Re: Probability Problem 2

In article <1177968891.530651.38320@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Aero <aerospyder@xxxxxxxxx> wrote:

As the problem is stated, it's hard for me to see how the answer isn't
as stated by Eric Sosman. I like the answer as given by "ken", but it
seems to be the answer to the question:

"Given three points on a line, if the points are randomly labeled 'A',
'B', and 'C', what is the probability that point A lies between points
B and C?"

Or something like that.

It seems reasonable that if the points are labeled randomly, each of the
6 possible orderings should be equally likely.

As 2 out of the 6 have A between B AND C, and 4 have it at one end or
the other, that would seem to be decisive.

(Alternately, it should be the case that each of the 3 has the same
chance of being in the middle)


Is that really the same as the question originally posed? No, I don't
believe it is. But at least the new question has a more satisfying
solution.

As an aside, I consider how the solution of the following question
relates to that of the question originally posted:

Given three independent draws from a normal distribution, what is the
probability that the first draw has a value between the other two?

If you examine the draws in turn, this is like the problem of the
original post, but instead you want the first point (rather than the
third) to lie between the other two. So, the first point would
partition the infinite line into two equal sections. Again, say the
points in turn are A, B, and C. Point A is a given, and B will fall
on one side of A or the other. So the real question, then, asks the
probability that C will fall on the side of A opposite B. This is
1/2.

I question the solution I gave above, because I'm not sure if a random
point on a line can really be said to effectively split the line in
half. Maybe it can if I decide to think of the line as the real
number line and assume point A to be at zero. Or maybe I don't to do
that. Anyone familiar enough with the subtleties of infinite sets/
intervals to help me out on this?

I think some of the controversy in problems like these lies in how you
consider order (if you consider it at all). I suppose this is related
to the difference between combinations and permutations. Any
thoughts?

Also, in response to "... That's just mapping -infinity .. +infinity
to 0 .. 1.":

I assume Richard Heathfield had a linear scale in mind. Then, if you
consider a unit length line segment, you are always dealing with
finite intervals and you get a satisfying answer (unlike the original
problem, IMO). If you map -inf .. +inf to 0 .. 1, you get a nonlinear
scale in which the probability of landing a point in, say, the
interval between 0.75 and 1 is not the same as landing between 0.5 and
0.75, even though the intervals have the same length in the new
domain. So, you are still dealing with the difficulties of infinite
v. finite intervals present in the original problem, just in a
different domain. Richard's problem is fundamentally different from
the original question no matter how you slice it (or map it).
.

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