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> There's an ever-decreasing but nonzero probability that you will keep getting heads.

But with enough flips, the odds getting heads on all of them comes close enough to zero that they are effectively zero.

"Effectively zero" times infinity is infinity. Infinity is like... really big.

Oh, I wish I could find it, but there's an excellent video by a mathematician explaining why an infinity doesn't work in your favor here. IIRC, it's because as the number of throws approach infinity, the odds also approach being infinitesimal, and you'll still lose.

This paradox is famously a case where the expected value of the game is infinite.

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