Comment by tailwind

"I wonder if it's also 60% likely to be Tails if I bet on Tails, and I might be tempted to check."

Only if you were clueless, or perhaps if the experimenter said "if you bet on heads it has a 60% chance of winning". Being unstated what would happen if you bet on tails, you might forget that the coin has know knowledge of how you bet, thus making it impossible for there to be any different outcome than a 60% chance of loss by betting on tails.