[kllvql]

I found myself struggling with the intuition of this similarly to amingilani until I made the connection you're mentioning to infinity. The thing that helped me understand the fallacy was understanding how truly large infinity is. If you've seen 10 coin flips, or even 10,000 or 10,000,000, those are all effectively nothing when compared against infinity. If you know ahead of time that a coin is fair (so you're not trying to test this hypothesis), you gain zero information about the long-term behavior of the coin from any finite number of flips.

As you mentioned, you don't see the convergence to 50% with any finite sample size, because that behavior is only guaranteed at infinity. All this to say, infinities are weird. 10,000,000 + infinity exactly equals infinity.