[balrog666]

To predict a coin with no worse than 50% chance, two strategies are available: first, pick the last outcome and second, pick the outcome which has dominated so far. Both are better than picking heads or tails at random.

I was going to suggest "tit for tat" selection.

[Doctor Stochastic]

Tit-for-tat would be the same as play-the-last. This is a winning strategy but not necessarily optimal. (Taking p as as the winning percentage, for example heads, and q=1-p as the losing, tails, with p>=1/2 because p corresponds to guessing right. The tails winning case is symmetrical.) You win for the cases HH and TT and lose with HT and TH. The probablities are P^2+q^2 to win and 2pq to lose. This wins as p^2=q^2-2pq = (p-q)^2 is greater than zero.

Play the winner would win with probability p which is superior in general.

Biased coins can be detected.