I was just thinking if it really is true that the probability of obtaining each number on a 6-faced fair dice always 1/6? If the die was previously rolled to a 3 for example, would the next die roll not be affected by the fact that it was in that particular orientation when we picked it up for this subsequent die roll?
Does such a bias really exist and can we eliminate it?
We assume the act of throwing the die sufficiently randomizes the position so that the previous roll does not matter. One can conceive of cases where this is not true. Imagine a die with a reservoir of a viscous liquid inside. If it sits with $3$ up for a long time, it will be heavy on the $4$ side and more likely to come up $3$. It is also possible some people are skilled enough to make a die come up preferentially the way they want. A mathematical die definitely has $\frac 16$ chance of coming up each number. Whether a particular physical die is perfect in this regard is an experimental question. You could roll the die a lot and see if numbers repeat more often than chance.