Doubt in relative frequency

[Himanshu]

I have tried to simulate some data in python. I have attached the screesnshot below.     

random.choice() is a function which takes a list as its argument and randomly chooses a single element from that list.

In first case if we have sample space of two elements {0,1}, then I found that that as number of trials increases relative occurence of 1 (No. times 1 occurs/Total number of trials) converges to probability of occurence of 1 (0.5).  

Similar is the case with 3 and 4 element sample space. I have not further increased the value of n beyond 15550000 because then it takes a lot of time to run the program.

But why the relative frequency converges to probabilty as number of trials increases? Intuitively it seems correct because physically probability is related to chance or likelihood. But the axioms of probability has nothing to do with the relative frequency? So why does this happen?