The richer structure you seek for those two coins is your distribution over their probabilities. They’re both 50% likely to come up heads, given the information you have. You should be willing to make exactly the same bets about them, assuming the person offering you the bet has no more information than you do. However, if you flip each coin once and observe the results, your new probability estimate for next flips are now different.
For example, for the second coin you might have a uniform distribution (ignorance prior) over the set of all possible probabilities. In that case, if you observe a single flip that comes up heads, your probability that the next flip will be heads is now 2⁄3.
The richer structure you seek for those two coins is your distribution over their probabilities. They’re both 50% likely to come up heads, given the information you have. You should be willing to make exactly the same bets about them, assuming the person offering you the bet has no more information than you do. However, if you flip each coin once and observe the results, your new probability estimate for next flips are now different.
For example, for the second coin you might have a uniform distribution (ignorance prior) over the set of all possible probabilities. In that case, if you observe a single flip that comes up heads, your probability that the next flip will be heads is now 2⁄3.
Yes, I understand that. This subthread started when cousin_it said
at which point I objected.