Filpping a coin infinitely many times

Question

If a coin was flipped an infinite number of times, is it guaranteed to be heads at least once?

Answer 1

If the probability of the coin landing on heads is greater than $0$, then, the probability that when tossed infinitely many times a heads will come up is $1$. That does not guarantee though that a heads will come up. In infinite probability spaces an event may have probability $0$ and still it may occur, and an event may have probability $1$, yet fail to occur.

Answer 2

It depends on what you mean by "guaranteed." The probability that all flips will be tails is 0 (hence the probability that there is at least one head is 1).

Answer 3

$P[\text{At least one head}]= \lim_{n \to \infty}(1-P[\text{tails in all n times}])\\=\lim_{n \to \infty}(1-(1-p)^n)=1$.

So if you have a coin with probability of head $p$ and tosses are independent.