In your scenario, where ten times as many people are brought into the room
each time double sixes aren't rolled, there is a 1/36 (.0278) chance that 100%
of the people will witness double sixes (if the first person is rolled double
sixes), a 35/1296 (.0270) chance that 90.09% of the people will witness double
sixes (if they are rolled for the second group), a 1225/46656 (.0263) that
90.09% (it's slightly less than the last amount, but it's approaching 90% as
the number of (increasingly unlikely) trials which take place increases) of
the people will observe double sixes, and so forth.
No matter when you're in the room to witness the roll of the dice, the chances
of the particular even of the (fair) rolled dice both coming up sixes will
always be 1/36. You make a link in your mind to the significance of the dice
coming up sixes and the rolling sessions ending, and you reason that since
approximately 90% of the people will observe this "significant" event, the
chances of the significant event occuring during any particular dice roll must
be 90%. Perhaps you will realize the flaw in your thinking if I point out
that the dice themselves are blithely unaware of your organizational schemes
and will roll just as randomly as always.
This is very related to the "midpoint of history" flaw, which I can't believe
any reasonable person would buy into. Why couldn't we be at the beginning of
things? Someone had to be! The dice rolling idea above and the "midpoint of
history" idea remind me of someone rolling 1000 dice, which are lined up in a
row and declaring that a miracle had occurred because the chances of the dice
coming up with that *particular* sequence of numbers is 1/6^1000 (almost
zero), yet it was rolled on the first try!
Hal, you also said, "My personal opinion is that a random person in the room
actually would see the dice roll as sixes, and that if I found myself in the
room I would bet on the double sixes if someone offered me this option. What
do you think?"
What do I think? I think I'll take you up on that bet! We'll run the
experiment and if the dice come up double sixes when you are in the room, then
you win, if it doesn't, then I win. Since you believe you have a 90% chance
of the dice coming up double sixes and I only have a 10% chance of winning,
it's only fair that the bet is made according to those odds (for example: if
you win, I pay you ten dollars, but if I win, you pay me ninety dollars). Is
it a deal?
- David Musick