Good for them--I'll give you 30 to 1 too (I'll bet $30 against your
$1 that they won't come up 6-6). You quite correctly point out that
/after the fact/, 90% of the people who bet my way will have lost
$30. But I'm not making the bet after the fact--I'm making it before
the dice roll. Time is a fairly simple concept, and I'm not sure
why you seem to have such confusion with it. After the party, if
some random participant came up to me and asked "what will you bet
me that I /saw/ 6-6?", he'd have to give me $10 to $1. Also, if I
were in the population of people from whom the experimental subjects
will be drawn, and I'm asked to bet on whether I /will/ see 6-6
whenever I'm drawn (cancelling the bet if I am not), then you'd have
to give me $10 to $1. But neither of those bets is the bet you asked
about: you asked about a single roll of fair dice, and that is 1/36.
In other words, you are setting up rules, and then asking that I
ignore certain ones where they apply, and apply others where they
do not. Just clarify your definitions and all is as it should be.
Your appeal to Bayesian analysis holds no water either, because you
set up the rules of the game clearly before you started playing.
Having set up the rules (which included "fair dice"), you are forbidden
from making deductions that assume otherwise. Take your other
coin-flipping example: after 100 heads, you are justified in thinking
that a 2-headed coin is a possibility, because you have not said
otherwise in setting up the conditions of the experiment--we are
given only the outcomes, so we can correctly state that the odds of
a fair coin are small, and the odds of a double-headed coin are
nearly 1. If you start the problem by saying "a fair coin is
flipped 100 times...", then you must conclude that the odds of heads
after 100 heads in a row is 1/2, because that's the definition you
gave. You don't get to make up rules and then ignore them when it's
convenient.
This problem isn't even as tricky as the Monte Hall problem, because
the missing assumptions are easier to identify.