If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
A scenario which occurred to me and I found strange at first glance: Consider a fair coin, and two people -- Alice who is 99.9% sure the coin is fair and who can update on evidence like a fine Bayesian, and Bob who says he's perfectly sure the coin is biased to show heads and does not update on the evidence at all.
Nonetheless the perfectly correct Alice (who effectively needs choose randomly and might as well always say 'heads') and the perfectly incorrect Bob (who always says 'heads' because he's always certain that'll be the correct answer) have the same chance (50%) to correctly predict the next coin's toss. Even when the experiment is repeated multiple times, its progress further confirming to Alice that she is right to believe the coin fair, Alice's predictive ability isn't improved over non-updating Bob's on a toss-by-toss basis.
I found that initially perplexing -- If we consider accuracy alone, Alice's more accurate beliefs can only be perceived if she's allowed to make predictions over large patterns (e.g. she'd expect a roughly equal number of heads to tails). If she's not given that ability, and if a third party is only told the number of times each of the participants were correct in their guesses, they couldn't tell who is who.
One more thing that distinguishes them: If Alice and Bob were allowed to bet on their guesses, Alice would accept only favorable odds, and Bob would soon go bankrupt...
The difference between them becomes apparent once they start betting on other things, like the number of tails in a series of 10 coinflips. The question is: what is special about betting on heads vs. tails of a fair coin that doesn't allow Alice to do any better than Bob?