You're right that the first coin is more likely to end up heads. But so is the second coin, and if both occur, that would invalidate the pair of tosses. Now, imagine you guessed tails despite the coin starting on heads. If the first toss lands tails, the second coin is still more likely to land heads, which keeps the pair valid.

In other words, whatever you gain by guessing the side that's up on the first coin, you lose on account of the second coin having that same higher probability of invalidating the pair.

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Using extreme numbers, in case that makes it more clear: imagine a coin that has a 99 % probability of ending up with the same side we start with, and – for simplicity of exposition – we always start with heads facing up before the toss.

If you guess heads, and the first coin lands heads, then there is a 1 % chance that you win, namely that when the second coin lands tails.

If you guess tails, and the first coin lands tails, then there is a 99 % chance that you win, namely that when the second coin lands heads.

The two outcomes of the first coin (99 % and 1 % respectively) perfectly balance out the two valid outcomes of the second coin.

1. keep flipping until you get HT (and so you choose 'heads') 2. keep flipping until you get TH (and so you choose 'tails')

Since HT and TH are equally likely, results 1 and 2 are equally likely, i.e. there's a 50% chance of choosing heads, 50% change of choosing tails.