When you are in the room with all the other people, you don't know if you are
among the 10% who won't see double sixes or among the 90% who will. You have
a 1/36 chance of being among the 90% of people who see the double sixes and a
35/36 chance of being among the 10%. This is because each time the dice don't
come up double sixes, ten times as many people are brought in to watch the
dice be rolled again. This next, larger group of people also have a 1/36
chance of seeing the dice come up double sixes and 35/36 chance that they
won't. And so on, until eventually one group of people will witness double
sixes being rolled, and this group of people has approximately nine times as
many people as all the other groups combined, who didn't see the dice come up
double sixes. But they only have nine times as many people because you keep
multiplying the number of people in the room by ten after each non-double
sixes. You are *guaranteeing* that there will be approximately nine times as
many people in the room when the event (which itself has only 1/36 chance of
happening) of rolling double sixes happens, as you had in the room when it
didn't happen. The event itself still has the same probability, you are just
making sure that when it *does* eventually happen, there will be many more
people witnessing the event than those who did not witness it. This
contrivance has no sort of magical influence over the dice; they will roll
just like always -- randomly.
Consider the situation this way: Most of the *groups* of people who are in
the rooms for each roll of the dice will not see it come up double sixes.
Each group has a 1/36 chance that the dice will come up double sixes while
that group is in the room. As a member of a group in the room, you have a
1/36 chance that your group will be the ones to witness the rolling of double
sixes. Again: Look at all the people in the room at any given dice roll as
one group. Each group has a 1/36 chance of the dice coming up double sixes,
regardless of the size of the group. As a member of any of the groups
witnessing the dice roll, you have the same chances of being in the room when
they come up double sixes as your whole group does, 1/36.
This concludes Mr. Hal Finney's math lesson for today. The homework for today
is to slap your hand against your forehead three times and say "Doh!" each
time. To avoid further embarassment, Mr. Finney is advised to direct whatever
questions he has regarding my explanation to me, privately.
- David Musick