This is your fundamental misunderstanding.
The statistics don't have to balance out in the way you think they do. They balance out, but in a way that gives you no predictive power.
Here's why they really balance out:
Suppose you flip a coin 10 times and it comes up heads only twice. 20% when it's meant to be 50% is way off. So you would say "keep playing and it has to balance out". Which is kind of true, but not for the reason you think.
If I play a thousand more times, I expect to get about 500 heads and 500 tails. That brings the totals to 502 heads and 508 tails.
Now 49.7% of the flips have been heads. Yay, it has "balanced out" from 20% heads to 49.7% heads even though we saw exactly 500 of both heads and tails.
And that's why it "balances out". Not because you can predict an excess of tails in the coming rolls to balance it, but because you can predict that the 2 vs 8 weirdness you saw at the start will become insignificant over time as you flip more and more times.