I have realized that your misunderstanding of the dice rolling problem is due
to the fact that you are considering only one instance of the experiment and
considering it to be a statistical sample, when statistics and probability
require *many* samples to become relevant. So, let's set up the experiment so
that it can be repeated many times, and then the actual odds of seing the
double sixes come up during any one particular trial will become much clearer
for you to understand.
Imagine we have an arbitrarily large number of people to select from, to
participate in the dice rolling experiment. We could assign each person a
group to be in, and each person will remain in their assigned group for each
trial. Group #1 would have one person in it. Group #2 would have ten people.
Group #3 would have one hundred people, and so on, so that any group #n has
10^(n-1) (10 to the n-1 power) number of people in it.
During each trial, group #1 will begin and have two *fair* dice rolled for it.
If double sixes don't come up for any particular group, the dice are rolled
for the next group. If double sixes come up for any group, the trial ends and
the next trial begins, starting with group #1 again.
Here are some probabilities for and other facts regarding various events:
Because the dice are fair dice, every time they are rolled, they have a 1/36
chance of coming up double sixes.
The person assigned to group #1 has a 100% chance of being included in the
trial and a 1/36 (approximately .27778, which is 2.8%) chance of the dice
coming up double sixes during that group's turn. The second group has the
remaining 35/36 (97.2%) chance that the dice *won't* come up double sixes for
the first group and thus that the second group will get a turn. They then
have a 1/36 chance that the dice will come up double sixes during their turn,
so their probability of seeing double sixes is 35/(36*36) (approximately
.02701, which is 2.7%). The third group has a 35/36 chance of the 35/36
chance that the second group had to get a turn (this is the probability that
both groups #1 and #2 will not see double sixes), which is (35*35)/(36*36) =
35^2/36^2 = 1225/1296 (94.5%), and of this chance, they only have a 1/36
chance of that, that they will see the dice come up double sixes. So the
probability of group #3 getting a turn AND having double sixes rolled on that
turn is 35^2/36^3 (approximately .02626, which is 2.6%). For any group #n,
the probability that the group will get a turn AND that the dice will come up
double sixes on that turn is 35^(n-1)/36^n (35 to the n-1 power, divided by
36 to the n power). Thus, the chances of the later groups getting a turn and
having double sixes come up becomes vanishingly small (group #100 has only a
0.17% chance of this happening to it).
After double sixes come up during any group's turn, the number of people in
that group will *always* have more than nine times the number of people in the
previous groups combined. Incidentally, that means that for each trial, more
than 90% of the participants will have been in the group that had double sixes
come up for it. However, that group's chances that they will get a turn and
see the double sixes come up next time the trial is run is very small. In
fact, for every group up to group #127, there is a greater chance that NONE of
the previous groups will see double sixes than there is that *that* group WILL
have double sixes come up. Also, during any group's turn, it is far more
likely (35/36 chance) that some later group will get a turn and have double
sixes come up then that *it* will have double sixes turn up (1/36 chance).
So, for any given participant, their chances are very slim that they will
actually get a turn AND that the dice will come up double sixes on their turn,
and even if they do get a turn, their chances are only 1/36 that they dice
will come up double sixes during their turn.
It is true that for each trial, more than 90% of the participants will have
the double sixes come up for them, but that doesn't mean that the same people
have more than a 90% chance of having it happen again next time, and it
*certainly* doesn't mean that *every participant* has more than a 90% chance
of having double sixes come up during their group's turn. You can't just look
at the outcome of a single trial and then say that it's representative of
every *possible* trial. For any given individual, it's most likely that they
won't see the dice come up double sixes, and even the more than 90% that do
see them come up will most likely be among the less than 10% that don't the
next time they get a turn. So, for any given individual, the safest bet is to
bet that they won't see double sixes come up during their group's turn.
You can track every individual among your arbitrarily large group of
participants during every trial, and you can note how often each particular
individual is in the group that saw double sixes come up, and you will see
that during the times that an individual's group actually gets a turn, that
individual typically sees the dice come up double sixes 1 out of every 36
times. This is true statistics and true probability. To determine the best
bet for each individual, the trial must be carried out *many* times.
The best bet for *every* individual in the trial is that double sixes will NOT
show up for them.
I hope you see that now; I have explained it about as thoroughly as possible
without sounding too condescending. Your "paradox" was trivial for me to
solve, but I enjoyed the *real* challenge of trying to explain the solution to
you in a convincing way that you couldn't avoid seeing it. Thank you for the
challenge. Although I can't help wondering if you have only been *pretending*
that you didn't understand, so that you could see what kinds of explanations
you'd get.
- David Musick