One roll is pure luck. A million rolls is a law — and the law is three and a half.
The first dozen rolls throw the average anywhere. Two sixes in a row and it reads 4.8; a run of ones drags it to 1.6. Early on the mean is all noise and no signal — the die owes the past nothing.
The six faces average (1+2+3+4+5+6)/6 = exactly 3.5. No single roll can land there — there is no half-pip — yet the running mean is dragged toward it, harder the longer you wait. That pull is the Law of Large Numbers.
The spread of the mean shrinks like 1/√N. Quadruple the rolls and you halve the wobble. The band drawn around 3.5 is that arithmetic made visible — a funnel the trace is forced, slowly, to thread.