I was erroneously counting the ways that it might have not been two heads in a row as belonging to the double-headed's claim.


Out of a total of 16 possible picks and flip sequences, 4 give two heads with the double headed coin. 3 give two heads with the single headed coins. Therefore the odds are 4/16 vs. 3/16 which reduces to 4:3 in favor of the double-headed coin. (I believe we must count the double-headed coin as having two distinct sides even though both sides give the same result).