Probability paradox: Is heads followed by heads or heads followed by tails more likely?

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You repeatedly flip a coin until you either get a heads followed by a tails, or two heads in a row. Which is more likely to happen first?

Solution 1: Both are equally likely, because each flip, there's a $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ of both the previous flip and this flip being heads, and a $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ of the previous flip being heads and this flip being tails.

Solution 2: Heads followed by tails is more likely, because if you want heads heads, flipping a heads followed by a tails makes you start over, but if you want heads tails, flipping a heads followed by another head gives you another chance of succeeding next turn.

Which reasoning is correct, and intuitively, why is the other one wrong?

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