The two states differ mathematically mainly with respect to how they update. In the first case, one is confident in the bias of the coin, so the probability will not shift much as new evidence comes in (like e.g. coinflips). In the second case, the probability will shift as new evidence comes in.
As a general rule, insofar as humans are well-described as thinking probabilistically, our probabilistic models are little parts of a big world model. Those little parts don’t just exist for e.g. one coin flip; they stick around after the coin is flipped and interact with the rest of the world model. So the way they update is an inherent part of their type signature; that’s why little models which update differently feel different.
The two states differ mathematically mainly with respect to how they update. In the first case, one is confident in the bias of the coin, so the probability will not shift much as new evidence comes in (like e.g. coinflips). In the second case, the probability will shift as new evidence comes in.
As a general rule, insofar as humans are well-described as thinking probabilistically, our probabilistic models are little parts of a big world model. Those little parts don’t just exist for e.g. one coin flip; they stick around after the coin is flipped and interact with the rest of the world model. So the way they update is an inherent part of their type signature; that’s why little models which update differently feel different.
Another difference would be expectations for when the coin gets tossed more than once.
With “Type 1” if I toss coin 2 times I expect “HH”, “HT”, “TH”, “TT”—each with 25% probability
With “Type 2” I’d expect “HH” or “TT” with 50% each.