As simon has already said, your math is wrong because the cases aren’t equiprobable. For k=2 you can fix this by doubling the cases of H since they’re twice as likely as the others (so the proper state space is Ω={H,H,HT,HT,TT,TT,TT,TT} with p(TT)=12.) For k=3 you’d have to quadruple H and double HT, which would give 4x H and 4x HT and 4x HTT and 8x TTT I believe, leading to 25 probability of TTT. (Up from 1x H, 2x HT, 4x HTT, 8x TTT.) In general, I believe the probability of only Ts approaches 0, as does the probability of H.
Regardless, are these the right betting odds? Yup! If we repeat this experiment for any k and you are making a bet every time you wake up, then these are the odds according to which you should take or reject bets to maximize profit. You can verify this by writing a simulation, if you want.
If you make the experiment non-repeating, then I think this is just a version of the presumptuous philospoher argument which (imo) shows that you have to treat logical uncertainty differently from randomness (I addressed this case here).
As simon has already said, your math is wrong because the cases aren’t equiprobable. For k=2 you can fix this by doubling the cases of H since they’re twice as likely as the others (so the proper state space is Ω={H,H,HT,HT,TT,TT,TT,TT} with p(TT)=12.) For k=3 you’d have to quadruple H and double HT, which would give 4x H and 4x HT and 4x HTT and 8x TTT I believe, leading to 25 probability of TTT. (Up from 1x H, 2x HT, 4x HTT, 8x TTT.) In general, I believe the probability of only Ts approaches 0, as does the probability of H.
Regardless, are these the right betting odds? Yup! If we repeat this experiment for any k and you are making a bet every time you wake up, then these are the odds according to which you should take or reject bets to maximize profit. You can verify this by writing a simulation, if you want.
If you make the experiment non-repeating, then I think this is just a version of the presumptuous philospoher argument which (imo) shows that you have to treat logical uncertainty differently from randomness (I addressed this case here).