[extropy-chat] what is probability?
Jef Allbright
jef at jefallbright.net
Thu Jan 18 02:02:46 UTC 2007
gts wrote:
> A train leaves at noon to travel a distance of 300km. It
> travels at a constant speed of between 100km/h and 300km/h.
> What is the probability that it arrives before 2pm?
>
> At least two solutions with different answers:
>
> 1. For the train to arrive before 2pm, its velocity must be
> greater than or equal to 150km/h. By the principle of
> indifference, all velocities between 100 and 300 km/h are
> equally likely. The probability of it being greater than or
> equal to 150km/h such that the train arrives before 2pm is
> therefore 3/4.
>
> 2. The train must arrive between 1pm and 3pm. 2pm is half way
> between these two. By the principle of indifference, the
> train is as likely to arrive before 2pm as after 2pm. So the
> probability must be 1/2.
Gordon, this is the same as the problem of the boxes.
With the box, your problem was underspecified because it didn't specify
which variable was to be considered (length, area, or volume.)
With the train, your problem is underspecified because you don't specify
what mechanism is to be considered for changing the speed of the train
(set the speed to a number between 100 and 300, or alternatively, set
the number of minutes to be offset (and then calculate the resulting
speed.)
The Principle of Indifference is correct and beautiful in its elegance.
But it doesn't claim to work across multiple spaces of possible states.
Maybe it would help you to remember that there's only a single TRUE
speed of the train. The puzzle here is not to try to give a range of
true speeds, but rather to give a range of uncertainty relative to a
specified subjective agent. You haven't specified the subjective
agent's viewpoint sufficiently, so you have ambiguity in your problem.
In theory, you could keep thinking up additional dimensions to this
problem, equally unspecified, but being 3D beings our imagination tends
to fail before we get that far.
- Jef
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