as the number of trials increases, the empirical probability will get closer and closer to the theoretical probability. This is what the law of large numbers states.

So are there any cases where empirical probability is greater than theoretical one during large trials?

I am not talking about 10 or 100, but million of tries.

Sure. Sampling deviations can go either way, right? If youâ€™re flipping a fair coin a million times, you could get 500,017 heads or 499,063 heads or even bigger deviations in either direction. Of course those numbers may not be a very likely in a million iterations. What ends up being â€ślargeâ€ť (as in â€śLarge Numbersâ€ť) depends on the nature of your experiment. Remember that in mathematics, â€ślargeâ€ť can get pretty far outside the bounds of our normal physical intuitions.

This is a possible outcome of 12 rolls but it doesnâ€™t seem to match what we expect. An equal distribution of dice values.

Is our imagined dice weighted?
Possibly but letâ€™s try 120000 ish trials but keep the difference between the categories by adding 20000 to each outcome.

When we look at the plot we see something that is much closer to what we expect which is about equal probability for each dice value.

Now if we scaled up and saw the same ratio between the dice values we might suspect that the dice might not be â€śfairâ€ť and our theory for this dice might not match reality.