 2:
Again, probability 0 doesn't mean absolutely impossible: see the other addendum.
Reply to the reply-answer posted by the OP
First of all, I suggest you post comments to the answer rather than an answer. AFAIK, you should always be able to comment on your own posts.
Secondly, my notation was inaccurate, and I corrected it. In general, if we have a sequence of sets $E_n$ which are all subsets of the same bigger set, then:
$$\limsup_{n\to\infty}E_n=\bigcap_{k=0}^\infty\left(\bigcup_{n=k}^\infty E_n\right).$$
What this means is: consider the sets $A_k$ given by the union of all sets $E_n$ with index $n\geq k$. Intersect all those sets.
What sets are we considering, and what big set? The big set here is the set of all infinite sequences of 0s and 1s (or $H$s and $T$s for heads and tails). The $E_n$ sets are the sets defined by their containing all the sequences which have $H$ in the $n$th place. The sequences represent the outcomes of sequences of flips, and the $E_n$s, containing all those sequences and only those sequences above described, are used to represent the event that the $n$th flip gives heads. Their limsup. If you unite the $E_n$s with $n\geq k$, i.e. consider $A_k$, you are considering sequences which have at least a $H$ in the flips from $k$ on. So $A_k$ is the event "the flips from number $k$ on get heads at least once". If you intersect all the $A_k$s, it means that if you fix any flip, there is at least one flip beyond that one where you get heads. But this clearly implies there are infinite heads. So $\limsup_{n\to\infty}E_n=\{\text{you get infinite heads}\}$.
Representing this via sets is useful because you want to apply the cited lemma to conclude this event has probability 1, so you can be almost certain that you will get infinite heads. The opposite, however, is not totally impossible, as joriki notes:

Note to the OP: An event is said to occur "almost surely" if its probability is 1. So the answers to your questions are: You cannot be sure to get infinitely many heads, but you can be almost sure, in a technical sense of that term. And it is possible to get only tails but this, too, is almost sure not to happen.

As a final word, you should learn some LaTeX. In your answer, I see "lim supn→∞{heads}". That is writing maths with unicode and no subscripts. In this case it is pretty straightforward, because the formula is simple, and quoted from my answer. But in other cases doing that might render the formula unreadable. The correct way to type that formula is by $\limsup_{n\to\infty}\{\text{heads}\}$, which you can see rendered above. \limsup is a command that typesets "limsup" in upright font. _ gives the subscript -- but only between dollars, as outside dollars that would be italics, because that's how Markdown behaves. \to and \infty give the arrow and the infinity symbol respectively. \{\} are braces. The backslashes are because braces are reserved for special purposes in LaTeX. \text gets me back to text mode, which is needed to avoid getting $heads$ instead of $\text{heads}$.
Update:
OP deleted his answer yesterday. For the record, it said more or less:

Thank you for your help, but I'm really inexperienced with math. What does 'lim supn→∞{heads} is 1' mean?

And perhaps another sentence.
