I am confused with the concept of completely random actions. I was thinking of a very common statistical experiment in which we have a device or black box which randomly chooses between 1 and -1. If we infinitely do this processes, we will have set of randomly chosen infinite 1's and -1's. If we sum up all the elements of this randomly generated set we will probably get 0 every time we perform this experiment. This implies that it is a biased random as it is following the constraint that sum of elements of randomly generated infinite set is 0 every time.
But I was wondering, since the above procedure of choosing between 1 and -1 is completely random, then the sum of all the elements of randomly generated set must be a random number instead of being zero every time. If the sum of elements is any random number, then only it must be called as a unbiased random number and hence confirming the performed experiment to be random.