the betting argument is not the same as what is being discussed.
to a halver, the likelihood of heads and tails are still each 1/2, but the expected winnings of betting tails would of course be larger.
it would be like saying i'll flip a coin, and if you guess correctly that it's tails i'll give you a million dollars. obviously i will guess that it's tails, even though i still believe each outcome is equally likely.
Of course, do you really think otherwise? If I promised to put a pound in your moneybox every time you correctly guessed the coin state on each awakening, would you really fall to sleep confident in a 50/50 guess strategy in your 1000 awakenings to 1 scenario?