A dice has a uniform distribution. 1/6 as you mention. But create a new random variable which is dice thrown a million times, this will by the central limit theorem approach a gaussian distribution.
With a gaussian distribution, the first point is false - every outcome is not equally likely.
What would happen if you got an unlikely outcome (ie something from the tail) - well nothing really. But if this unlikely outcome continues as you sample from your distribution, well then you are likely having a parameter issue (or having a wrong idea about the distribution) or a fake dice.
Remember you are speaking of statistics now. You are picking from the sample space. Make it simple, imagine a single dice - what is the chance for sum = 6?
Imagine two dice now, what is chance to get sum=12?
What is chance to get sum=7 with two dice?

I might have misunderstood you, but I believe your line of thinking is something like this; throw number one has 1/6 chance to get 1, throw number two has 1/6 chance to get 1 and so on.
Hence every outcome of throwing a million times is equally as likely as any other.
The issue with this is that you are not really asking for what you think you are asking for;
You are not asking for chance to get 1 on a given throw with the dice - you are asking for the chance to get 1 on a throw with the dice, given the condition that all previous other throws has also been 1.