I think it boils down to the thirder position.

P(monday,heads)=0,33 since P(heads)=0,5 and P(monday)=0,67.

This is because we are sampling awakenings, not coin tosses or beauties.

It's like betting on a coin toss, but with a side twist that the bet is evaluated twice (no new toss) when the coin is tails. Hence a majority of evaluations will be on tails:

  Guess, Coin, Profit
  H,     H,    +1
  H,     T,    -2
  T,     H,    -1
  T,     T,    +2

Analysis for betting heads: average is +1 + -2 = -1 Betting tails: average is -1 + +2 = +1.

I'd bet tails!