[xvedejas]

I think this is the correct way to phrase it. Just because the probabilities of each are both 50% doesn't mean it's more likely than not to get the same number of ones and zeros. It would just mean you're equally likely to get a few more ones as you are to get a few more zeros. But the counts are unlikely to be very far apart.

[082349872349872]

But in absolute (not relative terms) the counts will tend to diverge over time.

[eru]

Yes. You expect the absolute difference in numbers of 0s vs 1s to grow roughly with the square-root of the total number of digits produced.