That calculation shows that after 500 flips your expected profit is $10, which is not the same as the question that was asked.

because your calculation assumes that out of 100 flips, heads always gets 51 out of 100, which is not how probability works - it's not linear (for lack of a better word).

It's possible to be up $10 in just 10 flips, just as it's possible to be even at 1,000,000 flips (though neither very likely, but possible!). Also, the question asked "with a 99% certainty", which your answer doesn't take into account.

If you had a coin that when flipped had these odds, and you wanted to try this out, your answer of 500 would be a good rough estimate, but not the answer.

You didn't include the work required for the 99% confidence interval that the original post asked for. It requires statistics, not simple math.

Statistics is math.

It ain't simple.