Eric Cavalcanti writes:
> Suppose, in the room problem, that instead of a biased coin,
> everyone tossed a fair coin, as in Stathis original problem, and
> enters a room by the decision of the coin. If the number of people
> is large enough, it is highly likely that one of the rooms will
> be more crowded. But as you enter one of the rooms, you have no
> reason to believe that you are in the more crowded room, **even
> though you followed the same mechanism as everyone else**, in
> constrast with Stathis original problem (where there were already
> 1000/10 people in each room).
Actually you do have reason to believe you are in the more crowded
room, because your presence there makes it more likely that room is
more crowded. If the rooms happened to be a tie before you entered,
your room is now more crowded by virtue of your presence.
Although there is a 50-50 chance which room is more crowded, and a 50-50
chance which room you end up in, these two results are not independent.
The room you are in is (slightly) more likely to be the more crowded one.
If you imagine it as a betting game, where you want to know what the
odds are that you are in the more crowded room, then I think both of
these lines of reasoning, anthropic and probabilistic, will give the
same results. The odds are slightly in favor of your room being the more
crowded, and as the number of people increases, the advantage decreases.