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Basics
The law of large numbers
The law of large numbers says that as the number of trials grows, observed frequencies converge to their expected probabilities. It is the reason frequencies eventually look 'fair' — and it is routinely misunderstood as making the next draw predictable.
Convergence is slow and one-directional
The convergence happens by many future draws being ordinary, not by past imbalances being corrected. A number that is over-represented stays over-represented in absolute count forever; its *rate* approaches the expected value only as the denominator grows.
What a few hundred draws can and can't show
A few hundred draws is a tiny sample. The spread between the most- and least-drawn numbers at this scale is almost entirely noise — the randomness lab shows simulated histories producing the same spread.