If you flip a fair coin indefinitely, how likely is it that, at some point, you will have evidence that the coin is not fair?
What I mean is this: Take a fair coin and select a $p$-value. If we were to flip the coin $100$ times, there would be some threshold proportion $Q$ such that if the propotion of heads (or tails) were larger than $Q$, we would have evidence (given our selected $p$-value) that the coin was not fair. If we were to flip the coin $1000$ times, the threshold proportion would diminish. As the number of flips $N$ increases without bound, the $Q$ approaches $.5$.
If we flip the fair coin indefinitely, what is the probability that there will be some $N$, such that, if we had stopped flipping at $N$, we would have ended up with evidence against the coin's being fair (in terms of a $p$-value)?