On the physical probabilities: those have to do with the fraction of the initial phase space that is mapped to the specific final state. For things like symmetrical die, we can know the probability without doing statistics—if you paint the initial phase space with 6 colours corresponding to the die’s ultimate rolling outcome, you will get a lot of very very tiny regions, each colour occupying same hypervolume. We are very highly ignorant of which coloured area is the initial state in—given enough bounces, they’re tiny and really close together. Each bounce of the die works as a mapping operation that stretches and folds over that state, preserving the relative areas (due to symmetry).
The idea of finding an agreeable-probability lottery that is equivalent to a subjective-probability one, is very clever, I like it.